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## Monday, November 02, 2015
### Sudoku
It's hard to find a newspaper without one, and I personally find that Sudoku solutions are an interesting problem. Randomly filling a grid of 81 cells with numbers from 1-9 wouldn't take long, but the end result probably wouldn't be a Sudoku solution, where each row, each column, and each box, all need to have the numbers 1-9 exactly once.
Instead of doing it randomly, we could try a constrained approach: only pick numbers from the set that would form an allowable Sudoku solution. We could start off by defining the set of all values in row M, all values in column N, and all values in box MN. The union of all three sets would indicate the values that have already been picked; therefore they're the numbers we couldn't re-use if we were to make a valid Sudoku solution. $$R(m) \cup C(n) \cup B(m,n)$$ is the numbers that have been used, $$(R(m) \cup C(n) \cup B(m,n))'$$ are the numbers available to us (this last set is the equivalent of $$R(m)' \cap C(n)' \cap B(m,n)'$$).
Let's assume we're looking at an empty cell, in an empty grid. $$R(m)=\emptyset, C(n)=\emptyset, B(m,n)=\emptyset$$, therefore we can choose anything from $$\emptyset'$$. The world is our oyster. Let's pick 1. We could have picked anything, but 1's a good place to start. In the adjacent cell, $$R(m)$$ and $$B(m,n)$$ would both yield $$\{1\}$$ so we'd be forced to pick anything from $$\{1\}'$$. Let's pick 2. Continuing in ascending order (and increasing level of constraint) we'd finally reach the 9th cell where $$R(m)=\{1,2,3,4,5,6,7,8\}$$ and $$B(m,n)=\{7,8\}$$. The only number we can pick is 9. Our top row is complete.
Continuing at the first column of the second row, and proceeding in the same fashion, we keep filling cells. It's too easy, something's got to break.
Ah. On the second row, at the seventh colum, we reach Box(2,7). We've already written $$\{4,5,6,1,2,3\}$$ into the row, and $$\{7,8,9\}$$ into the box. What can we do now?
The answer is to backtrack. We didn't have to choose the lowest value from $$R(m)' \cap C(n)' \cap B(m,n)'$$ every time; in fact - more often than not - we had a wide array of options. So, we backtrack. That involves reverting the operation on the current cell (leaving it blank), moving to the previous cell, making sure it's blank (but we remembered which value(s) we'd already tried), and then trying another value. We don't need to limit ourselves to going back just one cell, either. In fact, in this case, we need to revert the last three cells. A second row that looks like $$\{4,5,6,7,8,9,1,2,3\}$$ is a valid partial solution to a Sudoku problem. We use backtracking even more on the third row, producing $$\{7,8,9,1,2,3,4,5,6\}$$. Still: a valid partial solution.
It turns out that our set-based constraints, plus the backtracking ability, give us the power to solve any Sudoku problem, even when the starting point is ambiguous (in the case of the empty grid).
1 2 3 4 5 6 7 8 9
4 5 6 7 8 9 1 2 3
7 8 9 1 2 3 4 5 6
2 1 4 3 6 5 8 9 7
3 6 5 8 9 7 2 1 4
8 9 7 2 1 4 3 6 5
5 3 1 6 4 2 9 7 8
6 4 2 9 7 8 5 3 1
9 7 8 5 3 1 6 4 2
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Copied to
clipboard
## G = Dic10⋊3D6order 480 = 25·3·5
### 3rd semidirect product of Dic10 and D6 acting via D6/C3=C22
Series: Derived Chief Lower central Upper central
Derived series C1 — C60 — Dic10⋊3D6
Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — Dic10⋊3D6
Lower central C15 — C30 — C60 — Dic10⋊3D6
Upper central C1 — C2 — C4 — D4
Generators and relations for Dic103D6
G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, cbc-1=a10b, dbd=a5b, dcd=c-1 >
Subgroups: 892 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12, C12 [×2], D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C3⋊C8, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, Q82S3 [×2], C2×D12, C3×C4○D4, C3×Dic5, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D4⋊D6, C5×C3⋊C8, C153C8, C5⋊D12, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×D12, D60, D4×C15, C2×S3×D5, D8⋊D5, C20.32D6, C20.D6, C15⋊SD16, C5×D4⋊S3, D4⋊D15, D5×D12, C3×D42D5, Dic103D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D4⋊D6, C2×S3×D5, D8⋊D5, D5×C3⋊D4, Dic103D6
Smallest permutation representation of Dic103D6
On 120 points
Generators in S120
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 106 11 116)(2 105 12 115)(3 104 13 114)(4 103 14 113)(5 102 15 112)(6 101 16 111)(7 120 17 110)(8 119 18 109)(9 118 19 108)(10 117 20 107)(21 81 31 91)(22 100 32 90)(23 99 33 89)(24 98 34 88)(25 97 35 87)(26 96 36 86)(27 95 37 85)(28 94 38 84)(29 93 39 83)(30 92 40 82)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 70 56 80)(47 69 57 79)(48 68 58 78)(49 67 59 77)(50 66 60 76)
(1 95 51)(2 84 52 10 96 60)(3 93 53 19 97 49)(4 82 54 8 98 58)(5 91 55 17 99 47)(6 100 56)(7 89 57 15 81 45)(9 87 59 13 83 43)(11 85 41)(12 94 42 20 86 50)(14 92 44 18 88 48)(16 90 46)(21 71 110 23 69 112)(22 80 111 32 70 101)(24 78 113 30 72 119)(25 67 114 39 73 108)(26 76 115 28 74 117)(27 65 116 37 75 106)(29 63 118 35 77 104)(31 61 120 33 79 102)(34 68 103 40 62 109)(36 66 105 38 64 107)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)
G:=sub<Sym(120)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,81,31,91)(22,100,32,90)(23,99,33,89)(24,98,34,88)(25,97,35,87)(26,96,36,86)(27,95,37,85)(28,94,38,84)(29,93,39,83)(30,92,40,82)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,70,56,80)(47,69,57,79)(48,68,58,78)(49,67,59,77)(50,66,60,76), (1,95,51)(2,84,52,10,96,60)(3,93,53,19,97,49)(4,82,54,8,98,58)(5,91,55,17,99,47)(6,100,56)(7,89,57,15,81,45)(9,87,59,13,83,43)(11,85,41)(12,94,42,20,86,50)(14,92,44,18,88,48)(16,90,46)(21,71,110,23,69,112)(22,80,111,32,70,101)(24,78,113,30,72,119)(25,67,114,39,73,108)(26,76,115,28,74,117)(27,65,116,37,75,106)(29,63,118,35,77,104)(31,61,120,33,79,102)(34,68,103,40,62,109)(36,66,105,38,64,107), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,81,31,91)(22,100,32,90)(23,99,33,89)(24,98,34,88)(25,97,35,87)(26,96,36,86)(27,95,37,85)(28,94,38,84)(29,93,39,83)(30,92,40,82)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,70,56,80)(47,69,57,79)(48,68,58,78)(49,67,59,77)(50,66,60,76), (1,95,51)(2,84,52,10,96,60)(3,93,53,19,97,49)(4,82,54,8,98,58)(5,91,55,17,99,47)(6,100,56)(7,89,57,15,81,45)(9,87,59,13,83,43)(11,85,41)(12,94,42,20,86,50)(14,92,44,18,88,48)(16,90,46)(21,71,110,23,69,112)(22,80,111,32,70,101)(24,78,113,30,72,119)(25,67,114,39,73,108)(26,76,115,28,74,117)(27,65,116,37,75,106)(29,63,118,35,77,104)(31,61,120,33,79,102)(34,68,103,40,62,109)(36,66,105,38,64,107), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,106,11,116),(2,105,12,115),(3,104,13,114),(4,103,14,113),(5,102,15,112),(6,101,16,111),(7,120,17,110),(8,119,18,109),(9,118,19,108),(10,117,20,107),(21,81,31,91),(22,100,32,90),(23,99,33,89),(24,98,34,88),(25,97,35,87),(26,96,36,86),(27,95,37,85),(28,94,38,84),(29,93,39,83),(30,92,40,82),(41,75,51,65),(42,74,52,64),(43,73,53,63),(44,72,54,62),(45,71,55,61),(46,70,56,80),(47,69,57,79),(48,68,58,78),(49,67,59,77),(50,66,60,76)], [(1,95,51),(2,84,52,10,96,60),(3,93,53,19,97,49),(4,82,54,8,98,58),(5,91,55,17,99,47),(6,100,56),(7,89,57,15,81,45),(9,87,59,13,83,43),(11,85,41),(12,94,42,20,86,50),(14,92,44,18,88,48),(16,90,46),(21,71,110,23,69,112),(22,80,111,32,70,101),(24,78,113,30,72,119),(25,67,114,39,73,108),(26,76,115,28,74,117),(27,65,116,37,75,106),(29,63,118,35,77,104),(31,61,120,33,79,102),(34,68,103,40,62,109),(36,66,105,38,64,107)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93)])
45 conjugacy classes
class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 15A 15B 20A 20B 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 8 8 10 10 10 10 10 10 12 12 12 12 12 15 15 20 20 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 10 12 60 2 2 10 20 2 2 2 4 4 20 12 60 2 2 8 8 24 24 4 10 10 20 20 4 4 4 4 4 4 8 8 8 8 12 12 12 12 8 8
45 irreducible representations
dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C8⋊C22 S3×D5 D4×D5 D4⋊D6 C2×S3×D5 D8⋊D5 D5×C3⋊D4 Dic10⋊3D6 kernel Dic10⋊3D6 C20.32D6 C20.D6 C15⋊SD16 C5×D4⋊S3 D4⋊D15 D5×D12 C3×D4⋊2D5 D4⋊2D5 C3×Dic5 C6×D5 D4⋊S3 Dic10 C4×D5 C5×D4 C3⋊C8 D12 C3×D4 Dic5 D10 C15 D4 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 2
Matrix representation of Dic103D6 in GL6(𝔽241)
1 0 0 0 0 0 0 1 0 0 0 0 0 0 52 1 240 188 0 0 240 190 52 51 0 0 233 125 0 240 0 0 234 124 0 240
,
240 0 0 0 0 0 0 240 0 0 0 0 0 0 16 0 148 225 0 0 164 0 148 77 0 0 15 169 0 0 0 0 200 57 148 225
,
0 240 0 0 0 0 1 240 0 0 0 0 0 0 51 1 0 0 0 0 51 190 0 0 0 0 0 0 1 0 0 0 52 1 189 240
,
1 240 0 0 0 0 0 240 0 0 0 0 0 0 52 1 188 240 0 0 240 190 51 52 0 0 0 0 240 0 0 0 51 1 240 0
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,233,234,0,0,1,190,125,124,0,0,240,52,0,0,0,0,188,51,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,16,164,15,200,0,0,0,0,169,57,0,0,148,148,0,148,0,0,225,77,0,225],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,51,51,0,52,0,0,1,190,0,1,0,0,0,0,1,189,0,0,0,0,0,240],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,52,240,0,51,0,0,1,190,0,1,0,0,188,51,240,240,0,0,240,52,0,0] >;
Dic103D6 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_3D_6
% in TeX
G:=Group("Dic10:3D6");
// GroupNames label
G:=SmallGroup(480,554);
// by ID
G=gap.SmallGroup(480,554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^6=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations
×
𝔽
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Lesson 9Sim CitySolidify Understanding
Rewrite the rational expression in the form .
Set
7.
For student body elections, a school’s statistics class samples and asks them who they will vote for. Of those surveyed, stated if the election were held today, they would vote for Juvenal for student body president. Create an interval of plausible values for the actual proportion of students in the entire school who would vote for Juvenal for president.
8.
If the statistics class were to double the sample size, what would happen to the interval of plausible values? Would the interval increase, decrease, or stay the same? Why?
9.
Based on your interval of plausible values (problem 7), is it likely that Juvenal will win the election? Why or why not?
10.
Suppose the actual proportion of students in the school who would vote for Juvenal is really . If the statistics students were to create a sampling distribution by taking repeated samples of size and identify the proportion of students who would vote for Juvenal for each sample, what would be the shape, center, and standard deviation of this distribution?
11.
If the actual proportion of students who would vote for Juvenal is really , what is the probability of taking a sample of and finding that or more of them would vote for Juvenal?
Go
A boat is being pulled towards a dock by means of a winch located above the deck of the boat. Let be the angle of elevation from the boat to the winch (in radians) and let be the length of the rope from the winch to the boat.
12.
Write as a function of .
Find when .
14.
Find when .
A spotlight is being set up so that its beam shines directly on the top of a flag pole. The light is located away from the bottom of the pole.
Solve for where .
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# Ancillary services (electric power)
Ancillary services are the services necessary to support the transmission of electric power from generators to consumers given the obligations of control areas and transmission utilities within those control areas to maintain reliable operations of the interconnected transmission system.
Ancillary services are specialty services and functions provided by actors within the electric grid that facilitate and support the continuous flow of electricity, so that the demand for electrical energy is met in real time. The term ancillary services is used to refer to a variety of operations beyond generation and transmission that are required to maintain grid stability and security. These services generally include active power control or frequency control and reactive power control or voltage control, on various timescales. Traditionally, ancillary services have been provided by large production units such as generators. With the integration of more intermittent generation and the development of smart grid technologies, the provision of ancillary services is extended to smaller distributed generation and consumption units.[1]
## Types of ancillary services
There are two broad categories of ancillary services:
• Frequency related: Inertia, Frequency Containment Reserve (FCR), and Automatic Frequency Restoration Reserve (aFRR)
• Non-frequency related: reactive power and voltage control and congestion management
Other types of ancillary services provision include:
• scheduling and dispatch
• loss compensation
• system protection
• energy imbalance
### Frequency control
Frequency control refers to the need to ensure that the grid frequency stays within a specific range of the nominal frequency. Mismatch between electricity generation and demand causes variations in frequency, so control services are required to bring the frequency back to its nominal value and ensure it does not vary out of range.[2]
If we have a graph for a generator where frequency is on the vertical axis and power is on the horizontal axis:
${\displaystyle slope=-R={\frac {\Delta f}{\Delta P_{m}}}}$
where Pm is the change in power of the system. If we have multiple generators, each might have its own R. Beta can be found by:
${\displaystyle \beta ={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+...+{\frac {1}{R_{n}}}}$
The change in frequency due to a change in power can be found with:
${\displaystyle \Delta f={\frac {\Delta P_{m}}{\beta }}}$
This simple equation can be rearranged to find the change in power that corresponds to a given change in frequency.[3]
### Reactive power and voltage control
Consumer loads expect voltage within a certain range, and the regulators require it be within a certain percent of the nominal voltage (for example, in the US it is 土5%).
Reactive power can be used to compensate the voltage drops, but must be provided closer to the loads than real power needs (this is because reactive power tend to travel badly through the grid). Notice that voltage can be controlled also using transformer taps and voltage regulators.[4]
### Scheduling and dispatch
Scheduling and dispatch are necessary because in most electrical systems energy storage is nearly zero, so at any instant, the power into the system (produced by a generator) must equal the power out of the system (demand from consumers). Since production must so closely match demand, careful scheduling and dispatch are necessary.
Usually performed by the independent system operator or transmission system operator, both are services dedicated to the commitment and coordination of the generation and transmission units in order to maintain the reliability of the power grid.
Scheduling refers to before-the-fact actions (like scheduling a generator to produce a certain amount of power the next week), while dispatch refers to the real-time control of the available resources.
### Operating reserves
Since production and demand must match so perfectly (see Scheduling and dispatch), operating reserves help make up the difference when production is too low.
An operating reserve is a generator that can quickly be dispatched to ensure that there is sufficient energy generation to meet load. Spinning reserves are generators that are already online and can rapidly increase their power output to meet fast changes in demand. Spinning reserves are required because demand can vary on short timescales and rapid response is needed. Other operating reserves are generators that can be dispatched by the operator to meet demand, but that cannot respond as quickly as spinning reserves, and grid battery storage that can respond within tens of milliseconds, generally faster than even spinning reserve.
## Renewable generation
The grid integration of renewable generation simultaneously requires additional ancillary services and has the potential to provide ancillary services to the grid. The inverters that are installed with distributed generation systems and roof top solar systems have the potential to provide many of the ancillary services that are traditionally provided by spinning generators and voltage regulators. These services include reactive power compensation, voltage regulation, flicker control, active power filtering and harmonic cancellation.[5] Wind turbines with variable-speed generators have the potential to add synthetic inertia to the grid and assist in frequency control.[6][7][8] CAISO tested the 131 MW Tule wind farm's synchronverter in 2018, and found it could perform some of the grid services similar or better than traditional generators.[9][10] Hydro-Québec began requiring synthetic inertia in 2005 as the first grid operator, demanding a temporary 6% power boost when countering frequency drop by combining the power electronics with the rotational inertia of a wind turbine rotor.[11] Similar requirements came into effect in Europe in 2016.[12]
## Electric vehicles
Plug-in electric vehicles have the potential to be utilized to provide ancillary services to the grid, specifically load regulation and spinning reserves. Plug-in electric vehicles can behave like distributed energy storage and have the potential to discharge power back to the grid through bidirectional flow, referred to as vehicle-to-grid (V2G). Plug-in electric vehicles have the ability to supply power at a fast rate which enables them to be used like spinning reserves and provide grid stability with the increased use of intermittent generation such as wind and solar. The technologies to utilize electric vehicles to provide ancillary services are not yet widely implemented, but there is much anticipation of their potential.[13]
## References
1. ^ Ribó-Pérez, David; Larrosa-López, Luis; Pecondón-Tricas, David; Alcázar-Ortega, Manuel (January 2021). "A Critical Review of Demand Response Products as Resource for Ancillary Services: International Experience and Policy Recommendations". Energies. 14 (4): 846. doi:10.3390/en14040846.
2. ^ Rebours, Yann G., et al. "A survey of frequency and voltage control ancillary services—Part I: Technical features." Power Systems, IEEE Transactions on 22.1 (2007): 350-357.
3. ^ Glover, J. Duncan, et al. Power System Analysis & Design. 6th ed., Cengage Learning, 2017.
4. ^ Ahmadimanesh, A., and M. Kalantar. "A novel cost reducing reactive power market structure for modifying mandatory generation regions of producers." Energy Policy 108 (2017): 702-711.
5. ^ Sortomme, Eric, and Mohamed A. El-Sharkawi. "Optimal scheduling of vehicle-to-grid energy and ancillary services." Smart Grid, IEEE Transactions on 3.1 (2012): 351-359.
6. ^ Lalor, Gillian, Alan Mullane, and Mark O'Malley. "Frequency control and wind turbine technologies." Power Systems, IEEE Transactions on 20.4 (2005): p. 1905-1913.
7. ^ "The utilization of synthetic inertia from wind farms and its impact on existing speed governors and system performance". ELFORSK. 2013. p. 6 (Summary). Archived from the original on 21 April 2017. Retrieved 18 April 2017. Installing wind turbines with synthetic inertia is a way of preventing this deterioration.
8. ^ Early, Catherine (30 September 2020). "How a Single UK Turbine Could Prove a New Use Case for Wind Power". www.greentechmedia.com. Archived from the original on 3 October 2020.
9. ^ Balaraman, Kavya (13 March 2020). "Wind plants can provide grid services similar to gas, hydro, easing renewables integration: CAISO". Utility Dive. Archived from the original on 2020-03-16.
10. ^
11. ^ Fairley, Peter (7 November 2016). "Can Synthetic Inertia from Wind Power Stabilize Grids?". IEEE. Retrieved 29 March 2017.
12. ^ "Network Code on Requirements for Grid Connection Applicable to all Generators (RfG)". ENTSO-E. April 2016. Retrieved 29 March 2017.
13. ^ Joos, G., et al. "The potential of distributed generation to provide ancillary services." Power Engineering Society Summer Meeting, 2000. IEEE. Vol. 3. IEEE, 2000.
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# Which of the following is a unit of electric charge?
30 views
in Physics
closed
Which of the following is a unit of electric charge?
1. coulomb
3. Ampere-second
4. All of the above
by (59.8k points)
selected by
Correct Answer - Option 4 : All of the above
The correct answer is option 4) i.e. All of the above
CONCEPT:
• Electric charge is a fundamental property of matter that causes the matter to experience a force when kept in electric and magnetic fields.
• There are two types of charges possessed by matter - negative charge and positive charge.
• The SI unit of electric charge is coulomb (C).
• Electric current is defined as the charge flowing per unit time.
It is given by:
$I = \frac{q}{t}$
Where I is current, q is the charge and t is the time.
EXPLANATION:
• The SI unit of electric charge is the coulomb.
Electric charge, q = Current (I) × time (t).
• The SI unit of current is ampere and that of time is second. Hence, ampere-second also denotes the electric charge.
• Capacitance is the ability of a capacitor to store electrical charge in it.
The capacitance (C) is related to the electrical charge Q and voltage V across them as:
$C =\frac{Q}{V}$
Therefore, Electrical charge (Q) = CV.
• The SI unit of capacitance is faraday and that of voltage is volt. Thus, faraday-volt is also a unit of electrical charge.
• Thus, the given all options are units of electrical charge.
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How to resample matrices to test for the robustness of their correlation?
I have several populations where I have morphology and diet for each individual. I am interested in the correlation between diet and morphological distances. However the number of individuals in each population ranges from 22 to 80 individuals. I have looked at the correlation diet-morphology for each population and (not surprisingly) the correlation coefficientis is highly correlated with the number of individuals per population.
I would like to resample (without replacement) the populations with 60-80 individuals and get random samplings of 30 individuals (1000 times). I would like to get a correlation coefficient distribution against which to test the original value of the correlation.
I guess it is possible do this in R, however I have never written a script in R and I am not familiar with resampling techniques at all. Any help with coding will be greatly appreciated
Thank you Camille
• Why is it not surprising that the correlation coefficient is highly correlated with the number of individuals per population? I see no obvious reason to expect it to be. – onestop Feb 17 '12 at 17:31
• We can answer your implicit question without any simulation: although the Pearson correlation coefficient is a biased estimator of the correlation, the bias is usually small, so the center of the sampling distribution you propose will be close to the correlation coefficient of the data from which you are sampling. – whuber Feb 17 '12 at 18:05
It would help if you could address @onestop's question, and I also agree with @whuber that using the correlation's you've found are probably fine. But I will try to provide some help with your question. Usually, when we resample, we do so with replacement, and we take a bootsample of the same size as the original sample. That is, if your sample is size 80, we resample with replacement to get 80 bootobservations in our bootsample (which will include some duplicates).
There are at least two packages in R that do bootstrapping, but I usually just do it myself. It's not that much more code and I like setting it up according to my specifications. Let's say that your data are in a matrix with 2 columns and 80 rows. You can bootstrap a correlation by sampling rows with replacement, calculating the correlation and storing it. Here's some sample R code:
set.seed(1) # this is for reproducability
X = matrix(rnorm(160), ncol=2) # a generic matix, note that I made no effort
# to correlate the data, this is just an example
rows = c(1:80) # row numbers to resample over
bootDist = c() # a vector to store the output
for(i in 1:1000) { # used a for loop, but it took 1 sec
bootRows = sample(rows, size=80, replace=T) # this gives me the rows I will use
bootDist[i] = cor(X[bootRows,1], X[bootRows,2]) # the cor for that bootsample
}
summary(bootDist)
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.5573 -0.3765 -0.2997 -0.2916 -0.2128 0.2336
• Thank you for your answer. It was very helpful. I use the following code: rows = c(1:2346) bootDist = c() for(i in 1:1000) { bootRows = sample(rows, size=435, replace=F) bootDist[i] = cor(tjornres[bootRows,1], tjornres[bootRows,2]) } summary(bootDist) hist(bootDist,) I hope this script will help other student. – Camille Feb 21 '12 at 11:44
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# Help Regarding the major product of the following sequence of reactions,(Hint, bulky groups will direct para...
###### Question:
help
Regarding the major product of the following sequence of reactions,(Hint, bulky groups will direct para rather than ortho position) 1. (CH3)3CCVAICI: 2. CH3CI, AICI: 3. KMnO4, H The IR spectrum will show a prominent peak(s) at: [Select) The 13C spectrum of the major product will show (Select] peaks
Which of the following reactions will make benzaldehyde? (two answers) Ozonolysis of styrene followed by DMS Reaction of benzylalcohol with MnO2 Reaction of phenylmagnesium bromide with oxirane (ethylene oxide) followed by PCC Reaction of toluene with Jones oxidant
Which of the following products you expect to be formed in highest yield and why? ОMe ОMe OMe Оме ОMe HNO3 H2SO4 + ON Oll and Ill since OMe is stronger EDG than ethyl I and IV since OMe is stronger EDG than ethyl land Il since ethyl is stronger EDG than Ome Il and Ill since ethyl is stronger EDG than Ome II and IV since OMe is stronger EDG than ethyl I and III since OMe is stronger EDG than ethyl Oll and I since ethyl is stronger EDG than Me I and II since OMe is stronger EDG than ethyl I and IV since ethyl is stronger EDG than Me I and III since ethyl is stronger EDG than Me
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# zbMATH — the first resource for mathematics
Polynomial operators. (English) Zbl 0762.46037
Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 410-444 (1989).
[For the entire collection see Zbl 0721.00014.]
The author gives a survey of basic properties of polynomials in (topological) vector spaces starting with works at the second quarter of this century (Part I). Part II contains remarks concerning Fréchet’s and Cauchy’s functional equations and remarks concerning polynomials defined on groups and semi-groups.
In details, let $$X$$ and $$Y$$ be vector spaces over a commutative field $$A$$ of characteristic zero. Consider different definitions of polynomials:
1) An operator $$P$$ from $$X$$ into $$Y$$ is called a polynomial of degree $$m$$ if $$P=P_ 0+\dots+P_ m$$, where $$P_ k$$ is a $$k$$-homogeneous polynomial $$(P_ m\neq 0)$$ defined via a $$k$$-linear mapping [due to S. Banach (1938) and A. D. Michal (1934)].
2) An operator $$P$$ from $$X$$ into $$Y$$ called a $$G$$-polynomial of degree $$m$$, if there are associated functions $$q_ 0,\dots,q_ m$$ ($$q_ m\neq 0$$) such that $P(x+\lambda z)=q_ 0(x,z)+\lambda q_ 1(x,z)+\dots+\lambda^ m q_ m(x,z)$ for all $$x,z\in X$$ and $$\lambda\in A$$ [due to Gateaux (1922), slightly generalized]. Among others, it is shown that both definitions are equivalent, the sums in the definitions are unique, and the polynomials satisfy the following functional equation. $${\overset{m+1} {\underset{x_ 1,\dots,x_{m+1}}\Delta}}F=0$$, where $${\overset {k}{\underset{x_ 1,\dots,x_ k}\Delta}}$$ is the iterated difference operator $$\bigl({\overset {1}{\underset{x_ 1} \Delta}}F)(x):=F(x+x_ 1)- F(x)$$.
3) Let $$X$$ and $$Y$$ be locally convex Hausdorff spaces, in additional. $$P$$ from $$X$$ into $$Y$$ is an $$F$$-polynomial of degree $$m$$ if $$P$$ is Gâteaux differentiable on $$X$$ and if its $$(m+1)$$st difference with arbitrary increments vanishes identically, while its $$m$$th difference does not. This definition is also equivalent to 1) and 2).
If $$X$$ and $$Y$$ are Banach spaces then the well-known relation between the uniform norms of a $$k$$-homogeneous polynomial $$P_ k$$ and the associated symmetric $$k$$-linear mapping $$P_ k^*$$ and their coincidence on Hilbert spaces [due to A. E. Taylor (1938), J. Kopec and J. Musielak (1955) and due to S. Banach (1938)]. Moreover, the author shows that the principle of uniform boundedness (resonance theorem) can be extended to multilinear mappings and homogeneous polynomials.
In section 4 it is shown that there are unique extensions of multilinear mappings and polynomials between real normed spaces to their complexifications. This is based on a work of A. E. Taylor (1938).
Part II: G. Van der Lijn (1939, 1940) has used the functional equation $${\overset {m+1} {\underset {x_ 1,\dots,x_{m+1}} \Delta}}F=0$$ to define polynomials of degree $$n$$ at most between abelian groups and obtained several results analogous to those discussed above.
D. Z. Djokvic (1969) deals with functions from an abelian semigroup into an abelian group and obtains a representation theorem for the difference operator of order $$n$$ with different increments in terms of the shift operator together with difference operators of order $$n$$ with the same increment.
The last section (Section 6) contains remarks on stability of Cauchy’s functional equation [investigated by the author (1941)] and Fréchet’s functional equation [studied by H. Whitney (1957, 1959) and the author (1961, 1983)].
##### MSC:
46G20 Infinite-dimensional holomorphy 47H99 Nonlinear operators and their properties
Cauchy, A. L.
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# zbMATH — the first resource for mathematics
New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients. (English) Zbl 1445.60052
Summary: We consider random trigonometric polynomials of the form $f_n(t):=\frac{1} {\sqrt{n}} \sum_{k=1}^na_k \cos (k t)+b_k \sin (k t),$ where $$(a_k)_{k\geq 1}$$ and $$(b_k)_{k\geq 1}$$ are two independent stationary Gaussian processes with the same correlation function $$\rho : k \mapsto \cos (k\alpha)$$, with $$\alpha \geq 0$$. We show that the asymptotics of the expected number of real zeros differ from the universal one $$\frac{2} {\sqrt{3}}$$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $$\varepsilon >0$$, for all $$\ell \in (\sqrt{2} ,2]$$, there exists $$\alpha \geq 0$$ and $$n\geq 1$$ large enough such that $\left |\frac{\mathbb{E} \left [\mathcal{N}(f_n,[0,2\pi ])\right]} {n}-\ell \right |\leq \varepsilon ,$ where $$\mathcal{N} (f_n,[0,2\pi ])$$ denotes the number of real zeros of the function $$f_n$$ in the interval $$[0,2\pi]$$. Therefore, this result provides the first example where the expected number of real zeros does not converge as $$n$$ goes to infinity by exhibiting a whole range of possible subsequential limits ranging from $$\sqrt{2}$$ to 2.
##### MSC:
60H99 Stochastic analysis 60G99 Stochastic processes 26C10 Real polynomials: location of zeros
Full Text:
##### References:
[1] [ADL] J.M Azaïs, F. Dalmao, J.R Leon, I. Nourdin and G. Poly, Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients, arXiv:1512.05583. [2] [ADP19] J. Angst, F. Dalmao and G. Poly, On the real zeros of random trigonometric polynomials with dependent coefficients, Proc. Amer. Math. Soc. 147 (2019), no. 1, 205-214. · Zbl 1406.26007 [3] [AP15] J. Angst and G. Poly, Universality of the mean number of real zeros of random trigonometric polynomials under a weak cramer condition, arXiv:1511.08750, 2015 [4] [AW09] J-M. Azaïs and M. Wschebor, Level sets and extrema of random processes and fields, Chap. 3 p71. · Zbl 1168.60002 [5] [DNV18] Y. Do, O. Nguyen and V. Vu, Roots of random polynomials with coefficients of polynomial growth, Ann. Probab., Vol 46, Number 5, 2407-2494, 2018. · Zbl 1428.60072 [6] [Dun66] J.E.A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3), 16:53-84, 1966 · Zbl 0141.15003 [7] [Far86] K. Farahmand, On the average number of real roots of a random algebraic equation, Ann. Prob., 14(2):702-709, 1986. · Zbl 0609.60074 [8] [FL12] K. Farahmand and T. Li, Real zeros of three different cases of polynomials with random coefficients, Rocky Mountain J. Math. 42 (2012), 1875-1892. · Zbl 1263.60049 [9] [Fla17] H. Flasche, Expected number of real roots of random trigonometric polynomials, Stochastic Processes and their Applications, pages -, 2017 · Zbl 1377.60063 [10] [IKM16] A. Iksanov, Z. Kabluchko and A. Marynych, Local universality for real roots of random trigonometric polynomials, Electron. J. Probab., Vol 21, paper no 63,19 pp, 2016 · Zbl 1361.30009 [11] [IM68] I.A. Ibragimov and N.B. Maslova, On the expected number of real zeros of random polynomials i. coefficients with zero means, Theory of Probability and Its Applications, 16(2):228-248, 1971. · Zbl 0277.60051 [12] [Mat10] J. Matayoshi, The real zeros of a random algebraic polynomial with dependent coefficients, Rocky Mountain J. Math. 42, No 3, pp. 1015-1034 · Zbl 1254.60076 [13] [Muk18] S. Mukeru, Average number of real zeros of random algebraic polynomials defined by the increments of fractional Brownian motion, J. Th. Prob., p.1-23, 2018 [14] [NNV15] H. Nguyen, O. Nguyen and V.Vu, On the number of real roots of random polynomials, Com. in Cont. Math. Vol 18, 1550052, 2015. [15] [Kac43] M. Kac. On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:314-320, 1943 · Zbl 0060.28602 [16] [Pir19a] A. Pirhadi, Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients, arXiv:1905.13349, 2019 [17] [Pir19b] A. Pirhadi, Real zeros of random cosine polynomials with palindromic blocks of coefficients, arXiv:1908.08154,to appear in Rocky Mountain J. Math., 2020 [18] [RS84] N. Renganathan and M. Sambandham, On the average number of real zeros of a random trigonometric polynomial with dependent coefficients, Indian Journal of Pure and Applied Mathematrics, 15(9):951-956, 1984 · Zbl 0553.60063 [19] [Sam78] M. · Zbl 0379.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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# Show Intermediate Solutions¶
## Introduction¶
During a long running solver session (job), we may want to compute intermediate results and show them to the end user as soon as they are available. Consider the following use cases:
1. The submitted job contains multiple decision subproblems, all of which are solved in one batch. Why wait for providing the solution of the first subproblem, while the job is already working on the second subproblem?
2. The optimization of a significant Mixed Integer Problem will compute several intermediate incumbents, and perhaps these incumbents are worth visualizing and studying further.
3. By showing intermediate solutions, the end user may decide that the last shown solution is good enough and decide to terminate the job.
## Difference with passing progress information¶
The differences between progress information and intermediate solutions are:
1. The intermediate solutions will be stored as cases such that they can be retrieved upon demand.
2. There is no limit to the amount of information that can be passed back to the data session.
3. The messages about the presence of the intermediate solutions are guaranteed to arrive at the data session; even if the data session is temporarily unreachable.
## Approach¶
The approach here involves passing information through three levels of execution.
1. The solver execution on the solver session.
Construct a new incumbent containing the entire solution. This is done as part of the incumbent callback mechanism.
2. The AIMMS execution on the solver session.
Retrieve incumbent solution or intermediate result from the solver, also as part of the incumbent callback mechanism. Store this retrieved solution as a case file on the AIMMS PRO storage.
3. The AIMMS execution on the data session.
Execute a procedure to retrieve the name of the case file generated in the previous step, and load the data from the case file into the data session for the end user’s viewing.
This approach is possible because both the data session and the solver session have access to the AIMMS PRO storage and storing all the different incumbent solutions as case files allows the user to access them when required to conduct further studies. The following image illustrates how AIMMS PRO storage is organized:
We will use the folder pro:/UserData/<environment>/<User>/Cases/<app>/ on AIMMS PRO storage.
## Implementation¶
There are two steps to communicate the information from the first to the third level.
### Step 1. From Solver (level 1) to solver session (level 2)¶
Step 1A Construct the incumbent solution on the solver session.
As we want to display/update a new incumbent solution as it becomes available, we use the mathematical program suffix CallbackNewIncumbent. A callback procedure is assigned to this suffix as below, and this executes the assigned procedure whenever a new incumbent solution is found by the solver. Include the below statement before the solve statement in your project.
FlowShopModel.CallbackNewIncumbent := 'NewIncumbentCallback';
Step 1B Retrieve the incumbent solution generated by the solver to the AIMMS solver session.
In the running example, the procedure NewIncumbentCallback first retrieves the incumbent solution from the solver session, transforms it to be displayed in the Ganttchart, creates a case file containing this data and saves it on the AIMMS PRO storage. The intermediate data transformation steps are specific to this example and hence are not explained here.
The predefined procedure RetrieveCurrentVariableValues retrieves the current values of the variables as the name suggests. It takes an argument specifying which variable values are to be retrieved and we use the predefined set AllVariables to get the values of all the variables in the model. TimeSpan is the objective function of the model and we are storing the current incumbent value with the assignment statement.
! Transfer the solution from the solver to AIMMS
empty JobSchedule;
RetrieveCurrentVariableValues(AllVariables);
TimeSpan := FlowShopModel.Incumbent;
Create a case file containing this solution.
AllCaseFileContentTypes += 'sIncumbentSolutionIdentifiers' ;
CurrentCaseFileContentType := 'sIncumbentSolutionIdentifiers' ;
sp_CaseFileName := FormatString( "Incumbent%i.data", pIncumbentNumber );
P_IncumbentNumber += 1 ;
sp_FullCaseFileName := "data/" + spCaseFileName ;
CaseFileSave( spCaseFileName, sIncumbentSolutionIdentifiers );
Now, save the case file on PRO storage and store the name of the case file (including location path) in a string parameter.
! Transfer the case from the data folder of the solver session to the AIMMS PRO storage user data folder.
! Transfer the GC solution from AIMMS to a case.
spFullProStorageName := "pro:/userdata/" + pro::GetPROEnvironment() +
"/" + pro::GetPROUserName() + "/Cases/" + pro::ModelName + "/" + spCaseFileName ;
Pro::SaveFileToCentralStorage(spCaseFileName, spFullProStorageName );
The AIMMS execution side is now triggered using the previously updated string parameters as arguments.
! Run the AIMMS execution on the data session
UpdateIncumbentToClient(spFullProStorageName);
### Step 2. From solver session (level 2) to data session (level 3)¶
The procedure UpdateIncumbentToClient is a simple loading case file execution using the predefined procedure, CaseFileLoad.
if pro::DelegateToClient(flags: 0) then
return 1;
endif ;
! From here on, only the client (data) session is running.
A copy of the flowshop model that is the result of this answer: Flow Shop - share intermediate.
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# Tag Info
12
The most straightforward and flexible approach to typesetting URL strings is to use the \url macro that's provided by the url and hyperref packages. I use the word "flexible" in part because \url{...} can usually find good line breaks -- an important consideration when dealing with long URL strings (which occur quite frequently, right?). Outside of ...
9
A macro definition is not executed at the point of definition, you can go \newcommand\foo{\any old \rubbish } and as long as {} match up \foo is defined. You may get an error later if you try to use \foo, but you get no error at this point, and if \any and \rubbish are defined by the time you use \foo there is no error. Conversely a box is typeset as it ...
8
It's your PDF viewer's fault: Mac OS X Preview (at least v7.0) recognises URLs and makes them clickable, whether they be typeset with a special package (hyperref, url) or not. To convince yourself, try compiling the following example and open the output in Preview; the URL will be clickable. So there's really nothing you can do about it on the author's ...
8
TeX assigns category code when an argument is grabbed and tokenized. As a result, inside your definition of \url, & has catcode 'tab alignment' (assuming normal rules apply). You can only \def an active char, so this step fails as you've observed. What you need to do is make sure that the definition of \url contains an active &: \begingroup ...
7
It is much better not to change catcodes mid document but instead change the mathcode (as then it works as much as possibe even in macro arguments) this is what url.sty does url.sty uses hardly any latex so works with plain tex with a bit of encouragement, or you could simply edit the file to remove the latex bits rather than defining stubs as here ...
7
Use eplain: \input eplain \beginpackages \usepackage{url} \endpackages \rightskip=10em minus 8em % avoid overfull box \url{http://tex.stackexchange.com/some/long/path/and?someBizarreLong=param&andYetAnotherSuchBizarreLong=param} \bye Your (simplistic) definition can be corrected with \begingroup \catcode`\&=\active % we want an active & ...
7
You should change the anything to do with the bibliography immediately before the bibliography, otherwise it would affect everything following the preamble as in your case. \documentclass[11pt,a4paper]{moderncv} % moderncv themes \moderncvtheme[blue]{classic} % character encoding \usepackage[utf8]{inputenc} \usepackage{bbding} % ...
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\documentclass{article} \usepackage{hyperref} \newcommand\rurl[1]{\xurl#1\empty\empty\empty\empty\empty\xurl} \def\xurl#1#2#3#4#5#6\xurl{% \def\tmp{#1#2#3#4#5}% \href{\ifx\tmp\xurlhttp \else http://\fi#1#2#3#4#5#6}% {\nolinkurl{#1#2#3#4#5#6}}% } \def\xurlhttp{http:} \begin{document} \rurl{ipython.org}\\ \rurl{http://ipython.org} \end{document}
7
6
since the purpose of this code is simply to list out the web sites, i have two suggestions: forget align*. instead use the enumerate environment and enter the urls with \verb+...+ as suggested by R. Schumacher. as previous, but instead of \verb, add \usepackage{url} and enter the site addresses as \url{...}. enumerate will take care of numbering and ...
6
\documentclass{article} \usepackage{url} \usepackage{pgffor} \usepackage{xparse} \usepackage{xstring} \usepackage[colorlinks=true]{hyperref} \NewDocumentCommand{\FormatLinks}{% s% #1 =* not used yet O{}% #2 = optional title m% #3 = Mandatory title m% #4 = URL Link }{% \par \hspace*{1.0cm}\href{#4}{#3\IfValueT{#2}{~#2}}% }% ...
6
EDIT: Added Title clickable (1. just the title cickable and 2. the whole reference clickable) 1. Just the Title reference clickable You can redefine the title macro and add the \href to the title using the DeclareFieldFormat. I edited the default definitions in the biblatex.def file. \DeclareFieldFormat{title}{\myhref{\mkbibemph{#1}}} \DeclareFieldFormat ...
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I think you are trying to print the paths. If so, you may need this: \documentclass{article} \usepackage[obeyspaces]{url} \begin{document} \path{\\server\folder\folder} \path{\\server\my folder\folder} \end{document}
6
csplain is (by default) sensitive to non UTF-8 codes in input. It uses encTeX for this in pdftex engine. The url.sty manipulates with non UTF-8 codes, encTeX is sensive to this more than we need and this yields this error. You can write \input utf8off at beginning of the document. After this, the pdftex treats its input normally as 8bit. If you are using ...
5
The \url command can be redefined the same way as hyperref. The following example first defines url command \guilurl, which uses single guillemots as angle brackets. Then \url is redefined using \guilurl: \documentclass{article} \usepackage[T1]{fontenc}% \guilsinglleft and \guilsinglright \usepackage{lmodern} \usepackage{amstext} \usepackage{hyperref} ...
5
url changes the font but you can set it to default to serif to match \href: \documentclass{book} \usepackage{xcolor} \usepackage{hyperref} \hypersetup{colorlinks=true,linkcolor=blue,urlcolor=blue} \urlstyle{rm} \begin{document} url: \url{www.yahoo.com} href: \href{http://www.yahoo.com}{www.yahoo.com} \end{document} Note that \urlstyle{} is from url ...
4
One way would be to use the background package: Code: \documentclass[10pt,a4paper]{article} \usepackage[all]{background} \usepackage{url} \usepackage{lipsum} \SetBgContents{\url{http://tex.stackexchange.com/}}% Set contents \SetBgPosition{0.25cm,-5.0cm}% Select location \SetBgOpacity{1.0}% Select opacity \SetBgAngle{90.0}% Select rotation of logo ...
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I usually use a combination of eso-pic, graphicx and rotating as it is easy to adjust in terms of location, angle and size. \documentclass[12pt]{scrartcl} \usepackage{blindtext} \usepackage{eso-pic, rotating, graphicx} \AddToShipoutPicture{\put(30,200){\rotatebox{90}{\scalebox{3}{Examiners copy}}}} \begin{document} \blindtext[4] \end{document}
4
LaTeX will refuse to break a 'word' if it contains, among a few other characters, a /. This has very solid reasoning that's outside the scope of this answer, but the url LaTeX package will handle these things very cleanly. Instead of typing in the file path directly, use Insert -> URL. This will (presumably) wrap the argument in \url{<my text ...
4
There is no need to escape a dollar symbol in a url. \documentclass[a4paper]{article} \usepackage[T1]{fontenc} \usepackage{hyperref} \begin{document} \url{test$test} \end{document} 4 Go to wikipedia and copy its URL as such; then add \ before each %: \documentclass{article} \usepackage{hyperref} \begin{document} \url{http://www.wikiwand.com/pl/Prawo_Lewisa-Mogridge\%E2\%80\%99a} \end{document} EDIT 1 Instead of \url, you can use \href, in case you want the hyperlink: \documentclass{report} \usepackage{hyperref} \begin{document} ... 4 If you help biblatex and hyperref figure out which protocol (HTTP, HTTPS, FTP, ...) to use, you should be fine. DOIs appear correctly since biblatex turns the raw doi into the proper hyperlink itself. So the obvious solution is to always specify the protocol properly as in @misc{Hohn.2013, author = {Höhn, Hans-Joachim}, title = {Theologie als ... 4 You have two errors in your hack. First, you leave urldate on stack after your if$ statement. This is how you get two dates. You need to use this instance with swap$instead of putting the third instance of urldate on stack: FUNCTION {format.urldate} { urldate duplicate$ empty${ pop$ "" } { "~(Accessed: " swap$* ")" * } if$ } However, if ...
4
How about ps. I do not think it is a good idea to post such a long url in the document. Use a small name and use that for the link. But if this is what you want: \documentclass[10pt,letterpaper]{article} \usepackage{hyperref} \usepackage{longtable} ...
4
You can use the user keys to store additional information. For example: \documentclass[12pt,BCOR=15mm]{scrbook} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[ngerman]{babel} \usepackage{hyperref} \usepackage[xindy,nonumberlist]{glossaries} \GlsSetXdyCodePage{duden-utf8} \makeglossaries \newglossaryentry{glossaries} { name=Glossaries, ...
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First of all I recommend you to use biblatex since it is very adaptable and processes information greatly! In the biblatex manual, they are defined as follows: howpublished: A publication notice for unusual publications which do not fit into any of the common categories url: The URL of an online publication. Thus, I recommend you in general to ...
4
Without hyperref no hyperlink is created by LaTeX; however, PDF previewers on Mac OS X have heuristics that try finding URL's (or, more generally, URI's) in the PDF files. Such heuristics often try being too smart and fail. :-( So it's not a LaTeX problem, but of the Apple library PDF viewers are based on (Preview, Skim and others). Since Adobe Reader is ...
4
If the urls are typeset with the url package, then you can load it as \usepackage[hyphens]{url} to allow breaks at hyphens. This is not the default, since these breaks can be confusing for the reader who doesn't know if the hyphen is actually part of the url or not. (With strings like implementing-operators-for-your-class.html that shouldn't be a ...
4
Package hypgotoe add support for embedded go-to actions (GoToE) to \href. This action type only works from and to PDF files. The package only supports destination names as link targets. For example they can be set via \hypertarget or extracted by package zref-xr or xr-hyper. More arbitrary bookmarks can be generated via package bookmark. Currently it does ...
4
The above answer by mhp may not provide a solution when PDFLaTeX complains with "Option clash for package url." The cause of this may be because of the package hyperref, which also loads the package. If hyperref is loaded before the url package it gives this error, because the url package is then loaded twice with different options: one without and one with ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# Fukushima impact is still hazy
### Subjects
Chaos and bureaucracy hamper assessment of nuclear crisis.
Schools such as this one in Fukushima City are a high priority for clean-up efforts. Credit: REUTERS/N. HAYASHI/GREENPEACE
Tatsuhiko Kodama began his 27 July testimony to Japan's parliament with what he knew. In a firm, clear voice, he said that the Radioisotope Center of the University of Tokyo, which he heads, had detected elevated radiation levels in the days following the meltdown of three reactors at the Fukushima Daiichi nuclear power station. But when it came to what wasn't known, he became angry. "There is no definite report from the Tokyo Electric Power Company or the government as to exactly how much radioactive material has been released from Fukushima!" he shouted.
Kodama's impassioned speech was posted on YouTube in late July and has received nearly 600,000 views, transforming him into one of Japan's most visible critics of the government. But he is not alone. Almost six months after an earthquake and tsunami triggered the meltdowns, other researchers say that crucial data for understanding the crisis are still missing, and funding snags and bureaucracy are hampering efforts to collect more. Some researchers warn that, without better coordination, clean-up efforts will be delayed, and the opportunity to measure the effects of the worst nuclear accident in decades could be lost. Kodama and a handful of Japanese scientists have become so frustrated that they are beginning grassroots campaigns to collect information and speed the clean-up.
Since the crisis began, the Tokyo Electric Power Company and the Japanese government have churned out reams of radiation measurements, but only recently has a full picture of Fukushima's fallout begun to emerge. On 30 August, the science ministry released a map showing contamination over a 100-kilometre radius around the plant. The survey of 2,200 locations shows a roughly 35-kilometre-long strip northwest of the plant where levels of caesium-137 contamination seem to exceed 1,000 kilobecquerels per square metre. (After the 1986 Chernobyl disaster in Ukraine, areas with more than 1,480 kilobecquerels per square metre were permanently evacuated by the Soviet authorities. In Japan, the high-radiation strip extends beyond the original forced evacuation zone, but falls within a larger 'planned evacuation zone' that has not yet been completely cleared.)
Exposure estimates
Japan's Nuclear and Industrial Safety Agency has also published new estimates of the total radiation released in the accident, based on models that combine measurements with what is known about the damage to the reactors. The latest figures, reported to the International Atomic Energy Agency in June, suggest that the total airborne release of caesium-137 amounts to 17% of the release from Chernobyl (see map). The government estimates that the total radiation released is 7.7 × 1017 becquerels, 5–6% of the total from Chernobyl.
Click for full map.
Yet "there are still more questions than definite answers", says Gerald Kirchner, a physicist at Germany's Federal Office for Radiation Protection in Berlin. High radiation levels make it impossible to directly measure damage to the melted reactor cores. Perhaps the greatest uncertainty is exactly how much radiation was released in the first ten days after the accident, when power outages hampered measurements. Those data, combined with meteorological information, would allow scientists to model the plume and make better predictions about human exposure, Kirchner says.
Several measurements suggest that some evacuees received an unusually high dose. Five days after the crisis began, Shinji Tokonami, a radiation health expert at Hirosaki University, and his colleagues drove several hundred kilometres from Hirosaki to Fukushima City, taking radiation measurements along the way. The results indicate that evacuees from Namie, a town some 9 kilometres north of the plant, received at least 68 millisieverts of radiation as they fled, more than three times the government's annual limit (http://dx.doi.org/10.1038/srep00087).
The dose is still safe, says Tokonami. Gerry Thomas, a radiation health expert at Imperial College London, adds that radiation exposures from Fukushima were far lower than those from Chernobyl. "Personally, I do not think that we will see any effects on health from the radiation, but do expect to see effects on the psychological well-being of the population," she says.
But Kodama says that residents of Namie and other towns inside the evacuation zone could have been better protected if the government had released its early models of the plume. Officials say they withheld the projections because the data on which they were based were sparse.
Hotspots
Many questions also remain about the radiation now in the environment. The terrain around Fukushima is hilly, and rainwater has washed the fallout into hotspots, says Timothy Mousseau, an ecologist at the University of South Carolina in Columbia who recently travelled to the Fukushima region to conduct environmental surveys. The plant, located on the Pacific coast, continues to spew radionuclides into the water, adds Ken Buesseler, an oceanographer from Woods Hole Oceanographic Institution in Massachusetts. During a cruise in mid-July, his team picked up low-level radiation more than 600 kilometres away. Ocean currents can concentrate the fallout into hotspots something like those on land, making the effect on marine life difficult to gauge.
Gathering more data is a struggle, say researchers. Tokonami says that overstretched local officials are reluctant to let his team into the region for fear that it will increase their workload. Buesseler and Mousseau add that Japan's famed bureaucracy has made it difficult for outside researchers to carry out studies. Funding has also been a problem. To complete his cruise, Buesseler turned to the Gordon and Betty Moore Foundation for a US$3.5-million grant. Mousseau got a biotech company to sponsor his trip and has since found funding through the Samuel Freeman Charitable Trust. There are still more questions than definite answers. , Some Japanese scientists have grown so frustrated with the slow official response that they have teamed up with citizens to collect data and begin clean-up. Because radiation levels can vary widely over small distances, the latest government maps are too coarse for practical use by local people, says Shin Aida, a computer scientist at Toyohashi University of Technology. Aida is proposing a more detailed map-making effort through 'participatory sensing'. Using the peer-to-peer support website 311Help (http://311help.com), Aida plans to have people gather samples from their homes or farms and send them to a radiation measuring centre, where the results would be plotted on a map. Kodama, meanwhile, is advising residents in Minamisoma, a coastal city that straddles the mandatory evacuation zone. Minamisoma has set aside ¥960 million ($12.5 million) for dealing with the nuclear fallout, and on 1 September it opened an office to coordinate the effort. "We needed to find out what's the most efficient and effective way to lower the risk," says one of the leading officials, Yoshiaki Yokota, a member of the local school board. The first job is to collect and bury soil at schools. Residents have learned to first roll the soil in a vinyl sheet lined with zeolite that will bind caesium and prevent it from seeping into the groundwater.
Farther northwest, in the city of Date, decontamination efforts are moving from schools to nearby peach farms. On 31 August, some 15 specialists started removing the top centimetre of soil at the farms with a scoop or with suction machines, trying not to damage the peach trees' roots. They hope to lower the radiation enough to produce marketable fruit next year.
After a sluggish start, the central government is launching two pilot clean-up projects for the region. One will focus on areas like Minamisoma, where radiation is less than 20 millisieverts per year on average but includes some hotspots. The other will look at 12 sites of radiation of more than 20 millisieverts per year.
Researchers are hopeful that the chaos immediately after the crisis will soon give way to a sharper picture of the fallout and its toll. The United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR), which conducted many studies after the Chernobyl disaster, is working with Japanese officials to collate the stacks of data collected since the crisis began. UNSCEAR is also studying the environmental effects of the accident and the exposure of workers and evacuees, and aims to have an interim report ready by next summer.
Clean-up is the top priority, but Fukushima also offers a unique research opportunity, says Mousseau, who has worked extensively at Chernobyl. Because of Soviet secrecy, researchers missed a crucial window of opportunity in studying the Ukrainian crisis. "Japan offers us an opportunity to dig in right off the bat and really develop a profound understanding," he says.
Authors
David Cyranoski reports from Tokyo and Geoff Brumfiel from London.
## Rights and permissions
Reprints and Permissions
Cyranoski, D., Brumfiel, G. Fukushima impact is still hazy. Nature 477, 139–140 (2011). https://doi.org/10.1038/477139a
• Published:
• Issue Date:
• DOI: https://doi.org/10.1038/477139a
• ### Public Exposure to U.S. Commercial Nuclear Power Plants Induced Disasters
• Dean Kyne
International Journal of Disaster Risk Science (2015)
• ### RETRACTED ARTICLE: Release, deposition and elimination of radiocesium (137Cs) in the terrestrial environment
• Ayesha Masood Khan
• Nor Kartini Abu Bakar
Environmental Geochemistry and Health (2014)
• ### Isotopic evidence of plutonium release into the environment from the Fukushima DNPP accident
• Jian Zheng
• Keiko Tagami
Scientific Reports (2012)
• ### The biological impacts of the Fukushima nuclear accident on the pale grass blue butterfly
• Atsuki Hiyama
• Chiyo Nohara
• Joji M. Otaki
Scientific Reports (2012)
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# colour.contrast.pupil_diameter_Barten1999#
colour.contrast.pupil_diameter_Barten1999(L: ArrayLike, X_0: ArrayLike = 60, Y_0: Optional[ArrayLike] = None) NDArrayFloat[source]#
Return the pupil diameter for given luminance and object or stimulus angular size using Barten (1999) method.
Parameters:
• L (ArrayLike) – Average luminance $$L$$ in $$cd/m^2$$.
• X_0 (ArrayLike) – Angular size of the object $$X_0$$ in degrees in the x direction.
• Y_0 (Optional[ArrayLike]) – Angular size of the object $$X_0$$ in degrees in the y direction.
Returns:
Pupil diameter.
Return type:
numpy.ndarray
References
[Bar99], [Bar03], [CKMW04], , [WY12]
Notes
• The Log function is using base 10 as indicated by [WY12].
Examples
>>> pupil_diameter_Barten1999(100, 60, 60)
2.7931307...
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# Programmable gear-based mechanical metamaterials
## Abstract
Elastic properties of classical bulk materials can hardly be changed or adjusted in operando, while such tunable elasticity is highly desired for robots and smart machinery. Although possible in reconfigurable metamaterials, continuous tunability in existing designs is plagued by issues such as structural instability, weak robustness, plastic failure and slow response. Here we report a metamaterial design paradigm using gears with encoded stiffness gradients as the constituent elements and organizing gear clusters for versatile functionalities. The design enables continuously tunable elastic properties while preserving stability and robust manoeuvrability, even under a heavy load. Such gear-based metamaterials enable excellent properties such as continuous modulation of Young’s modulus by two orders of magnitude, shape morphing between ultrasoft and solid states, and fast response. This allows for metamaterial customization and brings fully programmable materials and adaptive robots within reach.
## Main
Materials featuring tunable elastic properties1,2 offer tremendous possibilities for smart machines, robots, aircraft and other systems3,4,5. For example, robotic systems with variable stiffness can adapt to missions like grabbing6 and jumping, or maintain optimal performance in a changeable environment7. However, elastic properties of conventional materials are barely tunable even if phase changes are induced. Mechanical metamaterials8,9,10,11 are artificial architected materials that exhibit properties beyond those of classical materials12,13,14,15,16. Most existing metamaterials integrate monofunctional load-bearing elementary structures (such as rods, beams or plates) in specified topologies with fixed or hinged nodes (Fig. 1). Reconfigurable metamaterials open possibilities for drastic changes in properties17,18,19. When stimulated by stress, heat or electromagnetic fields, reconfigurations in these metamaterials are induced by the formation of new contacts, buckling20,21,22 or rotating hinges23,24,25. Due to node constraints, this permits the reshaping among only a few stable states26,27 and often includes unstable states, which limits the tunability. Reducing the connectivity28 or relaxing the constraints (for example, with chiral structures9 or by connecting elements with flexural traps29) can enable more states to improve the shape-changing capability, but this inevitably deteriorates the robustness and structural stability that are essential for most applications. Moreover, reconfiguration, including the shape-memory effect, usually involves large deformation that either leads to irreversible plastic deformation or adversely competes with the commonly required high stiffness18. Although the chemical-responsive materials30,31 enable some in situ tunability, the regulation process of their elastic properties is usually very slow, just like for thermal-responsive materials19. Assembling rods mounted with gears into special lattices can improve the stability while preserving the rotatable nodes32, but engineering practical and robust metamaterials with continuously tunable elasticity, especially with fast in situ tunability in service, remains a major challenge.
## Design concept
Overcoming these challenges requires an unprecedented design paradigm. First, tunability may be realized by assembling elements with built-in stiffness gradients. Second, the coupling between elements must comply with large deformation. Achieving tunable yet strong solids requires ensuring tunability under large force and robust controllability while avoiding plastic deformation in tuning. We find that such a mutable-yet-strong coupling can be realized with gear clusters. Gears provide an ideal mechanism to smoothly transmit rotation and heavy compressive loads thanks to the reliable gear engagement (meshing). Stiffness gradients can be built into an individual gear body or realized with hierarchical gear assemblies. Gear clusters can be assembled into manifolds and can, as metacells, be periodically arranged to form metamaterials (Fig. 1c). The proposed design concept is very general since there exist numerous architectures for gear assembly. Exotic functionality and flexible tunability can emerge from the diversity of gear types, built-in variability and cluster organization. We create several metamaterial prototypes with different gear clusters to demonstrate this.
## Metamaterial based on Taiji gears
The first prototype is created using compactly coupled periodic gears and two lattice frames (front and back) to arrange the gears into a simple quadratic pattern (Fig. 2a). The plane gears contain hollow sections. The outer part forms two elastic arms whose radial thickness smoothly varies with the rotation angle θ (Fig. 2b and Supplementary Figs. 1 and 2). Subject to compressive loading, deformation in the arms is dominated by bending (Fig. 2c). The effective stiffnesses of both an arm karm and a pair of arms Kp depend on the angle θ (Supplementary Fig. 3). The homogenized Young’s modulus of the metamaterial along the y axis Ey = Kp/B + Ef has contributions from the gears and frame (Methods, ‘Equivalent method’ section). Here, Ef is the stiffness of the frame and B denotes the gear width. Ey is continuously tunable by rotating the gears and dominantly depends on Kp since min(Kp/B) Ef = 2.06 MPa here.
The tunability relies on the shape of the built-in hollow section. Among diverse choices, the shape inspired by the Chinese Taiji diagram (Fig. 2b), characterized by a spiral direction, can give smooth variation and polarity. The angle difference between the two local coordinates is β (Fig. 2g). The spin rotations are opposite in any two meshing gears. Also, the spiral directions of the Taiji patterns on the front and the back faces are reversed. Therefore, the meshing mode of a pair of gears has two polarities. When the spiral directions of patterns are opposite (Fig. 2b), the polarity is positive, labelled as P+(β). The meshing pair in Fig. 2g features negative polarity, P(β).
We employ finite element analysis (FEA) to simulate the contact problem in gear-based metamaterials (Methods and Supplementary Figs. 4 and 5). Contact nonlinearity becomes apparent for high Kp (Supplementary Fig. 3). Young’s modulus is evaluated from the slope of the uniaxial stress–strain curves at relatively large strain ε. An all-metallic prototype, consisting of 5 × 5 copper gears and steel frames, is manufactured and assembled with P+(3°) and P(15°) metacells, respectively (Supplementary Fig. 2). The gear has 60 teeth, with the tooth thickness tto = 0.35π mm, diameter D = 42 mm and width B = 20 mm. Measured cyclic loading–unloading curves show some hysteresis (Fig. 2e). This is ascribed to the sliding friction between the meshed teeth (Supplementary Fig. 6). Figure 2f,g demonstrates that experimental results of Ey(θ) are in excellent agreement with FEA. The modulation period is 180° in both the P+(3°) and P(15°) cases. The smooth Ey(θ) curve indicates that the obtainable stable states are dense and that continuous tunability is achieved. Both polarity and β affect the tunable range and the correlation between the tunable properties and θ (Fig. 2d). For P+(3°) in Fig. 2f, the zigzag curve of Ey(θ) reaches the maximum value Emax = 7.67 GPa at θ = 78°, where the solid parts of the gears are in contact, and a minimal value Emin = 0.102 GPa at θ = 114°, where the meshing connects the forearms. This experimentally obtained modulation range of Emax/Emin = 75 demonstrates the spectacular reconfigurability. For P(15°), Ey(θ) is sombrero-shaped with a tunable range of 33 from Emin = 0.156 GPa to Emax = 5.13 GPa. Since Ex(θ) = Ey(θ + 90°), the anisotropy in orthogonal directions also changes with θ. For example, the P+(3°) metamaterial can be continuously modulated from Ex = Ey to the maximum ratio (Ey/Ex)max = 24.8. The latter gives a metamaterial with negligible lateral expansion upon compression (Supplementary Fig. 7). Compared to existing designs, the node constraints in gear-based metamaterials are relaxed, but the connection stability and reconfiguration robustness are maintained at any θ even under large compressive loads (Supplementary Video 1). The design is also robust to accommodate manufacturing inaccuracies when regarding the angle β as an indicator of the alignment error of the gears. Figure 2d shows that the programmability is preserved even for large β.
The all-metallic metamaterial introduced above is manufactured by assembling individual gears. For scale-up and miniaturization, it is desirable to avoid the assembly of individual parts. Next, we demonstrate that integrated gear-based metamaterials can be directly manufactured with three-dimensional (3D) printing, even on the microscale. The major challenge for such integrated manufacturing is to guarantee that the meshing teeth are not fused together but still reliably engaged. To tackle this problem, a small clearance is reserved between the surfaces of the meshing teeth in the assembled digital model to overcome manufacturing errors (Methods). Here we manufacture an integrated micro metamaterial consisting of 5 × 6 Taiji gears (Fig. 2h) by adopting the projection micro-stereolithography 3D printing technique. The diameter and tooth thickness of the Taiji gear are 3.6 mm and 235 µm, respectively; the thickest arm is 75 µm (Supplementary Fig. 8). The micro gears are arranged with P+(0°), and the reserved minimal clearance between the teeth is 32 µm. The sample is made of a photosensitive resin with a Young’s modulus of 3.5 GPa. As experimentally demonstrated in Fig. 2i, the equivalent modulus Ey(θ) of this micro specimen can be smoothly tuned by 35 times (from 8.3 MPa to 295 MPa). Using this integrated design strategy, gear-based metamaterials could be scaled up in size and number of gears with appropriate high-resolution large-scale 3D printing facilities. Modulation of such integrated metamaterials can be achieved with distributed drives or motors (Supplementary Video 2 and Supplementary Fig. 9a).
## Metamaterial based on planetary gears
Obviously, this first metamaterial is tunable only under compressive loading. The tensile load is carried by the frame, and the tensile modulus is Et = Ef. One may also aim at strong metamaterials whose compressive and tensile moduli are both tunable while preserving structural integrity. This can be achieved by organizing a planetary gear system as a metacell (Fig. 3a). In this example, the metacell contains six gears: an inner-toothed ring gear (Supplementary Fig. 10), a central sun gear and two pairs of planetary gears A1–A2 and B1–B2. Gear centres A1–O–A2 (and B1–O–B2) are colinear. Using this gear cluster, we create a hierarchical and strong metamaterial whose tunability emerges from the relative rotation of the gears inside the metacell. The thickness of the ring tr is uniform. Neighbouring rings are rigidly connected in a quadratic lattice, which ensures structural integrity. Planetary gears revolve along the ring when rotating the sun gear by θsun. Their position is given by the revolution angle θpr = θsunrsun/(Rin + rsun), where Rin and rsun are the radii of the ring and sun gears, respectively (Supplementary Table 1 for parameter values). The teeth prevent relative slippage between the two gears even under tension. The metamaterial elastic properties are given by the effective stiffness of the annulus ring supported by the planetary gears that act as fulcrums (Fig. 3b). The position θpr of the planetary gears determines the elastic properties. We adopt the orthogonal relation A1A2 B1B2, which gives a large tunable range and symmetric behaviour with a modulation period of 90°. The tunable range can be further modified using the angle A1OB1. For the assembled metamaterial, all sun gears are connected to transmission gears by shafts (Fig. 3a), and those transmission gears are compactly coupled. Thereby robust reconfiguration of all metacell patterns can be achieved by rotating transmission gears.
Fx and Fy denote the compressive loads in the x and y directions, respectively (Fig. 3a). Under uniaxial compression (Fy > 0, Fx = 0), only the pair of planetary gears with an angle smaller than that of the loading axial (min(YOA1, YOB1) < 45°) supports the load (Fig. 3b). Stress in the other pair is zero. Conversely, under uniaxial tension, only the other pair is load-bearing. The two pairs exchange roles at θpr = 45°, and the material is orthogonally isotropic. This metamaterial presents a more remarkable compressive nonlinearity because four pairs of meshing teeth in a metacell bear loads. Both the compressive and tensile moduli Ec and Et reach maxima at θpr = 0, but $$E_{{{\mathrm{c}}}}^{{{{\mathrm{max}}}}} \gg E_{{{\mathrm{t}}}}^{{{{\mathrm{max}}}}}$$, and thus the static compressive-tension symmetry is broken (Fig. 3g). Moreover, at θpr = 45°, no stress is transmitted to the planetary gears (Fig. 3b); both moduli reach minima there, and $$E_{{{\mathrm{c}}}}^{{{{\mathrm{min}}}}} = E_{{{\mathrm{t}}}}^{{{{\mathrm{min}}}}}$$.
We fabricate three kinds of specimens using this strategy. An all-steel macro metamaterial is manufactured by assembling 3 × 3 metacells (Fig. 3c) with lattice constants ax = ay = 27 mm. The steel gears have a small tooth thickness tto = 0.15π mm, and Rin = 12 mm, rsun = 6 mm and tr = 1 mm. Integrated manufacturing of this prototype is more challenging than that of the Taiji pattern because there are two layers and every metacell possesses eight pairs of meshing teeth. The integrated prototype can also be directly manufactured by 3D printing, at both macro and micro scales (Methods, Supplementary Figs. 11 and 12 and Supplementary Video 3 for more details). We print a 6 × 6 macro specimen (Fig. 3d) using a polymer with a Young’s modulus of 2.5 GPa, and print a 3 × 4 micro specimen (Fig. 3e) using a resin with a Young’s modulus of 3.5 GPa. The size and the number of metacells are limited by the capability of the 3D printer rather than the design strategy. The micro polymer sample (Fig. 3e) has Rin = 2.4 mm, rsun = 0.6 mm, tr = 0.3 mm and ax = ay = 5.4 mm, with a tooth width and height of 135 µm and 225 µm, respectively.
The experimental results are consistent with the FEA simulations for all specimens (Fig. 3g–i and Supplementary Fig. 11). In this strong hierarchical metamaterial, we can smoothly tune the compressive modulus Ec of the macro metallic specimen by 46 times (5.2–0.11 GPa), the macro polymer specimen by 55 times (69–1.25 MPa; Supplementary Fig. 12b) and the micro specimen by 25 times (100–4 MPa). Meanwhile, their tensile modulus Et can be tuned by 5 times (0.52–0.11 GPa), 5.6 times (7–1.25 MPa) and 5 times (20–4 MPa), respectively. In Fig. 3h, some differences between experiment and FEA near θpr = 45° arise from the boundary conditions (Supplementary Fig. 13). The in situ tunability combined with the reasonably large moduli in tension and compression as well as large shear rigidity makes this metamaterial design particularly robust and strong, yet tunable. Furthermore, the metamaterials can be synchronously controlled with distributed motors at both the macro and micro scales (Supplementary Video 4).
## Mechanisms for stability
Interestingly, the metamaterial in Fig. 2a (a discrete gear lattice with a very soft frame) remains stable under compressive stresses and shows large rigidity in shear. One of the contributing factors underpinning the observed stability stems from the non-uniform loading of the meshing teeth at different points, which leads to bending deformations that tightly grip the teeth together (Supplementary Fig. 5). The relatively large shear modulus of the metamaterial, G = Gg + Gf, is composed of the shear moduli generated by gears (Gg) and by frames (Gf = 1.04 MPa). Shear force induces both spin and the planetary rotation of gears. For a pair of gears, the relative planetary rotation leads to zero shear resistance, Gg = 0, giving a highly unstable state (Fig. 4a). However, in a group of four gears (shown in Fig. 4b), shear stress τ induces mutual locking of the planetary rotation by the opposite spin of the neighbouring gears, which is referred to as shear interlock. We calculate the shear stiffness of the metacell with periodic boundary conditions and the finite n × n gear lattice (Supplementary Figs. 1416). Owing to the shear interlock, the shear modulus is large but only marginally tunable (Supplementary Fig. 16), which is demonstrated by the measured generalized shear stiffness Kshear/B of the finite 3 × 3 architecture in Fig. 4c.
## Gear metamaterial for shape morphing
The programmability of gear-based metamaterials is not limited to elastic constants. Removing every second gear in every second row of the metamaterial in Fig. 2 can release the shear interlock (inset in Fig. 4f) to generate a state with Gg = 0. The effective shear modulus is then determined solely by the low stiffness frame, and the metamaterial can be considered as ultrasoft matter. The vanishing shear modulus enables complex deformation modes (Fig. 4d,e and Supplementary Video 5), conducive to shape morphing. To verify this, an ultrasoft prototype consisting of 4 × 4 metacells with rubber frames is manufactured and tested (Fig. 4f). The gears are made of aluminium alloy (Supplementary Fig. 17). Shear tests on the prototype give a tiny modulus of G = 21.52 kPa. Independent measurement of the frames gives Gf = 21.11 kPa, so that Gg = GGf is indeed negligibly small. Moreover, the modulus G remains tiny until the shear strain γ reaches 25%, at which point the semi-free gear interacts with two other gears, which builds a new meshing connection among the three gears. The new connection supports high shear stresses and leads to a sharp rise in G, switching the soft matter to a stiff solid. The resulting solid represents a geometrically interlocked state (Methods). Oscillations of the shear stress in Fig. 4c arise from the critical meshing state among the three gears before they interlock (Supplementary Fig. 18). The shear strain at which the geometrical interlock occurs can be adjusted by the size of the neighbouring gears, which in turn determines the limiting (strong) states of a shape-morphing structure. Previous metamaterial designs with vanishing shear modulus, like pentamode metamaterials33, show vanishing moduli only at small strains and in extremely fragile structures.
## Potential applications
Conventional machines generally rely on materials with constant stiffness and therefore show constant stiffness themselves. The designed stiffness is then a compromise among stability, safety, efficiency and performance, thus hindering the pursuit of the best performance and efficiency in variable environments. Programmable materials featuring tunable elastic properties, including active mechanical metamaterials, are much anticipated in intelligent machines and systems1,2. Here we offer a comparison of typical material designs from the literature2,19.
The response time, stability, force and energy required for property changes are all critical attributes for variable-stiffness structures4. We take the strain and response time required to accomplish a tunable period of material stiffness as metrics to position our gear-based metamaterials among the existing active materials (Fig. 5). Shape-morphing metamaterials18,34 enable tunability between or among two or a few stable states. The achieved tunability is non-continuous and requires a large deformation (ε ≈ 30%). Thermal-responsive composites35 made of shape-memory alloys or polymers may give a continuously tunable modulus. However, they require a long response time, and some suffer from nearly 100% strain36,37. Chemical-responsive materials30,31 containing hydrogels can offer continuous and in situ tunability, but also require hours of response time. Conventional magneto- or electro-responsive metamaterials38 based on elastomers or magnetorheological fluids can give fast, continuous, but narrow tunability, which usually requires a high active voltage (~5 kV) and complex facilities19. Our gear-based metamaterials in Figs. 2 and 3 can offer a fast response and the desired broad-range, continuous and in situ tunability of stiffness.
We propose several scenarios to showcase the broad application potential of the proposed gear-based metamaterials in Supplementary Figs. 1921 and Supplementary Table 2. For robots, a tunable-stiffness leg/actuator can offer high stiffness to stably support a heavy load while walking and a low stiffness for shock protection while jumping or running4,39 (Supplementary Figs. 19 and 20). A similar tunable-stiffness isolator is desired in the aero-engine pylon system to maintain the best performance and efficiency at different flight stages (Supplementary Fig. 21). Moreover, the fast-response gear-based metamaterial may give rise to a sensitive variable-stiffness skin, which has been attracting wide attention40. Furthermore, resonators with tunable stiffness are critical components in programmable metamaterials for wave manipulation41,42. Therefore, gear-based programmable metamaterials can aid in the realization of extensive intelligent machines. In contrast with conventional methods, the programmability enabled by a gear-based metamaterial does not require large deformation and heavy controlling systems, such as hydraulic/pneumatic or magnetic systems, and thus benefits the miniaturization and integration of machines and can even be used in harsh environments such as outer space.
## Conclusions
We show that gear-based mechanical metamaterials provide in situ tunability while preserving stability, strength and high load-bearing capacity. The programmability is robust and easily implementable. Gear clusters provide a vast design space that permits customizable performance of the metamaterials. Besides the demonstarted Young’s modulus, shape morphing and shock protection, the tunability can be extended to other elastic properties like shear modulus, Poisson’s ratio, strength, deformation modes and even damping coefficient (Supplementary Fig. 21). One can also envision 3D metamaterials by using bevel gears, assembling planar gears into hierarchical configurations as in Fig. 3 or synthesizing different types of gear (Supplementary Fig. 22). Integrated manufacturing bridges these tunable properties to produce robust multipurpose devices43. With the example of micro metamaterials, further miniaturization and an extension of gear-based metamaterials are possible with high-resolution and large-scale 3D printing.
In conclusion, this work proposes and demonstrates an unconventional design paradigm for programmable dynamic metamaterials via the mutable-yet-strong coupling and built-in variability of gears. We establish the general concept, conceive prototypes, conduct mechanical analyses, demonstrate the flexible tunability and integrated manufacturing at both the macro and micro scales and showcase the broad potential applications. The proposed design paradigm broadens the horizon for designing fully programmable materials, thus offering an impetus to their exploration for practical applications.
## Methods
### Integrated manufacturing
The printer used for the projection micro-stereolithography micro metamaterial fabrication is a BMF NanoArch S130, with a precision of about 5 µm. The material used in microscale 3D printing is a photosensitive resin with a Young’s modulus of about 3.5 GPa. The manufacturing process for the integrated micro metamaterial sample consisting of 5 × 6 Taiji gears follows three steps. First, the assembled gears are printed on a baseplate; those gears are adhered to the plate. Second, the sample is wrapped in a box to constrain the motion of the gears (Supplementary Fig. 9a). Last, everything including the box is removed from the plate. The box with a frame helps maintain the relative angle of the assembly in the removal process.
For the metamaterial based on planetary gears, the layer of the planetary gears is printed first (Supplementary Fig. 12a and Supplementary Video 3). Except for the preserved clearance, the connection shaft between the transmission gear and the sun gear is conical at both the macro and micro scales (Supplementary Fig. 10), which ensures that every printing part, especially the teeth of the transmission gears, is tightly attached on the formed structure. Otherwise, the teeth could move and then fuse together during the printing. At the macro scale, the integrated model is printed with two photosensitive resins using polymer injection with the printer Stratasys Objet260, with a precision of about 50 µm. The stiff model material is wrapped in the soft, soluble support material. The metamaterial acquires the targeted tunability after removing the support material. At the microscale, the material is immersed in the fluid resin during the printing. No support material/structure is required for this model owing to the conical shaft and high precision of projection micro-stereolithography. The sample shown in Fig. 3d is printed with a resin (polymer) with a Young’s modulus of 2.5 GPa.
In integrated manufacturing, the clearance reserved between the surfaces of meshing teeth in the assembled digital model depends on the precision of the printer, the structure and the materials. The minimal clearance Δ should be higher than the printer’s precision p (manufacturing errors) but much smaller than the tooth height (ht = 2.25m for standard gears), where the gear module m denotes the ratio between the gear diameter D and the number of teeth z, m = D/z (see Supplementary Text). Here, the minimal clearance between meshing teeth in all macro specimens printed with Objet260 is 86 µm. The minimal clearance for the micro metamaterial consisting of Taiji gears is set to 32 µm, and that for the micro planetary gear-based metamaterial is 21 µm. These clearances are sufficient to alleviate the manufacturing uncertainties to keep the meshing teeth separated but reliably engaged. Based on our 3D printers and tests, we suggest Δ > 1.5p and Δ ≤ ht/10 = 0.225m. This requires p < 0.15m, which helps us determine the required precision scale with a specified gear size.
### Actuation
As shown in Supplementary Video 2, we prepare a microscale sample consisting of 5 × 5 Taiji gears to show its actuation process. They are embedded into a box, and those gears connect to the frames through micro shafts. The sample is synchronously driven by four d.c. brushless motors (8 mm diameter) connected to the 1 × 1st, 1 × 4th, 4 × 1st and 4 × 4th gears. Here n × m denotes the position at the nth row and mth column in the array. As shown in Supplementary Video 4, the macro metamaterial in Fig. 3d is synchronously actuated by four-step motors whose diameter is 20 mm. These motors are synchronously controlled by an electronic controller. The revolving speed of the step motor depends on the impulse frequency generated by the controller. Similarly, the micro sample in Fig. 3e is put in a box and actuated by five micro step motors whose diameter is 5 mm. The controller is identical to the one used for the macro sample.
### FEA
FEA simulations are carried out with the commercial software ANSYS. We compare the accuracy of different finite element models, including two-dimensional (2D), 3D, linear and nonlinear models. The plane stress state is considered in the 2D model. In the linear models, the meshing points of gears are bonded by fixing together the two surfaces in contact, resulting in a linear stress–strain relationship. In the nonlinear models, the size of the contact area on the tooth surface at the meshing points depends on the load, and there is a relative sliding between the contact surfaces. The sliding induces frictional damping if the coefficient of friction is non-zero. We also use a simplified model by removing all teeth, where the contact between two gears becomes that between two cylinders. In principle, the 3D nonlinear model should be the most realistic representation of the experimental set-up. Supplementary Fig. 4 demonstrates that the 2D nonlinear model is in excellent agreement with the 3D nonlinear model. The two linear models produce a large discrepancy with the nonlinear ones, although they still can capture the general variation trend. The simplified model approximately presents the standard results. To enhance the simulation efficiency, we use the 2D nonlinear models in most cases. The 3D model is adopted only when considering the frictional contact.
Our metamaterials embrace a periodic architecture. To evaluate the homogenized elastic and shear moduli, ideal periodic boundary conditions are applied on the unit cell in the FEA. Boundary conditions depend on the deformation mode of the unit cell. The homogenized strain vector is ε = (εx, εy, γ). These strains are realized by enforcing the displacement fields (u, v) in the plane stress state.
As explained in Supplementary Fig. 14, two types of boundary condition are considered when calculating the shear modulus in the shear interlock state. To show the shear state of a finite n × n gear lattice, we fix the lower row/column of the gears and apply a displacement field to the upper row/column of the gears. As a second method, periodic boundary conditions are applied on a metacell to calculate the shear modulus. These periodic boundary conditions present the shearing state ε = (0, 0, γ). In both cases, the strain energy density W = 2/2 is extracted to evaluate the shear modulus G. For the n × n finite structure without periodic boundary conditions, although the equation of the generalized shear stiffness G′ = Kshear/B is the same as the formula for shear modulus G = τ/γ, the value of G′ may not equal the real shear modulus G (Supplementary Fig. 15) due to the free edge effects in finite structures (Supplementary Figs. 13a and 14a).
For the metamaterial based on a planetary gear system, the load is applied on the four blocks of the ring. For a metacell in FEA, we specify the uniaxial deformation v = εyay and make εx free for solving Ey.
### Equivalent method
For the metamaterial based on Taiji gears, the deformation mode for meshing gears can be represented by the overall stiffness of a pair of meshing elastic arms Kp = 1/(1/karm1 + 1/karm2 + 1/ktooth) (Supplementary Fig. 3 for their definitions). The stiffnesses of the two arms karm1 and karm2 are independent of the compressive deformation. As shown in Supplementary Fig. 5, the meshing of a pair of teeth features a line of contact on their surfaces. With compression, a small contact area is generated near the line where sliding occurs during the process. Therefore, the contact stiffness of the teeth ktooth depends on the contact pressure on the involute teeth. A high pressure leads to significant contact nonlinearity and results in a dependence of Kp on the displacement/load. By contrast, deformation mainly occurs in the elastic arm rather than the teeth if karmktooth, and Kp is constant in this case. The homogenized Young’s modulus in the y direction of the metamaterial is Ey = Kp/B + Ef. The equivalent methods for shear modulus are explained with Supplementary Fig. 14.
For the metamaterial consisting of a periodic planetary gear cluster, the Young’s modulus depends on the deformation of the ring. The influences of contact nonlinearity between teeth on Ey are the same as described above.
### Geometrical interlock
In a meshing pair, the rotation directions of the driving and driven gears are opposite. In a group of gears, if every meshing is viewed as a connection line, n gears form a closed polygon as shown in Fig. 4f. If n is odd, spin rotation is incompatible, leading to the locking among the gears. This meshing state is referred to as geometrical interlock.
### Mechanical tests for Young’s modulus
When measuring the Young’s modulus Ey in the metamaterial based on Taiji gears, a compressive load Fy is applied and released from the top of the prototype in Fig. 2a. We control the strain ε for different θ to overcome clearance nonlinearity while avoiding plastic deformation. The rotation angle θ is manually controlled. Similar cyclic loading–unloading tests are performed for the measurement of the shear modulus.
The experimental setting for the test on the metamaterial based on planetary gear systems is shown in Supplementary Fig. 9. When measuring the compressive modulus, a compressive load is applied on the top and the bottom blocks on the rings; when measuring the tensile modulus, we fix the tails on the sample to a pair of clamps and apply tensile loads through the tails.
As shown in Figs. 2e and 3g, the cyclic loading–unloading process features high repeatability, thus testifying to the experimental accuracy. Moduli Ey and G are both calculated as the slope around the maximum ε. The initial cycle is excluded when fitting Ey and G. The choice of the strain interval for the slope calculation affects the final modulus value. The error bars and the average values are evaluated by choosing different intervals along the curve.
### Mechanical tests for shear stiffness
For the metamaterial based on Taiji gears, a sample consisting of 3 × 3 gears and steel frames is manufactured for the measurement of the shear stiffness in the shear interlock state, as shown in Supplementary Fig. 15. A fixture apparatus is fabricated to obtain the shearing state. For the shape-morphing metamaterial, the sample is put in two right-angle grooves, and the load from the testing machine directly transfers to the sample.
## Data availability
The main data and models supporting the findings of this study are available within the paper and Supplementary Information. Further information is available from the corresponding authors upon reasonable request.
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## Acknowledgements
This research was funded by the National Natural Science Foundation of China (projects no. 12002371 and no. 11991032), the Hong Kong Scholars Program, the Fraunhofer Cluster of Excellence ‘Programmable Materials’ and the Excellence Cluster EXC 2082 ‘3D Matter Made to Order’ (3DMM2O) in Germany.
## Funding
Open access funding provided by Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V.
## Author information
Authors
### Contributions
X.F. and P.G. designed the study. X.F. conceived the idea and performed the experiments. X.F. and P.G. carried out the numerical simulations. L.C., D.Y. and H.Z. analysed the data. All authors interpreted the results. X.F., L.C., J.W. and P.G. wrote the manuscript with input from all authors. P.G. supervised the study.
### Corresponding authors
Correspondence to Xin Fang, Jihong Wen or Peter Gumbsch.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
## Peer review
### Peer review information
Nature Materials thanks Amir A. Zadpoor and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Supplementary information
### Supplementary Information
Supplementary Figs. 1–23 with explanatory text, Tables 1 and 2 and legends for Videos 1–5.
### Supplementary Video 1
Introduction of the metamaterial based on Taiji gears. This video shows the robust tunability and high stability of the proposed metamaterial under a large force.
### Supplementary Video 2
Actuation of the micro metamaterial consisting of Taiji gears.
### Supplementary Video 3
Integrated 3D printing process of the macro metamaterial consisting of a planetary gear system.
### Supplementary Video 4
Actuation of the macro metamaterial consisting of planetary gears.
### Supplementary Video 5
Shape-morphing metamaterial. This video shows the structure, manufacturing process and protected shape morphing of this metamaterial.
## Rights and permissions
Reprints and Permissions
Fang, X., Wen, J., Cheng, L. et al. Programmable gear-based mechanical metamaterials. Nat. Mater. (2022). https://doi.org/10.1038/s41563-022-01269-3
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• DOI: https://doi.org/10.1038/s41563-022-01269-3
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# matplotlib.axes.Axes.pcolorfast¶
Axes.pcolorfast(self, *args, alpha=None, norm=None, cmap=None, vmin=None, vmax=None, data=None, **kwargs)[source]
Create a pseudocolor plot with a non-regular rectangular grid.
Call signature:
ax.pcolorfast([X, Y], C, /, **kwargs)
This method is similar to pcolor and pcolormesh. It's designed to provide the fastest pcolor-type plotting with the Agg backend. To achieve this, it uses different algorithms internally depending on the complexity of the input grid (regular rectangular, non-regular rectangular or arbitrary quadrilateral).
Warning
This method is experimental. Compared to pcolor or pcolormesh it has some limitations:
• It supports only flat shading (no outlines)
• It lacks support for log scaling of the axes.
• It does not have a have a pyplot wrapper.
Parameters: Carray-like(M, N)The image data. Supported array shapes are: (M, N): an image with scalar data. The data is visualized using a colormap. (M, N, 3): an image with RGB values (0-1 float or 0-255 int). (M, N, 4): an image with RGBA values (0-1 float or 0-255 int), i.e. including transparency. The first two dimensions (M, N) define the rows and columns of the image. This parameter can only be passed positionally. X, Ytuple or array-like, default: (0, N), (0, M)X and Y are used to specify the coordinates of the quadrilaterals. There are different ways to do this: Use tuples X=(xmin, xmax) and Y=(ymin, ymax) to define a uniform rectangular grid. The tuples define the outer edges of the grid. All individual quadrilaterals will be of the same size. This is the fastest version. Use 1D arrays X, Y to specify a non-uniform rectangular grid. In this case X and Y have to be monotonic 1D arrays of length N+1 and M+1, specifying the x and y boundaries of the cells. The speed is intermediate. Note: The grid is checked, and if found to be uniform the fast version is used. Use 2D arrays X, Y if you need an arbitrary quadrilateral grid (i.e. if the quadrilaterals are not rectangular). In this case X and Y are 2D arrays with shape (M + 1, N + 1), specifying the x and y coordinates of the corners of the colored quadrilaterals. This is the most general, but the slowest to render. It may produce faster and more compact output using ps, pdf, and svg backends, however. These arguments can only be passed positionally. cmapstr or Colormap, default: rcParams["image.cmap"] (default: 'viridis')A Colormap instance or registered colormap name. The colormap maps the C values to colors. normNormalize, optionalThe Normalize instance scales the data values to the canonical colormap range [0, 1] for mapping to colors. By default, the data range is mapped to the colorbar range using linear scaling. vmin, vmaxfloat, default: NoneThe colorbar range. If None, suitable min/max values are automatically chosen by the Normalize instance (defaults to the respective min/max values of C in case of the default linear scaling). It is deprecated to use vmin/vmax when norm is given. alphafloat, default: NoneThe alpha blending value, between 0 (transparent) and 1 (opaque). snapbool, default: FalseWhether to snap the mesh to pixel boundaries. AxesImage or PcolorImage or QuadMeshThe return type depends on the type of grid: AxesImage for a regular rectangular grid. PcolorImage for a non-regular rectangular grid. QuadMesh for a non-rectangular grid. **kwargsSupported additional parameters depend on the type of grid. See return types of image for further description.
Notes
Note
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, every other argument can also be string s, which is interpreted as data[s] (unless this raises an exception).
Objects passed as data must support item access (data[s]) and membership test (s in data).
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每週問題 October 3, 2016
Let $A$ and $B$ be $n\times n$ Hermitian matrices. Suppose $A$ is invertible. Show that there exists a nonsingular matrix $P$ so that $P^\ast AP$ and $P^\ast BP$ are diagonal if and only if $A^{-1}B$ is diagonalizable and all its eigenvalues are real.
$P^\ast AP=D_1$$P^\ast BP=D_2$，其中 $D_1$$D_2$ 為對角矩陣。因為 $P^\ast AP$$P^\ast BP$ 都是 Hermitian，$D_1$$D_2$ 是實對角矩陣。寫出
$A^{-1}B=PD_1^{-1}P^\ast (P^\ast)^{-1}D_2P^{-1}=PD_1^{-1}D_2P^{-1}$
$D=\begin{bmatrix} \lambda_1I_{n_1}&&\\ &\ddots&\\ &&\lambda_kI_{n_k}\end{bmatrix}$
$\lambda_1,\ldots,\lambda_k$ 為兩兩互異的實數。因此，$BS=ASD$，即有 $S^\ast BS=(S^\ast AS)D$，其中 $S^\ast B S$$S^\ast AS$ 是 Hermitian。使用分塊矩陣表達，$S^\ast AS=[\tilde{A}_{ij}]$$S^\ast BS=[\tilde{B}_{ij}]$，這裡 $\tilde{A}_{ij}$$\tilde{B}_{ij}$$n_i\times n_j$ 階分塊，$i,j=1,\ldots,k$。使用 $\tilde{A}_{ij}=\tilde{A}_{ji}^\ast$$\tilde{B}_{ij}=\tilde{B}_{ji}^\ast$，且 $\tilde{B}_{ij}=\lambda_j\tilde{A}_{ij}$$i\neq j$，推得
$\lambda_j\tilde{A}_{ij}=\tilde{B}_{ji}^\ast=\overline{\lambda_i}\tilde{A}_{ji}^\ast=\lambda_i\tilde{A}_{ij}$
$S^\ast AS=\begin{bmatrix} \tilde{A}_{11}&&\\ &\ddots&\\ &&\tilde{A}_{kk}\end{bmatrix},~~S^\ast BS=\begin{bmatrix} \lambda_1\tilde{A}_{11}&&\\ &\ddots&\\ &&\lambda_k\tilde{A}_{kk}\end{bmatrix}$
$U=\begin{bmatrix} U_1&&\\ &\ddots&\\ &&U_k\end{bmatrix}$
\begin{aligned} P^\ast AP&=U^\ast S^\ast ASU=\begin{bmatrix} D_1&&\\ &\ddots&\\ &&D_k\end{bmatrix}\\ P^\ast BP&=U^\ast S^\ast BSU=\begin{bmatrix} \lambda_1D_1&&\\ &\ddots&\\ &&\lambda_kD_k\end{bmatrix}.\end{aligned}
This entry was posted in pow 二次型, 每週問題 and tagged , . Bookmark the permalink.
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# §27.7 Lambert Series as Generating Functions
Lambert series have the form
27.7.1 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.$ Symbols: $n$: positive integer and $x$: real number Referenced by: §27.7 Permalink: http://dlmf.nist.gov/27.7.E1 Encodings: TeX, pMML, png
If $|x|<1$, then the quotient $x^{n}/(1-x^{n})$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
27.7.2 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \divides n}f(d)x^{n}.$ Symbols: $d$: positive integer, $n$: positive integer and $x$: real number Referenced by: §27.7 Permalink: http://dlmf.nist.gov/27.7.E2 Encodings: TeX, pMML, png
Again with $|x|<1$, special cases of (27.7.2) include:
27.7.3 $\displaystyle\sum_{n=1}^{\infty}\mathop{\mu\/}\nolimits\!\left(n\right)\frac{x% ^{n}}{1-x^{n}}$ $\displaystyle=x,$ Symbols: $\mathop{\mu\/}\nolimits\!\left(n\right)$: Möbius function, $n$: positive integer and $x$: real number A&S Ref: 24.3.1 I.B Permalink: http://dlmf.nist.gov/27.7.E3 Encodings: TeX, pMML, png 27.7.4 $\displaystyle\sum_{n=1}^{\infty}\mathop{\phi\/}\nolimits\!\left(n\right)\frac{% x^{n}}{1-x^{n}}$ $\displaystyle=\frac{x}{(1-x)^{2}},$ 27.7.5 $\displaystyle\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}$ $\displaystyle=\sum_{n=1}^{\infty}\mathop{\sigma_{\alpha}\/}\nolimits\!\left(n% \right)x^{n},$ 27.7.6 $\displaystyle\sum_{n=1}^{\infty}\mathop{\lambda\/}\nolimits\!\left(n\right)% \frac{x^{n}}{1-x^{n}}$ $\displaystyle=\sum_{n=1}^{\infty}x^{n^{2}}.$
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# Compound path¶
Make a compound path -- in this case two simple polygons, a rectangle and a triangle. Use CLOSEPOLY and MOVETO for the different parts of the compound path
from matplotlib.path import Path
from matplotlib.patches import PathPatch
import matplotlib.pyplot as plt
vertices = []
codes = []
codes = [Path.MOVETO] + [Path.LINETO]*3 + [Path.CLOSEPOLY]
vertices = [(1, 1), (1, 2), (2, 2), (2, 1), (0, 0)]
codes += [Path.MOVETO] + [Path.LINETO]*2 + [Path.CLOSEPOLY]
vertices += [(4, 4), (5, 5), (5, 4), (0, 0)]
path = Path(vertices, codes)
pathpatch = PathPatch(path, facecolor='None', edgecolor='green')
fig, ax = plt.subplots()
ax.set_title('A compound path')
ax.autoscale_view()
plt.show()
## References¶
The use of the following functions, methods, classes and modules is shown in this example:
import matplotlib
matplotlib.path
matplotlib.path.Path
matplotlib.patches
matplotlib.patches.PathPatch
<function _AxesBase.autoscale_view at 0x7fba54b31c10>
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Q
# Solve it, - Chemical Bonding and molecular structure - NEET
The hybridization involved in complex $\left [ Ni \left ( CN \right )_{4} \right ]^{2-}$is.(At.No.Ni=28)
• Option 1)
$dsp^{2}$
• Option 2)
$sp^{3}$
• Option 3)
$d^{2}sp^{2}$
• Option 4)
$d^{2}sp^{3}$
106 Views
As we learnt in
Hybridisation -
The process of mixing of atomic orbitals belonging to the same atoms of slightly different energies so that a redistribution of energy takes place between them resulting in the formation of new set of orbital of equivalent energies and shape is called hybridisation.
- wherein
The new orbitals thus formed are called hybrid orbitals
In $\left [ Ni(CN) _{4}\right ]^{2-}, N\:is\:in +2\ O.S.$
$Ni^{2+}:\left [ A_{r} \right ]3d^{8}48^0$
or we could write
Since, CN- is a strong field ligand, so the electrons will pair up and hybridisation will be dsp2
Option 1)
$dsp^{2}$
This solution is correct
Option 2)
$sp^{3}$
This solution is incorrect
Option 3)
$d^{2}sp^{2}$
This solution is incorrect
Option 4)
$d^{2}sp^{3}$
This solution is incorrect
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# Difference between revisions of "LOADFONT"
The _LOADFONT function returns a font handle for a TrueType font (.TTF).
handle& = _LOADFONT (ttf_filename$, height[, "BOLD|, ITALIC|, UNDERLINE|, DONTBLEND|, MONOSPACE"]) ## Description • font_handle is the handle you want to use to represent the font. A return of -1 indicates a font loading failure. • TTF_filename$ is the filename of truetype fonts only. Can include the path to the font file.
• Windows users should find TTF(True Type) font files in the C:\WINDOWS\FONTS folder.
• Height is the height of the font. Font heights can be found using _FONTHEIGHT.
• Style parameter(s) used, if any, are literal(in quotes) or variable text parameters. Monospace has limited font selections.
• You can pass different font styles using different predefined STRING variable lists. You can include an empty style string!
• Font handles with values greater than 0 that are no longer needed should be freed using _FREEFONT. Font handle values of -1 (load failure) do not need to be freed! An error will occur if you try to free invalid handles!
• Check that the font values are greater than 0 before using them or illegal function errors may occur!
## Examples
Example 1: You need to know that if you are in a text mode (such as SCREEN 0 - the default) then you will only be able to use mono-spaced (fixed width) fonts.
font$= "C:\WINDOWS\Fonts\cour.ttf" 'TTF file in Windows style$ = "monospace, italic, bold" 'font style f& =_LOADFONT(font$, 30, style$) _FONT f& PRINT "Hello!"
Hello!
Note: 30 means each row of text (including vertical spacing) will be exactly 30 pixels high. This may make some program screens larger. If you don't want a style listed just use style$= "" if using a STRING variable for different calls. Example 2: In a 32-bit graphics mode you can alpha blend onto the background: i& =_NEWIMAGE(800,600,32) SCREEN i& COLOR &HC0FFFF00,&H200000FF f& =_LOADFONT("C:\Windows\Fonts\times.ttf", 25) 'normal style PRINT "Hello!" Hello! Note: You can load a fixed width font file without using the "monospace" option and it will be treated as variable width. This can be useful because LOCATE treats the horizontal position as an offset in pixels for variable width fonts. QB64 Font SUBs and Functions: SUB _FREEFONT font_handle SUB _FONT font_handle[,image_handle] FUNC _FONT (function) font_handle = _FONT[(image_handle)] SUB _PRINTSTRING (x,y), text_to_print$[,image_handle]
FUNC _PRINTWIDTH width_in_pixels = _PRINTWIDTH(text_to_print\$[,image_handle])
SUB _PRINTMODE _FILLBACKGROUND/_KEEPBACKGROUND/_ONLYBACKGROUND[,image_handle]
FUNC _PRINTMODE (function) 1 keepbackground/2 onlybackground/3 fillbackground =_PRINTMODE[(image_handle)]
FUNC _FONTHEIGHT character_height_in_pixels = _FONTHEIGHT[(font_handle)]
FUNC _FONTWIDTH character_width_in_pixels = _FONTWIDTH[(font_handle)]
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# N-Gram Model - Basics
The n-gram model is an approach in the language model to determine the most probable word sequence with several word sequences given. By means of a probability model, it is possible to compute the probability of each possible word sequence. The desired word sequence is the sequence with the greatest probability.
# 1 Motivation
Generally, the language model is divided into two steps. In the first step, the phoneme sequence generated by the acoustic model is used to determine very probable word sequences. Since there are several possible words for each time step, there are many possible word combinations, which might have been spoken. The task of the second step is to choose that sentence from all possible word sequences, which is most probable. For this, there are different approaches in the literature. One approach is based on the underlying grammatical structure, which is known as grammar model. The grammatical rules of a language are used to choose the word sequence which is grammatically correct. In case that there are two possible word sequences: "I ball" and "I play". Then the grammar model decides for the word sequence "I play", since it knows that it is more probable that a verb follows the word "I " than a noun. A more common approach is the so-called stochastic language model. In this case, the probability of each possible word sequence is computed. Then the word sequence with the greater probability was most probably spoken by the user of the speech recognition software. The probability of each word sequence is determined by means of n-grams. N-grams are a word sequence of n-words. The basic concept of the n-grams is described in this article.
# 2 The n-gram Model
Before providing more information on the n-gram model, we first want to consider how to calculate the probability of a sentence. Assume that the sentence $\mathbf{W}$ consists of N words $w_1,&space;w_2,&space;\cdots,&space;w_N$. Then the probability $p(\mathbf{W})$ can be determined as follows:
$\dpi{100}&space;p(\mathbf{W})&space;&=&space;p(w_1,&space;w_2,&space;\cdots,&space;w_N)&space;=&space;p(w_1)&space;\cdot&space;p(w_2|w_1)&space;\cdot&space;p(w_3|w_1,w_2)&space;\cdots&space;p(w_n|w_1,&space;w_2,&space;\cdots,&space;w_{N-1})&space;\\&space;&=\prod\limits_{i=1}^{N}&space;p(w_i|w_1,w_2,&space;\cdots,&space;w_{i-1})$
The probability $p(w_i|w_1,w_2,&space;\cdots,&space;w_{i-1})$ is very hard or even impossible to determine, since many word sequences do not appear often or are unique. It sounds reasonable to assume that the word $w_i$ only depends from its last (n-1)-words:
$p(w_i|w_1,w_2,&space;\cdots,&space;w_{i-1})&space;\approx&space;p(w_i|w_{i-1},&space;w_{i-2},&space;\cdots,&space;w_{i-n+1})$
This approach is called n-gram model in the literature. In earlier times n-grams models with order $n&space;=&space;2$ were very common, which are also called bigrams. Nowadays, trigrams ($n&space;=&space;3$) or even greater $n$ are used since the computation power has increased in the last years. A comparison between unigrams ($n&space;=&space;1$), bigrams and trigrams is shown under the following link. Note that only bigrams will be used in the further article, since the extension from the bigram model to the trigram model and n-gram model is straightforward.
Using the n-gram model, the probability of a sentence yields:
$p(\mathbf{W})&space;=&space;p(w_1,&space;w_2,&space;\cdots,&space;w_N)&space;=&space;\prod\limits_{i=1}^{N}&space;p(w_i|w_{i-1},&space;w_{i-2},&space;\cdots,&space;w_{i-n+1})$
For example, the probability of the sentence "Anne studies in Munich" can be calculated by
$p(\textit{Anne&space;studies&space;in&space;Munich})&space;=&space;p(Anne|)&space;\cdot&space;\\&space;\cdot&space;p(studies|Anne)&space;\cdot&space;P(in|studies)&space;\cdot&space;p(Munich|in)&space;\cdot&space;p(|Munich)$
Note, that the $$ token is used as a sentence start such that the probability $p(w_1|w_0)$ is defined. $$ symbolizes the end of a sentence, such that the probability of all word sequences is one.
# 3 Probabilities of n-grams:
First at all, a training set is needed to calculate the probability of the n-grams compared to a given set of word sequences. The training set usually consists of many million words. The probability of the n-grams can be estimated as follows:
$p(w_i|w_{i-1},w_{i-2},\cdots,w_{i-n+1})&space;=&space;\frac{C(w_{i-n+1},\cdots,w_{i-2},w_{i-1},&space;w_i)}{C(w_{i-n+1},\cdots,w_{i-2},w_{i-1})}$
where $C(w_{i-n+1},\cdots,w_{i-2},w_{i-1},&space;w_i)$ is the number of the word sequence $w_{i-n+1},\cdots,w_{i-2},w_{i-1},&space;w_i$ in the training set. Using the above definition the probability for a bigram yields:
$p(w_i|w_{i-1})&space;=&space;\frac{C(w_{i-1},&space;w_i)}{C(w_{i-1})}$
# 4 A small example
Let's consider a small example: We know that the sentences "Anne studies in Munich" and "Anton studies in Munich" are probably spoken by some speaker. Now it is of interest, which of them was most probably spoken. The procedure is as follows. First, we determine the probability of each sentence, $p(\textit{Anne&space;studies&space;in&space;Munich})$ and $p(\textit{Anton&space;studies&space;in&space;Munich})$ using the bigram model. Then it is decided, which sentence was spoken by the speaker by comparing both probabilities. The sentence with the greater probability was more probably spoken.
In this example the training set consists of the following three sentences: "Anne studies in Munich. Anton studies in Nuremberg. Anne studies electrical engineering". The training set is used to determine the probabilities of each bigram:
$p(Anne|)&space;=&space;\frac{C(,Anne)}{C()}&space;=&space;\frac{2}{3}$
$p(Anton|)&space;=&space;\frac{C(,Anton)}{C()}&space;=&space;\frac{1}{3}$
$p(studies|Anne)&space;=&space;\frac{C(Anne,studies)}{C(Anne)}&space;=&space;\frac{2}{2}$
$p(studies|Anton)&space;=&space;\frac{C(Anton,studies)}{C(Anton)}&space;=&space;\frac{1}{1}$
$p(in|studies)&space;=&space;\frac{C(studies,in)}{C(studies)}&space;=&space;\frac{2}{3}$
$p(Munich|in)&space;=&space;\frac{C(in,Munich)}{C(in)}&space;=&space;\frac{1}{2}$
$p(|Munich)&space;=&space;\frac{C(Munich,)}{C(Munich)}&space;=&space;\frac{1}{1}$
These bigram probabilities are used to calculate the probability of each of the both sentences:
$p(\textit{Anne&space;studies&space;in&space;Munich})&space;=&space;p(Anne|)&space;\cdot&space;p(studies|Anne)&space;\cdot&space;p(in|studies)&space;\cdot&space;p(Munich|in)&space;\cdot&space;p(|Munich)&space;=&space;\frac{2}{3}&space;\cdot&space;\frac{2}{2}&space;\cdot&space;\frac{2}{3}&space;\cdot&space;\frac{1}{2}&space;\cdot&space;\frac{1}{1}&space;\approx&space;0.222$
$p(\textit{Anton&space;studies&space;in&space;Munich})&space;=&space;p(Anton|)&space;\cdot&space;p(studies|Anton)&space;\cdot&space;p(in|studies)&space;\cdot&space;p(Munich|in)&space;\cdot&space;p(|Munich)&space;=&space;\frac{1}{3}&space;\cdot&space;\frac{1}{2}&space;\cdot&space;\frac{2}{3}&space;\cdot&space;\frac{1}{2}&space;\cdot&space;\frac{1}{1}&space;\approx&space;0.111$
Consequently, the speech recognition software assumes that the spoken sentence was "Anne studies in Munich" since the probability $p(\textit{Anne&space;studies&space;in&space;Munich})&space;\approx&space;0.222$ is greater than $p(\textit{Anton&space;studies&space;in&space;Munich})&space;\approx&space;0.111$.
# 5 Problem of the N-Gram Model
In this section, the example of the previous section is used to show the main problem of the n-gram model.
For example, it should be no problem to determine the probability $p(\textit{Anne&space;studies&space;electrical&space;engineering&space;in&space;Munich.})$ using the training set from above. Though it seems reasonable that the probability is unequal to zero, it is always zero since $p(in|engineering)=&space;0$ using the training set from above. This problem does not only exist in this small example. Even in very large training sets, it is not possible to determine the probability of each word combination since many word combinations do not exist due to the sparsity of the training set, especially if trigrams are used instead of bigrams.
There are many approaches which try to compensate this problem. Examples are the Backoff Approach, the Katz Smoothing Approach, the Kneser-Ney Smoothing, the Good Turing Smoothing, or the Laplace Smoothing.
# 6 References
[1] Huang, X & Deng, L. (2009) An Overview of Modern Speech Recognition, Chapter 15 of the book "Handbook of Natural Language Processing"
[2] Gales, M. (2008). The application of hidden Markov models in speech recognition. Foundations and Trends in Signal Processing
[3] Adami A. Automatic Speech Recognition: From the Beginning to the Portuguese Language
[4] Chen, S. & Goodman, J. (1998) Empirical Study of Smoothing Techniques for Language Modeling
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In the figure to the right, if point C is the center of the Go to page: 1, 2 Tags: Difficulty: 700-Level, Geometry, Source: Manhattan GMAT
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If x and y are both prime, is xy = 323?
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15 Jun 2015, 09:52
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Are at least 10% of the employees at ABC Corporation who are
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911
06 Dec 2013, 08:56
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If x is a member of the set {a, b, c, d}
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2384
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What is the median number of employees assigned per project Go to page: 1, 2 Tags: Difficulty: 700-Level, Statistics and Sets Problems, Source: Official Guide
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# Thread: Bivariate Normal Distribution: Joint distribution of functions of random variables
1. ## Bivariate Normal Distribution: Joint distribution of functions of random variables
Hi, I need your help with this problem: Suppose (X, Y)' follows a Bivariate Normal Distribution with parameters μ1 ,μ2, σ1^2, σ2^2, and ρ. Let U = X + Y and V = X - Y. Considering that X and Y are not independent random variables, how will I get the joint distribution of U and V. Thanks in advance!
Hey Mach.
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# Solve the following
Question:
Add vectors $A, B$ and $C$ each having magnitude of 100 unit and inclined to the $X$-axis at angles $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively.
Solution:
Vectors $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are oriented at $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively.
$|A|=|B|=|C|=100$ units
Let $A=A_{x} \mathbf{i}+A_{y} \mathbf{j}+A_{z} \mathbf{k}, B=B_{x} \mathbf{i}+B_{y} \mathbf{j}+B_{z} \mathbf{k}$, and $C=C_{x} \mathbf{i}+C_{y} \mathbf{j}+C_{z} \mathbf{k}$, and we can write that, $A_{x}=C_{x}=100 \cos \left(45^{\circ}\right)=100 / \sqrt{2}$, by considering their components
$B_{x}=-100 / \sqrt{2}$
Now $A y=100 \sin \left(45^{\circ}\right)=100 / \sqrt{2}$
By $=100 \sin \left(135^{\circ}\right)=100 / \sqrt{2}$
Similarly, Cy= $=-100 / \sqrt{2}$
Net $x$ component $=100 / \sqrt{2}+100 / \sqrt{2}-100 / \sqrt{2}=100 / \sqrt{2}$
Net $y$ component $=100 / \sqrt{2}+100 / \sqrt{2}-100 / \sqrt{2}=100 / \sqrt{2}$
$R^{2}=x^{2}+y^{2}=100^{2}$
$R=100$ and $\tan \phi=(100 / \sqrt{2}) /(100 / \sqrt{2})=1$, and $\phi=45^{\circ}$
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# Definition:Coreflexive Relation
## Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
### Definition 1
$\mathcal R$ is coreflexive if and only if:
$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$
### Definition 2
$\mathcal R$ is coreflexive if and only if:
$\mathcal R \subseteq \Delta_S$
where $\Delta_S$ is the diagonal relation.
## Linguistic Note
Coreflexive is pronounced co-reflexive, not core-flexive.
## Also see
• Results about reflexivity of relations can be found here.
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# Is it possible to determine shape and scale for a gamma distribution from a mean and confidence interval?
Having the 95% confidence interval and mean for a distribution and knowing nothing else (other than the data is skewed and will likely follow a gamma distribution) is there any way to determine the shape and scale of that gamma distribution? If not, what are the minimum data you would need to determine these?
• Do you mean the 95% confidence interval centered on the mean? – Jack M Aug 6 '18 at 11:00
• In general, if you have two unknowns, you need two independent equations to form a system to solve them. In your case, if you have functions of point estimatiors (the confidence limits) and equate to the realizations, you can solve the point estimates of the parameters out. – BGM Aug 6 '18 at 12:47
• @BGM can you expand on that a bit? Lets say I know the SD. Where would I go from there? – Munki Fisht Aug 6 '18 at 14:04
• @JackM Yes. The CI is centered on the mean. – Munki Fisht Aug 6 '18 at 14:05
If you know the mean is $\mu$ and the standard deviation is $\sigma$, then the shape parameter of a Gamma distribution is $\dfrac{\mu^2}{\sigma^2}$ and the scale parameter is $\dfrac{\sigma^2}{\mu}$, making the corresponding rate parameter $\dfrac{\mu}{\sigma^2}$
As an illustration of what is possible, suppose you knew that the mean is $40$ and you had an interval of $[30,50]$ representing about $2$ standard deviations either side of the mean
Then the standard deviation is about $\frac{50-40}{2}=5$, and the variance is therefore about $5^2=25$
For a Gamma distribution with shape parameter $k$ and scale parameter $\theta$, the mean would be $k\theta$ and the variance $k\theta^2$, suggesting with these numbers that $\theta \approx \frac{25}{40} = 0.625$ (equivalent to a rate of $1.6$) and $k \approx \frac{40^2}{25}=64$
As a check, we can look at the corresponding interval for these parameters in R
> pgamma(50,shape=64,scale=0.625) - pgamma(30,shape=64,scale=0.625)
[1] 0.9553145
> c(qgamma(0.025,shape=64,scale=0.625),qgamma(0.975,shape=64,scale=0.625))
[1] 30.80487 50.37773
which shows this approach is not exact, but is not that far away.
$k=59.3749$ and $\theta=0.66312$ would get you closer to the confidence interval with $2.5\%$ each side but at the cost (due to the asymmetry of the Gamma distribution) of a corresponding mean of $39.372$ rather than $40$
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## Stream: new members
### Topic: Product of continuous functions
#### Heather Macbeth (May 07 2020 at 05:00):
Hello, I am still finding my way around mathlib. Where would I find that the product of two continuous functions from R to R (or in whatever greater generality) is continuous?
#### Johan Commelin (May 07 2020 at 05:04):
I think it's called continuous_mul
#### Johan Commelin (May 07 2020 at 05:05):
src/topology/algebra/monoid.lean:lemma continuous.mul [topological_space β] {f : β → α} {g : β → α}
#### Johan Commelin (May 07 2020 at 05:05):
@Heather Macbeth It has a . in the name now...
#### Johan Commelin (May 07 2020 at 05:06):
You would use it as hf.mul hg, where hf : continuous f and hg : continuous g.
Thank you!
#### Heather Macbeth (May 07 2020 at 05:21):
Would this have sufficient generality to prove that for topological vector spaces E, F and continuous functions $f : \mathbb{R} \to E$ and $g : \mathbb{R} \to F$, the tensor product $f \otimes g : \mathbb{R} \to E \otimes F$ is continuous?
#### Johan Commelin (May 07 2020 at 05:23):
Nope, it's only for functions into a topological monoid
#### Johan Commelin (May 07 2020 at 05:23):
I'm afraid that the lemma you want is not yet there. Although @Sebastien Gouezel might have proved it
#### Johan Commelin (May 07 2020 at 05:24):
I don't know that part of the library too well
#### Johan Commelin (May 07 2020 at 05:24):
We have a whole bunch of stuff on continuous multilinear maps... but that's not exactly what you want
#### Heather Macbeth (May 07 2020 at 05:25):
I guess there is a monoid (in particular, graded algebra) consisting of arbitrarily iterated tensor products of E and F? Who knows if this is sensibly topological.
#### Johan Commelin (May 07 2020 at 05:41):
I think this is not a "new members" question. You might have more luck in the "maths" stream. Hopefully an expert can help you over there.
#### Johan Commelin (May 07 2020 at 05:41):
Some people don't read this stream, because there is lot's of traffic here.
#### Johan Commelin (May 07 2020 at 05:42):
@Heather Macbeth you might want to ping Yury Kudrashov and/or Sébastien Gouëzel and/or Patrick Massot, in your new question.
#### Heather Macbeth (May 07 2020 at 05:57):
Done! Thank you.
Last updated: May 17 2021 at 21:12 UTC
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### Home > APCALC > Chapter 11 > Lesson 11.1.3 > Problem11-38
11-38.
Multiple Choice: A racer with a $5$-foot head start runs with an acceleration of $a(t) = 6t \text{ ft} / \sec^2$. At $t = 4$ seconds, her velocity is $50$ ft/sec and she finishes the race in $9$ seconds. How long was the race? Homework Help ✎
1. $734$ ft
1. $737$ ft
1. $747$ ft
1. $750$ ft
1. $752$ ft
$v(t)=\int 6tdt=3t^2+C$
Use $v\left(4\right) = 50$ to solve for $C$.
$d(t)=\int(3t^2+2)dt=t^3+2t+C$
Use $d\left(0\right) = 5$ to solve for C.
Use your equation for $d\left(t\right)$ to evaluate $d\left(9\right)$.
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## likelihood-free inference by ratio estimation
Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on September 9, 2019 by xi'an
“This approach for posterior estimation with generative models mirrors the approach of Gutmann and Hyvärinen (2012) for the estimation of unnormalised models. The main difference is that here we classify between two simulated data sets while Gutmann and Hyvärinen (2012) classified between the observed data and simulated reference data.”
A 2018 arXiv posting by Owen Thomas et al. (including my colleague at Warwick, Rito Dutta, CoI warning!) about estimating the likelihood (and the posterior) when it is intractable. Likelihood-free but not ABC, since the ratio likelihood to marginal is estimated in a non- or semi-parametric (and biased) way. Following Geyer’s 1994 fabulous estimate of an unknown normalising constant via logistic regression, the current paper which I read in preparation for my discussion in the ABC optimal design in Salzburg uses probabilistic classification and an exponential family representation of the ratio. Opposing data from the density and data from the marginal, assuming both can be readily produced. The logistic regression minimizing the asymptotic classification error is the logistic transform of the log-ratio. For a finite (double) sample, this minimization thus leads to an empirical version of the ratio. Or to a smooth version if the log-ratio is represented as a convex combination of summary statistics, turning the approximation into an exponential family, which is a clever way to buckle the buckle towards ABC notions. And synthetic likelihood. Although with a difference in estimating the exponential family parameters β(θ) by minimizing the classification error, parameters that are indeed conditional on the parameter θ. Actually the paper introduces a further penalisation or regularisation term on those parameters β(θ), which could have been processed by Bayesian Lasso instead. This step is essentially dirving the selection of the summaries, except that it is for each value of the parameter θ, at the expense of a X-validation step. This is quite an original approach, as far as I can tell, but I wonder at the link with more standard density estimation methods, in particular in terms of the precision of the resulting estimate (and the speed of convergence with the sample size, if convergence there is).
## vector quantile regression
Posted in pictures, Statistics, University life with tags , , , , , , , on July 4, 2014 by xi'an
My Paris-Dauphine colleague Guillaume Carlier recently arXived a statistics paper entitled Vector quantile regression, co-written with Chernozhukov and Galichon. I was most curious to read the paper as Guillaume is primarily a mathematical analyst working on optimisation problems like optimal transport. And also because I find quantile regression difficult to fathom as a statistical problem. (As it happens, both his co-authors are from econometrics.) The results in the paper are (i) to show that a d-dimensional (Lebesgue) absolutely continuous random variable Y can always be represented as the deterministic transform Y=Q(U), where U is a d-dimensional [0,1] uniform (the paper expresses this transform as conditional on a set of regressors Z, but those essentially play no role) and Q is monotonous in the sense of being the gradient of a convex function,
$Q(u) = \nabla q(u)$ and $\{Q(u)-Q(v)\}^\text{T}(u-v)\ge 0;$
(ii) to deduce from this representation a unique notion of multivariate quantile function; and (iii) to consider the special case when the quantile function Q can be written as the linear
$\beta(U)^\text{T}Z$
where β(U) is a matrix. Hence leading to an estimation problem.
While unsurprising from a measure theoretic viewpoint, the representation theorem (i) is most interesting both for statistical and simulation reasons. Provided the function Q can be easily estimated and derived, respectively. The paper however does not provide a constructive tool for this derivation, besides indicating several characterisations as solutions of optimisation problems. From a statistical perspective, a non-parametric estimation of β(.) would have useful implications in multivariate regression, although the paper only considers the specific linear case above. Which solution is obtained by a discretisation of all variables and linear programming.
Posted in R, Statistics, University life with tags , , , , , , , , , on January 5, 2011 by xi'an
Yves Atchadé presented a very recent work on the fundamental issue of estimating the asymptotic variance estimation for adaptive MCMC algorithms, with an intriguing experimental observation that a non-converging bandwidth with rate 1/n was providing better coverage than the converging rate. (I always found the issue of estimating the asymptotic variance both a tough problem and an important item in convergence assessment.) Galin Jones showed new regeneration results for componentwise MCMC samplers, with applications to quantile estimation. The iid structure produced by the regeneration mechanism allows rather naturally to introduce an adaptive improvement in those algorithms, if regeneration occurs often enough. (From the days of my Stat’Sci’ paper on convergence assessment, I love regeneration techniques for both theoretical and methodological reasons, even though they are often difficult to efficiently implement in practice.) Matti Vihola summarised several of his recent papers on the stability and convergence of adaptive MCMC algorithms, pursuing the Finnish tradition of leadership in adaptive algorithms! One point I found particularly interesting was the possibility of separating ergodicity from the Law of Large Numbers, thus reducing the constraints imposed by the containment condition. In the afternoon, Dawn Woodard discussed the convergence rate of the Gibbs sampler used for genomic motif discovery by Liu, Lawrence and Neuwald (1995). Scott Schmidler concluded the workshop by a far-ranging talk distinguishing between exploration and exploitation in adaptive MCMC algorithms, ie mixing vs burning, with illustrations using the Wang-Landau algorithm.
Thus, as in the previous editions of Adap’ski, we have had a uniformly high quality of talks about the current research in the area of adaptive algorithms (and a wee further). This shows the field is very well active and expanding, aiming at reaching a wider audience by providing verifiable convergence conditions and semi-automated softwares (like Jeff Rosenthal’s amcmc R code we used in Introducing Monte Carlo Methods with R). Looking forward Adap’ski 4 (Adap’skiV?!), hopefully in Europe and why not in Chamonix?! Which could then lead us to call the next meeting Adap’skiX…
## Back to Philly
Posted in Statistics, Travel, University life with tags , , , , , , , on December 15, 2010 by xi'an
Today and tomorrow, I am attending a conference in Wharton in honour of Larry Brown for his 70th birthday. I met Larry in 1988 when visiting Cornell for the year—even using his office in the Math department while he was away on a sabbatical leave—and it really does not feel like that long ago, nor does it feel like Larry is any close to 70 as he looks essentially the same as 22 years ago! The conference is reflecting Larry’s broad range of research from decision-theory and nonparametrics to data analysis. I am thus very glad to celebrate Larry’s birthday with a whole crowd of old and more recent friends. (My talk on Rao-Blackwellisation will be quite similar to the seminar I gave in Stanford last summer [except that I have to talk twice as fast!])
## València 9 snapshot [3]
Posted in Statistics, University life with tags , , , , on June 7, 2010 by xi'an
Today was somehow a low-key day for me in terms of talks as I was preparing a climb in the Benidorm backcountry (thanks to the advice of Alicia Quiròs) and trying to copy routes from the (low oh so low!) debit wireless at the hotel. The session I attended in the morning was on Bayesian non-parametrics, with David Dunson giving a talk on non-parametric classification, a talk whose contents were so dense in information that it felt like three talks rather than one, especially when there was no paper to back it up! Katja Ickstadt modelled graphical dependence structures using non-parametrics but also mixtures of normals across different graph structures, an innovation I found interesting if difficult to interpret. Tom Loredo concluded the session with a broad and exciting picture of the statistical challenges found in spectral astronomy (even though I often struggle to make sense of the frequency data astronomers favour).
The evening talk by Ioanna Manolopoulou was a superbly rendered study on cell dynamics with incredible 3D animations of those cell systems, representing the Langevin diffusion on the force fields in those systems as evolving vector fields. And then I gave my poster on the Savage-Dickey paradox, hence missing all the other posters in this session… The main difficulty in presenting the result was not about the measure-theoretic difficulty, but rather in explaining the Savage-Dickey representation since this was unknown to most passerbys.
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Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Normal-gamma distribution ▷ Mean
Theorem: Let $x \in \mathbb{R}^n$ and $y > 0$ follow a normal-gamma distribution:
$\label{eq:ng} x,y \sim \mathrm{NG}(\mu, \Lambda, a, b) \; .$
Then, the expected value of $x$ and $y$ is
$\label{eq:ng-mean} \mathrm{E}[(x,y)] = \left[ \left( \mu, \frac{a}{b} \right) \right] \; .$
Proof: Consider the random vector
$\label{eq:rvec} \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \\ y \end{array} \right] \; .$
According to the expected value of a random vector, its expected value is
$\label{eq:mean-rvec} \mathrm{E}\left( \left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[ \begin{array}{c} \mathrm{E}(x_1) \\ \vdots \\ \mathrm{E}(x_n) \\ \mathrm{E}(y) \end{array} \right] = \left[ \begin{array}{c} \mathrm{E}(x) \\ \mathrm{E}(y) \end{array} \right] \; .$
When $x$ and $y$ are jointly normal-gamma distributed, then by definition $x$ follows a multivariate normal distribution conditional on $y$ and $y$ follows a univariate gamma distribution:
$\label{eq:ng-def} x,y \sim \mathrm{NG}(\mu, \Lambda, a, b) \quad \Leftrightarrow \quad x \vert y \sim \mathcal{N}(\mu, (y \Lambda)^{-1}) \quad \wedge \quad y \sim \mathrm{Gam}(a,b) \; .$
Thus, with the expected value of the multivariate normal distribution and the law of conditional probability, $\mathrm{E}(x)$ becomes
$\label{eq:mean-x} \begin{split} \mathrm{E}(x) &= \iint x \cdot p(x,y) \, \mathrm{d}x \, \mathrm{d}y \\ &= \iint x \cdot p(x|y) \cdot p(y) \, \mathrm{d}x \, \mathrm{d}y \\ &= \int p(y) \int x \cdot p(x|y) \, \mathrm{d}x \, \mathrm{d}y \\ &= \int p(y) \left\langle x \right\rangle_{\mathcal{N}(\mu, (y \Lambda)^{-1})} \, \mathrm{d}y \\ &= \int p(y) \cdot \mu \, \mathrm{d}y \\ &= \mu \int p(y) \, \mathrm{d}y \\ &= \mu \; , \end{split}$
and with the expected value of the gamma distribution, $\mathrm{E}(y)$ becomes
$\label{eq:mean-y} \begin{split} \mathrm{E}(y) &= \int y \cdot p(y) \, \mathrm{d}y \\ &= \left\langle y \right\rangle_{\mathrm{Gam}(a,b)} \\ &= \frac{a}{b} \; . \end{split}$
Thus, the expectation of the random vector in equations \eqref{eq:rvec} and \eqref{eq:mean-rvec} is
$\label{eq:ng-mean-qed} \mathrm{E}\left( \left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[ \begin{array}{c} \mu \\ a/b \end{array} \right] \; ,$
as indicated by equation \eqref{eq:ng-mean}.
Sources:
Metadata: ID: P237 | shortcut: ng-mean | author: JoramSoch | date: 2021-07-08, 09:40.
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Question Solved1 Answercalassfiy each signal Classify the following signals as power, energy, or neither! Find the power or the energy 1) $$x(t)=e^{-t} u(t)$$ 2) $$x(t)=5 \cos ^{2}(\pi t)$$ 3) $$x(t)=10 e^{0.1|-t|}$$ 4) $$x(t)=2 t u(t-1)$$
K0GVPC The Asker · Electrical Engineering
calassfiy each signal
Transcribed Image Text: Classify the following signals as power, energy, or neither! Find the power or the energy 1) $$x(t)=e^{-t} u(t)$$ 2) $$x(t)=5 \cos ^{2}(\pi t)$$ 3) $$x(t)=10 e^{0.1|-t|}$$ 4) $$x(t)=2 t u(t-1)$$
More
Transcribed Image Text: Classify the following signals as power, energy, or neither! Find the power or the energy 1) $$x(t)=e^{-t} u(t)$$ 2) $$x(t)=5 \cos ^{2}(\pi t)$$ 3) $$x(t)=10 e^{0.1|-t|}$$ 4) $$x(t)=2 t u(t-1)$$
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2. "2 times row one of $A^{-1}$ minus row two of $A^{-1}$ plus 3 times row 3 of $A^{-1}$" is simply $\begin{bmatrix}2 & -1 & 3\end{bmatrix}A^{-1}$. But that is the second row of A. Multiplying the second row of A by $A^{-1}$ gives the second row of the identity matrix which is $\begin{bmatrix}0 & 1 & 0\end{bmatrix}$.
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# A Copper Wire of Radius 0.1 Mm and Resistance Kω is Connected Across a Power Supply of 20 V. - Physics
Sum
A copper wire of radius 0.1 mm and resistance kΩ is connected across a power supply of 20 V. (a) How many electrons are transferred per second between the supply and the wire at one end? (b) Write down the current density in the wire.
#### Solution
Given:-
Radius of the wire, r = 0.1 mm = 10-4 m
Resistance, R = 1 kΩ = 103 Ω
Voltage across the ends of the wire, V = 20 V
(a) Let q be the charge transferred per second and n be the number of electrons transferred per second.
We know:-
$i = \frac{V}{R}$
$\Rightarrow i = \frac{20 V}{{10}^3 \Omega}$
$\Rightarrow i = 20 \times {10}^{- 3} = 2 \times {10}^{- 2} A$
$q = it$
$\Rightarrow q = 2 \times {10}^{- 2} \times 1$
$\Rightarrow q = 2 \times {10}^{- 2} C$
Also,
q = ne
$\Rightarrow n = \frac{q}{e} = \frac{2 \times {10}^{- 2}}{1 . 6 \times {10}^{- 19}}$
$\Rightarrow n = 1 . 25 \times {10}^{17}$
(b) Current density of a wire,
$j = \frac{i}{A}$
$\Rightarrow j = \frac{2 \times {10}^{- 2}}{3 . 14 \times {10}^{- 8}}$
$\Rightarrow j = 6 . 37 \times {10}^5 A/ m^2$
Concept: Current Density
Is there an error in this question or solution?
#### APPEARS IN
HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 10 Electric Current in Conductors
Q 11 | Page 198
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Today when I try to move a file using shutil.move() on my Windows machine, I encounter an error message:
PermissionError: [WinError 32] The process cannot access the file because it is being used by another process
In this post, I will write about what I have learned from this error.
# How to move files correctly on Windows
On Windows, before moving a file, you must close it. Or, you will see the above error message.
Suppose that we want to move images in a child directory images/ to another child directory small_image/ if the width of an image is below a threshold. On Windows system, the correct way to do it is like the following:
from glob import glob
from PIL import Image
all_images = glob("images/*.jpg")
for i, im_path in enumerate(all_images):
im = Image.open(im_path)
width = im.width
# we must close the image before moving it to another directory
im.close()
if width < 15:
shutil.move(im_path, 'small_images/')
On Linux, you are not required to close the file before moving it, i.e., you can move a file even if it is opened by another process.
# How to move a file if a file with the same name already exists in the destination directory?
On both Linux and Windows, when you try to move a file using shutil.move(src, dst) with dst set as a directory path, you will encounter the following error message if a file with the same name already exists under dst:
shutil.Error: Destination path ‘./test.txt’ already exists
The solution is to use the full file path in dst, i.e., a complete path to the new file. If a file with the same name exists under the destination folder, it will be silently replaced. If that behaviour is not what you want, you may consider renaming the file under the new directory.
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## Tuesday, August 23, 2016
### Recommended Economics Writing: Link Exchange
Sraffa's archival material is a gift to the science of economics. Scott Carter has been working in the Sraffa archives for a while, and this is an update about his efforts (among others) in getting the Sraffa archives published digitally.
The History of Economic Thought Website, I'm excited to say this has been brought back from the dead. The website's author has returned, so this is the official version.
The State of Macro Is Sad (Wonkish); Paul Krugman manifesting what looks like cognitive dissonance about the state of Macroeconomics.
How to Think about Own Rates of Interest, Version 2.0. When Hayek wrote Prices and Production, Sraffa wrote a scathing review about it. In the review, Sraffa used the concept of "own rates of interest" in a unique manner. Unfortunately it seems that Keynes had a very similar but distinct notion of "own rates of interest" in chapter 17 of his General Theory. I've recently become curious about the subject, so this blog-post has piqued my interest (even though I vehemently disagree with just about everything the author has written).
## Tuesday, February 23, 2016
### CEPA's "History of Economic Thought"
So, the History of Economic Thought site appears to be back and stable at its new location, but I'm not so sure it will last.
I've backed it up using `wget`. I will probably translate the site to Markdown, then put the result up on Github. As one might expect, the links to various external web-pages are out of date or completely broken...so just fixing the references might be a full time job.
Addendum (August 23, 2016): it seems the author of the History of Economic Thought has returned from...wherever he was, and has hosted the site elsewhere. It seems back up for good now.
## Saturday, May 3, 2014
### Logical Structure of Austrian Economics
1. Introduction. We will attempt to reconstruct the Austrian approach to economics using first-order logic.
We observe (in section 3) Austrian economists confuse deduction with introducing logically independent propositions. The general reasoning is "This doesn't directly contradict our foundational axiom, therefore it must logically follow from it" promoting the non-sequitur from fallacy to rule of inference.
Nevertheless, we bravely continue, and in section 4 discover it's impossible to deduce marginal utility from the action axiom. This spells disaster for any marginal analysis in the Austrian school.
# Definition and Explication
2.1. Axiom ("Action Axiom"). Murray Rothbard's The Logic of Action One: Method, Money, and the Austrian School (1997) describes the "action axiom" as:
Praxeology rests on the fundamental axiom that individual human beings act, that is, on the primordial fact that individuals engage in conscious actions toward chosen goals. This concept of action contrasts to purely reflexive, or knee-jerk, behavior, which is not directed toward goals. The praxeological method spins out by verbal deduction the logical implications of that primordial fact. In short, praxeological economics is the structure of logical implications of the fact that individuals act. This structure is built on the fundamental axiom of action, and has a few subsidiary axioms, such as that individuals vary and that human beings regard leisure as a valuable good. Any skeptic about deducing from such a simple base an entire system of economics, I refer to Mises's Human Action. Furthermore, since praxeology begins with a true axiom, A, all the propositions that can be deduced from this axiom must also be true. For if A implies B, and A is true, then B must also be true. (58--59)
This outlines the Austrian methodology fairly faithfully (I hope).
In order to make heads or tails out of it, lets first refine the meaning of "action" (since "humans act" is ambiguous at the moment).
2.2. Definition (Action). Ludwig Mises' Human Action itself defines "action" rather vaguely:
Human action is purposeful behavior. Or we may say: Action is will put into operation and transformed into an agency, is aiming at ends and goals, is the ego's meaningful response to stimuli and to the conditions of its environment, is a person's conscious adjustment to the state of the universe that determines his life. Such paraphrases may clarify the definition given and prevent possible misinterpretations. But the definition itself is adequate and does not need complement of commentary.
Personally, I find this unsatisfactory, but I will resign myself to accept the definition of "action" as "physical and psychological processes which render a specific state". (Even then, I'm nervous.)
If it makes much of a difference, Rothbard insists that All action in the real world, furthermore, must take place through time; all action takes place in some present and is directed toward the future (immediate or remote) attainment of an end (59). I thought this went without saying, but it is good to be explicit.
# Immediate "Deductions"
3.1. Corollary. Rothbard continues:
Let us consider some of the immediate implications of the action axiom. Action implies that the individual's behavior is purposive, in short, that it is directed toward goals. Furthermore, the fact of his action implies that he has consciously chosen certain means to reach his goals. (59)
Well, is "the ego's meaningful response to stimuli" necessarily "consciously chosen"? Wasn't that the point of Pavlov's dogs?
OK, lets overlook this and continue analyzing the consequences of the action axiom. (I mean, real and meaningful consequences, not tautological statements.)
3.2. Corollary. Rothbard tries to pull a fast one, insisting
Furthermore, that a man acts implies that he believes action will make a difference; in other words, that he will prefer the state of affairs resulting from action to that from no action. (59)
How does this logically follow at all? The actor's belief in his success seems irrelevant to the supposition the actor "acts" (in the Austrian sense). It seems Rothbard assumes "conscious actions toward chosen goals" implies that choosing a goal requires first belief in succeeding at accomplishing that goal. So without that prior belief in success, we would have no action?
So, if I had doubt or no belief whatsoever in my success to bring about a desired state, and I resign myself to this fate, am I still "acting"?
This is so stupid a point to make, because this has no bearing on anything at all in Austrian economics. But Rothbard insists on making it! So, I should say Rothbard will say two things: first, that I am not acting (otherwise he immediately contradicts himself); and second, I am acting, because I have belief in my success in my resignation.
My own personal belief is that this point should be disregarded, as it has no bearing on Austrian economics...nor does it illuminate the action axiom (or any other proposition "shown").
Fine, I'm willing to expand the definition of "action" to include the condition "The actor consciously believes in his or her own success".
3.3. Corollary (Uncertainty). Rothbard continues in his analysis, suggesting:
Action therefore implies that man does not have omniscient knowledge of the future; for if he had such knowledge, no action of his would make any difference. Hence, action implies that we live in a world of an uncertain, or not fully certain, future. Accordingly, we may amend our analysis of action to say that a man chooses to employ means according to a technological plan in the present because he expects to arrive at his goals at some future time. (59)
The proposition "Humans live in a world of uncertain future" is compatible with the Action axiom, but in no way does it logically follow. That is, there are no rules of inference which gets us from the Action axiom to this Uncertainty proposition. (Why? Because they're independent propositions!)
At the same time, there is no rule of inference denying this Uncertainty proposition. The two (the Action axiom and this uncertainty proposition) are compatible, like the Continuum hypothesis and ZFC set theory.
But the term "technological plan" here (introduced for the first time) Rothbard does not define.
3.4. Corollary (Scarcity). Rothbard's fifth conclusion:
The fact that people act necessarily implies that the means employed are scarce in relation to the desired ends; for, if all means were not scarce but superabundant, the ends would already have been attained, and there would be no need for action.
So if I want to read Mises' Human Action, that is only possible provided there is scarcity? This does not logically follow from anything stated thus far.
The proof Rothbard gives is a proof by contradiction, which is worse than useless.
Rothbard attempts clarifying this proposition, Stated another way, resources that are superabundant no longer function as means, because they are no longer objects of action (60). So the argument basically boils down to "Because the current state of the world is not the end-state desired by an action, there must be scarcity." This is a non-sequitur.
3.5. Observation. The "logic" Rothbard uses appears to be "Here's a proposition B. It's logically compatible with the action axiom. (But in no way does the action axiom logically imply B or its negation.) Therefore we deduce B must be true."
This is an invalid rule of inference. Why? Because you're not proving anything! You don't have a statement "If A, then B."
Instead you have a statement "We have A. And here's an independent proposition B. Therefore A implies B."
# Marginal Utility
4.1. Scarcity. Thorsten Polleit "deduces"
Human action implies employing means to the fulfillment of ends, and the axiom of human action implies that means are scarce. For if they were not scarce, means would not serve as objects of human action; and if means were not scarce, there would be no action — and that is unthinkable.
But nothing in the definition of "action" necessitates the existence for any "means to the fulfillment of ends". Having such "means" exist is not necessary for the definition of "action".
If we change the definition for "action" to "employing some 'means' to achieve some 'end'", then we have problems: we have introduced two undefined terms. We can handwave 'end' as "The state of the world after the action is done" to arbitrary precision (specify how long afterwards, etc. etc. etc.).
But the term "means" here is completely ambiguous. If we take it as "physical objects", then the axiom of action collapses on itself: the argument "Trying to refute the action axiom is a contradiction" becomes false, and all the preceding "deductions" in section 3 become false.
If we weaken the meaning for "means" as both physical objects and mental processes, then we still have problems: the claim for scarcity has a metaphysical statement that needs to be shown (namely, "Mental processes are scarce").
If we ignore everything except the proposition if means were not scarce, there would be no action, then...this claim still needs to be demonstrated. Why? Because it is the contrapositive of the claim "If there is action, then the means are scarce" which has not been shown.
So, in short, this nice-sounding couple of sentences is ambiguous.
4.2. Can Scarcity Be Deduced? The statement concerning scarcity's existence (or non-existence) is necessarily an a postereori claim, since it is an empirical statement.
If we buy into this Kantian taxonomy of propositions (a priori vs a postereori, analytic vs synthetic), then there is no way to deduce an a postereori proposition from an a priori one...otherwise it would be, by definition, a priori.
Consequently, by definition, it is impossible to "deduce" anything about scarcity's existence.
4.3. Scholium. What's the consequence of this? Any proposition in Austrian economics dependent on scarcity's existence has no logical grounding. So, basically, all of Austrian economics has no logical grounding.
# Conclusion
We have examined the action axiom and the definition of "action". We found it mildly ambiguous, but workable.
We have seen the "immediate consequences" are really just independent propositions that are not logically linked to the action axiom.
We tried to reconstruct the inference "action implies scarcity", and found this to be impossible (trying to deduce a postereori from the a priori is always impossible). Consequently, all Austrian economics depending on scarcity has no logical grounding.
Future research might include analyzing the Austrian business cycle, or other macroeconomic theories.
12 May 2014, 8:23AM (PST). It dawns on me the "Action axiom" isn't a priori --- it's based on the observation that people "act", and the observation attempts to refute it are "actions". No one really cares about this in Austrian circles nowadays, it seems, as no one seriously defends Mises peculiar Kantian inclinations.
I wonder about the "synthetic-ness" of the "Action axiom", too.
NB: the fact that the "Action axiom" is a proposition that's neither a priori nor synthetic doesn't seriously alter anything in Austrian economics. Fundamentally, it's an "axiom" in the modern mathematical sense rather than the Kantian sense: a specification we expect to hold while making "deductions" (in some vague sense).
12 May 2014, 8:48AM (PST). After thinking deeply about a priori synthetic statements (in the Kantian sense), it dawns on me that Kant used Aristotlean logic. Theoretically, Austrian economics cannot use first-order logic because of their Kantian underpinnings. I suppose it would be an interesting philosophical project to re-cast Austrian economics in rigorous first-order logic, and see what happens. A project, I hope, I will not commit myself to...
But it does mean the proposition "Man acts" is not a valid proposition for Aristotlean logic. Rothbard's proposition individuals engage in conscious actions toward chosen goals is invalid within Aristotlean logic. Mises' Human action is purposeful behavior. likewise is invalid. Hence it's invalid to consider it either a priori or a postereori, analytic or synthetic. Being charitable, perhaps a better form of the action axiom would be "All humans are 'actors'". But this only confirms the previous point: this is clearly not a priori.
## Wednesday, February 5, 2014
### The Definition of Value
So, what is a theory of value? In this post, these are just my notes defining "value" in some suitably abstract way, such that every paradigm has its own theory of value. In this way, we can meaningfully discuss theories of value from different paradigms on equal terms...or so I hope will be the case (eventually)!
1. Definition. In one sense, value is a mapping from commodities to numbers. That is to say, value is some mapping $\mathrm{value}:\mathbf{Commodities}\to \mathbb{R}$ where $\mathbf{Commodities}$ is the module of commodities over the integers (we interpret a negative quantity of commodities as a debt to be repaid), or perhaps a vector space over the rationals[1]. The basis is formed by the different "species" of commodities (e.g., iron, corn, wheat, tobacco, computers, cars, etc.).
2. Remark. Value has to be linear. Why? Because we expect, e.g., This is half the condition for linearity. We also expect or more generally, the value of any linear combination of commodities is precisely the sum of the constituents of that basket of goods. This would be sufficient to imply linearity. (End of Remark)
3. Remark (Theories of Value). The main contention between different paradigms in economics (notably the Neoclassical, Ricardian & Neo-Ricardian, and I think post-Keynesian paradigms) has to do with how we determine the $\mathrm{value}$ function.
I.5.1. Labour, therefore, is the real measure of the exchangeable value of all commodities.
I.5.2. The real price of every thing, what every thing really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What every thing is really worth to the man who has acquired it, and who wants to dispose of it or exchange it for something else, is the toil and trouble which it can save to himself, and which it can impose upon other people.
[...]
I.5.7. Labour alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price; money is their nominal price only.
David Ricardo refines this approach (Principles, Ch 1, Paragraphs 9–10), noting Smith's inconsistencies using corn as a standard of value at some times, then labor at other times:
“The real price of every thing,” says Adam Smith, “what every thing really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What every thing is really worth to the man who has acquired it, and who wants to dispose of it, or exchange it for something else, is the toil and trouble which it can save to himself, and which it can impose upon other people.” “Labour was the first price—the original purchase-money that was paid for all things.” Again, “in that early and rude state of society, which precedes both the accumulation of stock and the appropriation of land, the proportion between the quantities of labour necessary for acquiring different objects seems to be the only circumstance which can afford any rule for exchanging them for one another. If among a nation of hunters, for example, it usually cost twice the labour to kill a beaver which it does to kill a deer, one beaver should naturally exchange for, or be worth two deer. It is natural that what is usually the produce of two days’, or two hours’ labour, should be worth double of what is usually the produce of one day’s, or one hour’s labour.”*
That this is really the foundation of the exchangeable value of all things, excepting those which cannot be increased by human industry, is a doctrine of the utmost importance in political economy; for from no source do so many errors, and so much difference of opinion in that science proceed, as from the vague ideas which are attached to the word value.
I will refrain from reviewing the history of theories of value, as Dobb's Theories of Value and Distribution since Adam Smith does this in far better detail. But I will make note of a few other approaches.
The Neoclassical approach determines value from a microeconomic point of view using supply & demand curves.
I've discussed the Neo-Ricardian approach elsewhere (see, e.g., my notes on Sraffa's Production).
3.1. Questions to Self. David Ricardo notes how the price of a given commodity is expressing its value in terms of the money commodity. Does the Neoclassical approach do likewise?
In other words, is the concept of "value" an adequate abstraction such that each paradigm has their own theory of value? (Or, equivalently, no paradigm lacks a theory of value.)
4. Then from this mapping we induce an equivalence relation between commodities. That's the whole point of introducing value: to determine how much a given quantity of a given good will exchange for. We want to figure out $x$ in the equation It tells us how much 1 ton of steel commands in the wheat market.
4.1. We will say that this is the expression of the value for steel in terms of wheat. When we express all commodities in terms of some "standard unit" (say, wheat), then we have some money-commodity (for us: wheat, since we chose it as the standard unit).
The function of money (how it gets value, etc.) is a completely different subject (why, it's the theory of money!). Each paradigm likewise has its own theory of money.
5. Value is a function of time (or parametrized by time). This is the difficulty with measuring value. When we see the value of a commodity change in time, we are uncertain if: (1) the value of a given commodity is fluctuating, (2) everything else is fluctuating, or (3) the value of money is fluctuating. (Or, worse, some combination of the three!)
More explicitly, we have ${\mathrm{value}}_{t+dt}\left(x\phantom{\rule{thickmathspace}{0ex}}\mathrm{units}\phantom{\rule{thickmathspace}{0ex}}A\right)=c\phantom{\rule{thinmathspace}{0ex}}{\mathrm{value}}_{t}\left(x\phantom{\rule{thickmathspace}{0ex}}\mathrm{units}\phantom{\rule{thickmathspace}{0ex}}A\right)$ where $c>0$ is some real number. This describes a change in value for $x$ units of commodity $A$.
5.1. Remark. We don't measure this variation directly. We gauge it from how the value at time $t+\mathrm{d}t$ for $x$ units $A$ equates to other goods, ${y}^{\prime }$ units of $B$ at time $t+\mathrm{d}t$, etc. Then consider the value of $x$ units $A$ in terms of $y$ units of $B$ at time $t$. We suppose the ratio ${y}^{\prime }/y$ describes the change in value of $A$.
This should be viewed as problematical, since the values for every commodity fluctuates over time. So it may not be practical to consider ${y}^{\prime }/y$ as the defining factor for fluctuation.
### Endnotes
[1] Technically, one could view it as a category - in the sense of category theory. This gets really complicated really quick if we try to interpret a negative quantity of commodities, since negative numbers haven't been adequately (vertically) categorified yet.
## Monday, August 5, 2013
### Revising Portions of my Notes on Sraffa
I realize, looking back through my notes on Sraffa's Production, I begin to get a little sloppy with chapter 7 or chapter 8. [Just updated chapter 7's notes .]
Perhaps this is because I am having difficulty grasping various concepts, and I don't know which ones they are!
Consequently, I am going to pause, go back to these chapters, then revise them to a higher standard. I'll also revise chapter 9's notes, too, despite publishing them today!
I'm in the middle of moving, so this will take a while, but bear with me as I revise an otherwise incoherent summary...
### Notes on Sraffa's Production, Chapter 9
#### 66. Quantity of labour embodied in two commodities jointly produced by two processes
• So what results from single-product systems generalize over to joint-product systems?
• One rule we should study: when the rate of profits is zero, the relative value of each commodity is proportional to the quantity of labor which (directly and indirectly) has gone to producing them (§14).
• For joint-products, there is no obvious criterion apportioning the labor among individual products. It seems doubtful whether it makes any sense to speak of a "separate" quantity of labor as having gone to produce one among many jointly produced commodities.
• We get no help from the "Reduction" approach, where we sum the various dated labor inputs weighted by the product of rate of profits. (This is further discussed in §68)
• With the system of single-product industries, we had an alternative (if less intuitive) approach using the method of "Sub-systems" (Sraffa discusses this in his Appendix A).
It was possible to determine — for each of the commodities composing the net product — the share of aggregate labor which could be regarded as directly or indirectly entering its production.
• This method (with appropriate adaptation) extends to joint-products, so the conclusion about the quantity of labor "contained" in a commodity and its proportionality to value (at zero profits) can be generalized to joint products.
• Consider two commodities jointly produced through each of two processes in different proportions.
Instead of looking separately at the two processes and their products, lets consider the system as a whole and suppose quantities of both commodities are included in the net product for the system.
• We further assume the system is in a self-replacing state, and whenever the net product is changed...the self-replacing state is preserved (i.e., immediately restored through means of suitable adjustments in the proportions of the processes composing it).
• We also note: it is possible to change (within certain limits) the proportions in which two commodities are produced if we alter the relative sizes of the two processes producing them.
• If we wish to increase the quantity which a commodity enters the net product of the system (while leaving all other components unchanged), we normally must increase the total labor employed by society.
It's natural to conclude we must increase labor for producing the commodity in question. This may go directly (i.e., directly into the process in question) or indirectly (i.e., producing the means of production).
• The commodity added will (at the prices corresponding to zero rates of profits) be equal in value to the additional quantity of labor.
• This conclusion seems to hold for commodities jointly produced, as it holds for single-product systems.
• The conclusion appears to hold even when we change the quantities of the means of production, since any additional labor needed to produce the latter is included as indirect labor in the quantity producing the addition to the net product.
• Footnote. Since joint-products are present, the contraction for some processes might occur, and thus we might fall into the awkward "negative industries" scenario again...but even then, the adjustments noted include them!
This can be avoided, provided the initial increase for the commodity in question is supposed to be "sufficiently small", and the net product for the system is assumed to comprise at the start "sufficiently large quantities" of all products...so any necessary contraction may be absorbed by existing processes, without the need for any of them having to receive a negative coefficient.
#### 67. Quantity of labour embodied in two commodities jointly produced by only one process
• Similar reasoning holds for the case when two commodities ('a' and 'b') are jointly produced through only one process...but are used as means of production (in different relative quantities) through two processes, each produes singly the same commodity 'c'.
• So we have two processes of the form ${q}_{1,a}a+{q}_{1,b}b\to {q}_{1,c}c$ and ${q}_{2,a}a+{q}_{2,b}b\to {q}_{2,c}c$ where ${q}_{1,a}/{q}_{1,b}\ne {q}_{2,a}/{q}_{2,b}$, and none of the coefficients vanish.
• We can't change the proportions which 'a' and 'b' appear in the output of their production processes (i.e., the processes producing them). But we can (through altering the relative size of the two processes using them) vary the relative quantities in which they enter as means for producing a given quantity of 'c'.
We can vary the relative quantities of 'a' and 'b' this way, and this by itself alters the relative quantities in which they enter the net social product. (The relative quantities in which the two enter the gross product are fixed.)
• Remark. As a childish example, we could have $\begin{array}{r}a+2b\to c\\ 3a+b\to c\end{array}$ So, suppose we have for our toy example ${q}_{a}={q}_{b}$ (there is a one-to-one ratio between the quantity of 'a' and 'b' produced).
The relative quantities of 'a' and 'b' seems like a strange term to me. We could consider enlarging the first process and keep the second process constant: $\begin{array}{rl}f\left(\stackrel{⃗}{x}\right)+{L}_{a}& \to 5a\\ g\left(\stackrel{⃗}{y}\right)+{L}_{b}& \to 5b\\ 2\left(a+2b\right)& \to 2c\\ 3a+b& \to c\end{array}$ For simplicity, the production of 'a' and 'b' are blackbox functions which takes "some vector" of inputs. We have combined $5a+5b\to 3c$. The relative quantities of 'a' and 'b' are, literally, one-to-one. Observe the surplus is 3 c...and we had ${L}_{a}+{L}_{b}$ contribute.
But if we change how we produce things, say use only the first process, then we have $\begin{array}{rl}f\left(0.4\stackrel{⃗}{x}\right)+0.4{L}_{a}& \to 2a\\ g\left(0.8\stackrel{⃗}{y}\right)+0.8{L}_{b}& \to 4b\\ 2a+4b& \to 2c\end{array}$ and hence we have the surplus be 2c. The relative proportion which 'a' and 'b' enter production change; is this what Sraffa means? We varied the size of the processes producing 'a' and 'b', without deforming the processes (i.e., changing the proportions of the coefficients, just reduced the ratio to produce a lesser amount).
The amount of labor also changed from ${L}_{a}+{L}_{b}$ to $0.4{L}_{a}+0.8{L}_{b}$.
• It is thus possible (through an addition to total labor) to arrive at a new self-reproducing state, where a quantity for one of the two products (say 'a') is added to the net product, while all other components of the latter remain unchanged.
We can conclude the addition to labor is the quantity which directly and indirectly is required to produce the additional amount of 'a'.
#### 68. Reduction to dated quantities of labour not generally possible
• Sraffa claims there is no equivalent (in the case of joint-products) to the "alternative method", i.e., Reduction to a series of dated labor terms. Sraffa explains the "essence" of Reduction is that each commodity should be (a) produced separately, (b) by only one industry, and (c) the whole operation consists in tracing back the successive stages of a single-track production process.
• Remark. I am very suspicious of this claim, and I don't follow the reasoning given. After all, consider the system given as $\left(1+r\right)A\stackrel{⃗}{p}+w\stackrel{⃗}{L}=\stackrel{⃗}{p}$ where A is the input-output matrix, $\stackrel{⃗}{p}$ is the price-vector, w wage, $\stackrel{⃗}{L}$ the labor vector, and r the rate of profits. Then we have $\left(I-\left(1+r\right)A\right)\stackrel{⃗}{p}=w\stackrel{⃗}{L}$ where I is the identity matrix. This gives us $\stackrel{⃗}{p}=\left(I-\left(1+r\right)A\right)\cdot w\stackrel{⃗}{L}=\left(\sum _{0}^{\mathrm{\infty }}\left(1+r{\right)}^{n}{A}^{n}\right)w\stackrel{⃗}{L}$ Isn't this a Reduction-type equation?
If so, it could be suitably generalized in the straightforward way for a joint-product. Provided the joint-product system satisfies the conditions Sraffa gives (basically, the general linear algebraic conditions that a solution exists).
• Remark (Cont'd). Now, we are dealing with a slightly more general situation, specifically: $\left(1+r\right)A\stackrel{⃗}{p}+w\stackrel{⃗}{L}=B\stackrel{⃗}{p}$ where the matrix B is necessary for joint-products. Without loss of generality, we may assume it is an invertible matrix. Thus we re-write this system as $\left(1+r\right){B}^{-1}A\stackrel{⃗}{p}+w{B}^{-1}\stackrel{⃗}{L}=\stackrel{⃗}{p}$ or if we introduce new symbols to stress the similarity to the previous case: $\left(1+r\right)\stackrel{˜}{A}\stackrel{⃗}{p}+w{\stackrel{⃗}{L}}^{\prime }=\stackrel{⃗}{p}.$ We should observe this becomes the previous situation.
• Sraffa suggests we should have to give a negative coefficient to one of the two joint-production equations and a positive coefficient to the other, thus eliminating one of the products while retaining the other in isolation.
Some of the terms in the Reduction equation would represent negative quantities of labor, which Sraffa insists "no reasonable interpretation could be suggested."
• Sraffa insists the series would contain both positive and negative terms, so the "commodity residue" wouldn't necessarily be decreasing at successive stages of approximation. Instead, it might show steady or even widening fluctuations — the series might not converge!
• Sraffa will investigate this in §79 ("Different depreciation of similar instruments in different uses").
• Reduction could not be attempted if the products were jointly produced by a single process, or by two processes in the same proportions, since the apportioning of the value and of the quantities of labor between the two products would depend entirely on the way the products were used as means of production for other commodities.
#### 69. No certainty that all prices will remain positive as the wage varies
• Sraffa urges us to reconsider another proposition considered earlier: if the prices of all commodities are positive at any one value of the wage between 1 and 0, no price could become negative as a result of varying the wage within those limits (§39).
• Sraffa denies the possibility we could generalize this proposition to joint-product systems.
• Recall, the premise underpinning this proposition: the price of a commodity could only become negative if the price for some other commodity (one of its means of production) had become negative first — so no commodity could ever be the first to do so.
• But for joint-products, there is a way around and the price for one of them may become negative...provided the balance was restored by a rise in the price of its companion product sufficient to maintain the aggregate value of the two products above that of their means of production by the requisite margin.
#### 70. Negative quantities of labour
• Sraffa suggests his conclusion is "not in itself very startling". He interprets the situation quite simple. Sraffa notes in fact all prices are positive...but a change in the wages may create a situation which necessarily requires prices to become negative. Since this is unacceptable, those methods giving negative prices would be discarded in favor of those giving positive prices.
• When we consider this with the previous section (concerning the quantity of labor entering a commodity), the combined effect requires some explaining...
• What's involved is not merely something like "In the remote contingency of the rate of profits falling to zero, the price of such a commodity would (if other things remain equal) have to become negative"...but we conclude in the actual situation, with profits at the perfectly normal rate of (say) 6%, that particular commodity is in fact produced by a negative quantity of labor.
• Caution: We will work supposing 6% is the "normal rate of profits" throughout this section, so bear that in mind...
• Sraffa says "This looks at first as if it were a freak result of abstraction-mongering that can have no correspondence in reality." He has such a way with words, sometimes!
• If we apply it to the test employed for the general case in §66, where — under the supposed conditions — the quantity of such a commodity entering the system's net product is increased (the other components remaining constant), we shall find as a result the aggregate quantity of labor society employs has diminished.
• Nevertheless! Since the change in production occurs while the "ruling rate of profits" is 6%, and the system of prices is the one appropriate to that rate, Sraffa argues "nothing abnormal will be noticeable".
In effect the diminution in the expense for labor will be more than balanced by an increased charge for profits, the addition to net output will entail a positive addition to the cost of production.
• So, what happened? In order bring about the required change in the net product, one of the two joint-production processes must be expanded while the other contracted.
In the case under consideration, the expansion of the former employs (either directly or through "other processes as it carries in its train the ensure full replacement") a quantity of labor which is smaller...but means of production which at the prices appropriate to the given rate of profits are of greater value — and thus attracts a heavier charge for profits — than the contraction of the latter process "under a similar proviso".
• Sraffa concludes "It seems unnecessary to show in detail that what has been said in this section concerning negative quantities of labor can be extended (on the same lines as was done for positive quantities in §67) to the case in which two commodities are jointly produced by only one process, but are used as means of production by two distinct processes both producing a third commodity."
#### 71. Rate of fall of prices no longer limited by rate of fall of wages
• Sraffa has one further proposition about prices which needs reconsideration for the case of joint products.
• We have seen (§49) for single-product industries, when the wage falls in terms of the Standard commodity that no product can fall in price at a higher rate than does the wage.
• The premise underpinning this: were a product able to do so, it must be owing to one of its means of production falling in price at a still higher rate.
Since this could not apply to the product that fell at the highest rate of all, that product itself could not fall at a higher rate than wage.
• With one of a group of joint products, there is the alternative possibility the other commodities jointly produced with it should rise in price (or suffer only a "moderate" fall) with the fall of wage so as to make up — in the aggregate product of the industry — for any excessive fall of the first commodity's price.
To such a rise, there is no limit...and thus there is none to the rate at which one of the several joint products may fall in price.
• But as soon as it is admitted the price of one (out of two or more joint products) can fall at a higher rate than does the wage, it follows even a singly produced commodity can do so...provided it employs — as one of its means of production, and to a sufficient extent — the joint product so falling.
#### 72. Implication of this
• This possibility — price may fall faster than the wage — has some noteworthy consequences...
• First we have an exception to the rule "The fall of wage in any Standard involves a rise in the rate of profits."
• Suppose a 10% fall in the Standard wage entails (at a certain level) a larger proportionate fall — say 11% — in the price of 'a' as measured in the Standard product.
• This means labor has risen in value by about 1% relative to the commodity 'a'.
• Remark. I think the ratio would be $90/89\approx 1.01123595505$ or the rise of value of labor relative to 'a' is about 1.12%.
• If we were to express the wage in terms of commodity 'a', a fall in such a wage over the same range would involve a rise in the Standard wage and consequently a fall in the rate of profits.
• Moral. We can't speak of a rise or fall in the wage unless we specify the standard, for what is a rise in one standard may be a fall in another.
• For the same reasons, it becomes possible for the wage-line and price-line of a commodity 'a' to intersect more than once as the rate of profits varies
• Figure 5: Several intersections are possible in a system of multiple-product industries.
• As a result, to any one level of the wage in terms of commodity 'a', there may correspond several alternative rates of profits.
• In figure 5, the several points intersective the solid black curve — representing the price of 'a' — with the dashed wages curve represent equality in value between a unit of labor and a unit of commodity of 'a'...i.e., the same wage in terms of 'a'.
Of course, they represent different levels of wage in terms of the Standard commodity.
• On the other hand, as in the case of the single-products system, to any one level of the rate of profits there can only correspond one wage, whatever the standard in which the wage is expressed.
## Friday, August 2, 2013
### Recommended Economics Writing: Link Exchange
Stock Simulation in Clojure, a basic introduction to modeling using software. Fairly mainstream, but I work with clojure professionally, so there it is.
The meaning of short and long-term and the natural rate (Naked Keynesianism)
Brief Thoughts on the Real Bills Doctrine (Unlearning Economics)
Rate of Profits And Value Of Stock Independent Of Workers Saving (Robert Vienneau)
The Time Bernanke Got It Wrong (Floyd Norris)
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Assertion A: Enol form of acetone
Question:
Assertion A: Enol form of acetone
$\left[\mathrm{CH}_{3} \mathrm{COCH}_{3}\right]$ exists in $<0.1 \%$ quantity. However, the enol form of acetyl acetone
$\left[\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{OCCH}_{3}\right]$ exists in approximately $15 \%$ quantity.
Reason $\mathrm{R}$ : enol form of acetyl acetone is stabilized by intramolecular hydrogen bonding, which is not possible in enol form of acetone.
Choose the correct statement:
1. $A$ is false but $R$ is true
2. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
3. Both $A$ and $R$ are true but $R$ is not the correct explanation of $\mathrm{A}$
4. $\mathrm{A}$ is true but $\mathrm{R}$ is false
Correct Option: , 2
Solution:
enol from of acetone is very less $(<0.1 \%)$
Administrator
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# Testing hillclimbing and simulated annealing
November 09, 2019 in
In the previous two posts, I described hill climbing and simulated annealing as ways of breaking substitution ciphers where we can't make good initial guesses of the key. I said that simulated annealing, compared to hillclimbing, is more likely to find a good solution and is less likely to get stuck on some locally-good but globally-poor solution. In this post, I'll test the accuracy of those claims.
# Experimental design
The basic idea is simple: I'll take a random chunk of text from the Complete Works of Sherlock Holmes, create a random mapping of plaintext letters to ciphertext letters (a random key), and encipher the text with that key. I'll then see if hillclimbing or simulated annealing are able to recover the original text from the given ciphertext.
I'll do that many, many times to see how often each algorithm succeeds. As always, you can find the code for this on Github.
## Measuring success
One slight wrinkle in this experiment is determining what counts as "success". The cipher breaking systems evaluate their scores with a simple n-gram language model. However, there can be occasions where the best-scoring substitution isn't the right one. For instance, if a plaintext contains many uses of the word majesty and no or few zs, the algorithms will try to decipher that word as mazesty because mazes occurs more often in English than majes.
This means we have to judge success of of the decipherment by looking at the key it produces rather than the proposed plaintext. But how can we score the key?
The key is a mapping from plaintext letters of the alphabet to ciphertext letters of the alphabet. That gives us something like
Plaintext alphabet a b c d ...
Actual ciphertext alphabet g e p j ...
Proposed ciphertext alphabet g e l j ...
So we can score the proposed ciphertext alphabet by counting how many letters are in the same order as in the actual ciphertext alphabet. Conveniently, this is implemented as the the Kendall rank correlation measure (also known as Kendall's τ), and implemented as the function kendalltau in the scipy.stats library.
## The experiments
There are a few more things we can experiment with, in addition to the core experiment of hillclimbing vs simulated annealing.
While we can use Kendall's τ to tell whether we've found the right key, the codebreaking algorithm only knows about the n-gram score to determine the best key. That means we can see which n-gram score is better; I'll compare unigrams with trigrams.
In simulated annealing, the algorithm will sometimes choose a lower-scoring solution over its existing solution. How likely this is depends on both how much worse the alternative is, and the current temperature used by the algorithm. Higher temperatures make the algorithm more likely to choose a worse solution; this could lead the algorithm better to explore the range of possible cipher keys, or it could prevent the algorithm making any progress in the early part of its run.
Another variation in the algorithm is how the existing key is changed. We can generate a new key from the current one by swapping two letters. But which two letters to swap? One method is to pick two letters uniformly at random, regardless of where they are in the key. Another method is to assume that the current key is mostly correct, and that we should swap two nearby letters. In these experiments, I chose "nearby" by sampling from a Gaussian (normal, bell-shaped) distribution.
A final thing to look at is the initial key. The key is the complete mapping from each plaintext letter to each ciphertext letter. We could create this mapping entirely randomly, with all mappings equally likely. Or, we could attempt to give a head start to the search for the key, but mapping the most frequent plaintext letters to the most frequent ciphertext lettters.
This gives us five parameters to vary, and we can test combinations of all of them to see which works best.
• Hillclimbing vs simulated annealing
• Unigram vs trigram scoring
• High or low initial temperature (for simulated annealing)
• Uniform vs Gaussian swaps
• Random or guided initial alphabet
As we're trying to find out if simulated annealing is better than hillclimbing, I'll look at the other four parameters and compare simulated annealing and hillclimbing in each situation.
Both hillclimbing and simulated annealing are stochastic methods (they use randomness in the algorithm). That means that different runs are likely to produce different results. Therefore, I run the algorithm with ten workers in each situation, each working independently. The algorithm will return the best fitness of all the workers, but I show graphs of the scores of each worker.
I use the same ciphertext for all experiments.
## Results
### Unigram scoring
The first combination to look at will give us a baseline we can use to compare the performance of the other algorithm/parameter combinations. The simplest thing we can do is give the algorithms a random starting key, uniformly select swaps, and use unigram scoring for measuring fitness.
The results show that hillclimbing (on the left) quickly finds one solution and sticks with it: the fitness score doesn't deviate much. But looking at the τ score shows that it rejects slightly better keys as they yield lower fitness scores. The simulated annealing traces (on the right) show that the algorithm doesn't stabilise on a single solution until the end of the run, but the variation reduces over time; that's what we'd expect as the temperature drops from high to low.
But even though all runs of both algorithms eventually find similar solutions (same fitness score, very similar τ scores), the low τ score shows that this is a poor solution.
It seems that unigram scoring is not the way to go.
### Trigram scoring
What if we repeat the previous experiment, but use trigram scoring?
The first thing to note is that both algorithms do well. We can see that from the τ scores: both algorithms end with most or all of the runs ending at a τ score of 1.0, showing that they've found the correct key. However, every simulated annealing worker finds the correct key, while only about half the hillclimbing workers find it. This suggests that simulated annealing is more likely to find the best solution.
Another interesting feature is the time scale of finding the solution. Hillclimbing has essentially stabilised by 2500 iterations: only one of the workers progresses after this point. Simulated annealing, on the other hand, shows an increase in fitness throughout the run. But a look at the τ plot shows that it's only the last 8000 or so iterations where the correct key is being found. It seems likely that the annealing temperature before this is too high for any potential solution to survive.
### Guessing the key
Before I look at the temperature to use with simulated annealing, I'll do another comparison between hillclimbing and simulated annealing. Does the initial guess of the key make any difference?
The experiments above used a random initial key, but we could seed the algorithm with our best guess of the key, based on unigram letter frequencies. This is quick and easy to compute, and is likely to be somewhat close to the final solution.
If we compare these graphs to the ones above, we see that the initial fitness and τ scores are higher than with a random alphabet. With hillclimbing, these scores increase. But the simulated annealing traces show that the initial boost in fitness and τ is soon erased by the algorithm choosing worse solutions in order to explore more of the solution space.
However, it should come as no surprise that both algorithms find the correct solution.
### Limiting swaps
If we're giving the algorithm a mostly-correct initial key, we would expect that the best changes would be to swap nearby letters in the key, as these are more likely to be better; swaps between letters in very different parts of the key is likely to be harmful.
(This makes sense if we keep the mapping in order of highest to lowest frequency of letter, so rather than a mapping that looks like this:
Plaintext alphabet a b c d ...
Ciphertext alphabet q m g c ...
, with the plaintext alphabet in alphabetical order, we have a mapping that looks like this:
Plaintext alphabet e t o a ...
Ciphertext alphabet t f b q ...
, where the plaintext alphabet is in order of frequency of that letter in normal English.)
As you can see, it makes very little difference. The hillclimbing results are just about identical with the previous runs. With simulated annealing, the fitness score seems to be slightly higher between 1000 and 8000 iterations, but there's basically nothing in it.
### Simulated annealing temperature
The final parameter to vary is the initial temperature of the simulated annealing algorithm. It's clear from the experiments above that the simulated annealing effectively scrambles the initial key and doesn't begin to stabilise on a good solution until about the last 7000 iterations. Perhaps a lower starting temperature would work better?
This shows that the initial reasonable guess at the key is preserved: the top right graph shows that the trigram fitness only increases from the initial score of around -8500, and the bottom right graph shows that the τ score is consistently higher when the temperature starts lower. It's also clear that the lower-temperature run settles to a nearly-optimal solution earlier.
That's what happens when the simulated annealing is given a good guess at the initial key. What happens if we go back to an earlier condition and give it a random initial key (using uniformly-selected letter swaps)?
The results on the left are familiar from above. Compared to them, it's clear that the lower temperature run quickly moves towards a cipher key that's about as good as the letter-frequency inspired best-guess and then the run continues similarly to the other low-temperature run.
## Conclusions
The main takeaway from these experiments is that monoalphabetic substitution ciphers are easy to break. Even a straightforward hillclimbing algorithm, combined with trigram scoring to evaluate possible breaks, is capable of breaking these ciphers, given about 2000 characters of ciphertext.
Simulated annealing does slightly better, but only in the sense that an individual worker is more likely to find the correct solution while using simulated annealing than hillclimbing. But this advantage is eliminated when using several workers and picking the best result from the pool.
There are several parameters to vary with these algorithms: the initial guess at the key, how swaps are performed to change the key, the starting temperature of the simulated annealing, and so on. What these experiments show is that these parameters don't really affect the overall outcome: monoalphabetic substitution ciphers are easy to break, however you decide to do it.
## Code
The code for these experiments is on Github, in the hillclimbing-results directory.
## Credits
Cover photo by Nick Karvounis
Next Post
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# Support Vector Machines (and Kernel Methods in general)
## Presentation on theme: "Support Vector Machines (and Kernel Methods in general)"— Presentation transcript:
Support Vector Machines (and Kernel Methods in general)
Machine Learning March 23, 2010
Last Time Multilayer Perceptron/Logistic Regression Networks
Neural Networks Error Backpropagation
Today Support Vector Machines
Note: we’ll rely on some math from Optimality Theory that we won’t derive.
Maximum Margin Perceptron (and other linear classifiers) can lead to many equally valid choices for the decision boundary Are these really “equally valid”?
Max Margin How can we pick which is best?
Maximize the size of the margin. Small Margin Large Margin Are these really “equally valid”?
Support Vectors Support Vectors are those input points (vectors) closest to the decision boundary 1. They are vectors 2. They “support” the decision hyperplane
Support Vectors Define this as a decision problem
The decision hyperplane: No fancy math, just the equation of a hyperplane.
Support Vectors Aside: Why do some cassifiers use or
Simplicity of the math and interpretation. For probability density function estimation 0,1 has a clear correlate. For classification, a decision boundary of 0 is more easily interpretable than .5.
Support Vectors Define this as a decision problem
The decision hyperplane: Decision Function:
Support Vectors Define this as a decision problem
The decision hyperplane: Margin hyperplanes:
Support Vectors The decision hyperplane: Scale invariance
Support Vectors The decision hyperplane: Scale invariance
Support Vectors The decision hyperplane: Scale invariance
This scaling does not change the decision hyperplane, or the support vector hyperplanes. But we will eliminate a variable from the optimization The decision hyperplane: Scale invariance
What are we optimizing? We will represent the size of the margin in terms of w. This will allow us to simultaneously Identify a decision boundary Maximize the margin
How do we represent the size of the margin in terms of w?
There must at least one point that lies on each support hyperplanes Proof outline: If not, we could define a larger margin support hyperplane that does touch the nearest point(s).
How do we represent the size of the margin in terms of w?
There must at least one point that lies on each support hyperplanes Proof outline: If not, we could define a larger margin support hyperplane that does touch the nearest point(s).
How do we represent the size of the margin in terms of w?
There must at least one point that lies on each support hyperplanes Thus: And:
How do we represent the size of the margin in terms of w?
There must at least one point that lies on each support hyperplanes Thus: And:
How do we represent the size of the margin in terms of w?
The vector w is perpendicular to the decision hyperplane If the dot product of two vectors equals zero, the two vectors are perpendicular.
How do we represent the size of the margin in terms of w?
The margin is the projection of x1 – x2 onto w, the normal of the hyperplane.
Aside: Vector Projection
How do we represent the size of the margin in terms of w?
The margin is the projection of x1 – x2 onto w, the normal of the hyperplane. Projection: Size of the Margin:
Maximizing the margin Goal: maximize the margin
Linear Separability of the data by the decision boundary
Max Margin Loss Function
If constraint optimization then Lagrange Multipliers Optimize the “Primal”
Max Margin Loss Function
Optimize the “Primal” Partial wrt b
Max Margin Loss Function
Optimize the “Primal” Partial wrt w
Max Margin Loss Function
Optimize the “Primal” Partial wrt w \frac{\partial L(\vec{w},b)}{\partial \vec{w}}&=&0\\\vec{w} - \sum_{i=0}^{N-1}\alpha_it_i\vec{x_i}&=&0\\\vec{w}&=&\sum_{i=0}^{N-1}\alpha_it_i\vec{x_i}\\ Now have to find αi. Substitute back to the Loss function
Max Margin Loss Function
Construct the “dual” W(\alpha) = \sum_{i=0}^{N-1}\alpha_i - \frac{1}{2}\sum_{i,j=0}^{N-1}\alpha_i\alpha_j t_it_j(\vec{x_i}\cdot\vec{x_j})
Dual formulation of the error
Optimize this quadratic program to identify the lagrange multipliers and thus the weights There exist (extremely) fast approaches to quadratic optimization in both C, C++, Python, Java and R
If Q is positive semi definite, then f(x) is convex. If f(x) is convex, then there is a single maximum.
Support Vector Expansion
New decision Function Independent of the Dimension of x! When αi is non-zero then xi is a support vector When αi is zero xi is not a support vector
Kuhn-Tucker Conditions
In constraint optimization: At the optimal solution Constraint * Lagrange Multiplier = 0 \alpha_i(1-t_i(\vec{w}^T\vec{x_i} + b))=0 Only points on the decision boundary contribute to the solution!
Visualization of Support Vectors
Interpretability of SVM parameters
What else can we tell from alphas? If alpha is large, then the associated data point is quite important. It’s either an outlier, or incredibly important. But this only gives us the best solution for linearly separable data sets…
Basis of Kernel Methods
The decision process doesn’t depend on the dimensionality of the data. We can map to a higher dimensionality of the data space. Note: data points only appear within a dot product. The error is based on the dot product of data points – not the data points themselves.
Basis of Kernel Methods
Since data points only appear within a dot product. Thus we can map to another space through a replacement The error is based on the dot product of data points – not the data points themselves.
Learning Theory bases of SVMs
Theoretical bounds on testing error. The upper bound doesn’t depend on the dimensionality of the space The lower bound is maximized by maximizing the margin, γ, associated with the decision boundary.
Why we like SVMs They work Easily interpreted.
Good generalization Easily interpreted. Decision boundary is based on the data in the form of the support vectors. Not so in multilayer perceptron networks Principled bounds on testing error from Learning Theory (VC dimension)
SVM vs. MLP SVMs have many fewer parameters
SVM: Maybe just a kernel parameter MLP: Number and arrangement of nodes and eta learning rate SVM: Convex optimization task MLP: likelihood is non-convex -- local minima R(\theta)=\frac{1}{N}\sum_{n=0}^N\frac{1}{2}\left(y_n-g\left(\sum_k w_{kl}g\left(\sum_jw_{jk}g\left(\sum_iw_{ij}x_{n,i}\right) \right)\right)\right)^2
Soft margin classification
There can be outliers on the other side of the decision boundary, or leading to a small margin. Solution: Introduce a penalty term to the constraint function
Soft Max Dual Still Quadratic Programming!
W(\alpha) = \sum_{i=0}^{N-1}\alpha_i - \frac{1}{2}\sum_{i,j=0}^{N-1}t_it_j\alpha_i\alpha_j(x_i\cdot x_j)
Soft margin example Points are allowed within the margin, but cost is introduced. Hinge Loss
Probabilities from SVMs
Support Vector Machines are discriminant functions Discriminant functions: f(x)=c Discriminative models: f(x) = argmaxc p(c|x) Generative Models: f(x) = argmaxc p(x|c)p(c)/p(x) No (principled) probabilities from SVMs SVMs are not based on probability distribution functions of class instances.
Efficiency of SVMs Not especially fast. Training – n^3 Evaluation – n
Quadratic Programming efficiency Evaluation – n Need to evaluate against each support vector (potentially n)
Good Bye Next time: The Kernel “Trick” -> Kernel Methods or
How can we use SVMs that are not linearly separable?
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# If y = |x + 5|-|x - 5|, then y can take how many integer
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Joined: 11 Jun 2019
Posts: 33
Location: India
Re: If y = |x + 5|-|x - 5|, then y can take how many integer [#permalink]
### Show Tags
11 Sep 2019, 20:15
For Y = | X+5 | - | X - 5 |, we can identify critical points as -5 and 5.
So X must be in following ranges. X < -5 or -5 <= X < 5 or X > 5
1. Considering X < -5, we can modify modulus as
Y = -(X + 5) + (X - 5)
= -X -5 + X -5
Y = - 10...….(1 integer solution of Y for X < -5)
2. Considering X >= 5, we can modify modulus as
Y = (X +5 ) - (X - 5)
= X + 5 - X + 5
Y = 10...……(1 integer solution of Y for X >= 5)
3. Considering 3rd and last possible range of X. -5 <= X < 5,
Y = (X + 5) + (X -5)
Y = 2X...….Thus X can take any value between -5 till 4.9. But need integer values of Y.
Y= 2X can give integer values of Y only when X is an integer or X is a fraction with .5 at the end. So the value X can take are ( -5, -4.5, -4, -3.5, -3, -2.5, -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5) so total 20 values. But X = -5 will give Y to be -10. We have already considered Y = -10 in 1st range. hence in this range we get 19 possible integer values of Y.
Thus total possible integer values of Y are 1 + 1 + 19 = 21
hence (E)
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer [#permalink]
### Show Tags
22 Sep 2019, 08:27
Bunuel wrote:
If $$y=|x+5|-|x-5|$$, then $$y$$ can take how many integer values?
A. 5
B. 10
C. 11
D. 20
E. 21
Kudos for a correct solution.
Given: $$y=|x+5|-|x-5|$$,
Asked: $$y$$ can take how many integer values?
When x = 5; y = 10
when x=-5; y = -10
When x <-5; y= -10
When x>5; y =10
-5<=x<=5; -10<=y<=10; 21 integer values
IMO E
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Re: If y = |x + 5|-|x - 5|, then y can take how many integer [#permalink] 22 Sep 2019, 08:27
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Changes between Version 25 and Version 26 of WRF4G2.0/WRFAPP
Ignore:
Timestamp:
May 23, 2016 12:42:40 PM (6 years ago)
Comment:
--
Legend:
Unmodified
v25 = WRF configuration = To use WRF model in WRF4G2.0, you have to configure app variable in the [wiki:WRF4G2.0/Experiment experiment.wrf4g] file. By default, WRF4G2.0 provides a bundle with all you need to simulate an WRF experiment, configure as follows : To use WRF model in WRF4G2, you have to configure the app variable in the [wiki:WRF4G2.0/Experiment experiment.wrf4g] file. By default, WRF4G2 provides a bundle with all you need to simulate an WRF experiment, configure as follows : {{{ * If WRF is already configured on your system, you do not have use app variable. [[NoteBox(warn, As you have probably noticed\, we did not mention anything about WRF and WPS configuration files (e.g. CAM_ABS_DATA\, CAM_AEROPT_DATA\, co2_trans\, etc). For that purpose\, it is highly recommendable to create a tree directory file configuration under [wiki:WRF4G2.0/WRF4GFiles wrf4g_files] directory per experiment.)]] [[NoteBox(warn, As you have probably noticed\, we did not mention anything about WRF and WPS configuration files (e.g. CAM_ABS_DATA\, CAM_AEROPT_DATA\, co2_trans\, etc). For that purpose\, it is highly recommendable to create a tree directory file configuration under [wiki:WRF4G2.0/WRF4GFiles wrf4g_files] directory per experiment.)]]
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Welcome to MathMeetings.net! This is a list for research mathematics conferences, workshops, summer schools, etc. Anyone at all is welcome to add announcements.
## Know of a meeting not listed here? Add it now!
Additional update notes are available in the git repository (GitHub).
# Upcoming Meetings
## February 2021
### Combinatorial Algebraic Geometry
ag.algebraic-geometry co.combinatorics
2021-02-01 through 2021-05-07
ICERM
Providence, RI; USA
Meeting Type: thematic research program
Contact: see conference website
### Description
Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.
Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.
In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of polytopes are consequences of geometric theorems applied to associated toric varieties. In other cases geometry is a source of inspiration. For instance, long-standing conjectures about matroids have recently been resolved by proving that associated algebraic structures behave as if they are cohomology rings of smooth algebraic varieties.
Much research in combinatorial algebraic geometry relies on mathematical software to explore and enumerate combinatorial structures and compute geometric invariants. Writing the required programs is a considerable part of many research projects. The development of new mathematics software is therefore prioritized in the program.
The program will bring together experts in both pure and applied parts of mathematics as well mathematical programmers, all working at the confluence of discrete mathematics and algebraic geometry, with the aim of creating an environment conducive to interdisciplinary collaboration. The semester will include four week-long workshops, briefly described as follows.
• A 'boot-camp' aimed at introducing graduate students and early-career researchers to the main directions of research in the program.
• A research workshop dedicated to geometry arising from flag manifolds, classical and quantum Schubert calculus, combinatorial Hodge theory, and geometric representation theory.
• A research workshop dedicated to polyhedral spaces and tropical geometry, toric varieties, Newton-Okounkov bodies, cluster algebras and varieties, and moduli spaces and their tropicalizations.
• A Sage/Oscar Days workshop dedicated to development of programs and software libraries useful for research in combinatorial algebraic geometry. This workshop will also feature a series of lectures by experts in polynomial computations.
## April 2021
### Midwest Topology Seminar
at.algebraic-topology
2021-04-08 through 2021-04-08
Wayne State University
Detroit, Michigan; USA
Meeting Type: conference
Contact: Dan Isaksen
### Description
The Midwest Topology Seminar in Winter/Spring 2021 will be held in an online format. Three 45-minute talks will be separated by two 30-minute coffee breaks. The breaks will be structured to provide an opportunity for small group conversation in breakout rooms. Participants will be able to move themselves between rooms, and will be able to see who is in other rooms. (Note: Breakout room functionality works imperfectly with a smartphone or tablet. It’s best to use a computer for this.)
The speakers are Kristine Bauer (Calgary), Jeremy Hahn (MIT), and Kate Ponto (Kentucky).
### Unlikely intersections, Diophantine geometry, and related fields
lo.logic nt.number-theory
2021-04-12 through 2021-04-16
Meeting Type: conference
Contact: see conference website
none
### Front Range Number Theory Day Spring 2021
ag.algebraic-geometry nt.number-theory
2021-04-24 through 2021-04-24
Online via Zoom; USA
Meeting Type: conference
Contact: Amie Bray, Sarah Arpin
### Description
The University of Colorado at Boulder and Colorado State University in Fort Collins will be hosting Front Range Number Theory Day on April 24th 2021 on Zoom. Registration information can be found on our website. The goal of the FRNTD is to provide a venue for faculty, graduate students, and undergraduates on the Front Range who are interested in number theory to meet, learn, and collaborate. For more information and to register, please visit our website.
Invited Speakers: Lian Duan (CSU), Patrick Ingram (YorkU), Chole Martindale (Bristol)
### Towards a mod p Langlands correspondence
nt.number-theory rt.representation-theory
2021-04-26 through 2021-04-30
University of Duisburg-Essen
Essen; Antarctica
Meeting Type: spring school
Contact: see conference website
none
## May 2021
### Harmonic Analysis and Analytic Number Theory
nt.number-theory
2021-05-03 through 2021-08-20
Hausdorff Research Institute for Mathematics
Bonn; Germany
Meeting Type: conference
Contact: see conference website
none
### eCHT Mini-course
at.algebraic-topology
2021-05-04 through 2021-05-13
Wayne State University
Detroit, Michigan; USA
Meeting Type: minicourse
Contact: Dan Isaksen
### Description
4-hour online mini-course, offered on consecutive Tuesdays and Thursdays at 11:30am Eastern Time.
Speaker: Sander Kupers, University of Toronto Title: An introduction to homological stability
### Noncommutative Hodge-to-de Rham degeneration
ag.algebraic-geometry at.algebraic-topology kt.k-theory-and-homology
2021-05-07 through 2021-05-21
University of Warwick
Coventry; UK
Meeting Type: Online short course
### Description
I warmly invite you to attend the following short course:
Title: Noncommutative Hodge-to-de Rham degeneration.
Speaker: Dmitry Kaledin (Steklov Institute).
Abstract: I am going to give an overview of the proof of what is known as "non-commutative Hodge-to-de Rham Degeneration Theorem": for any smooth and proper DG algebra over a field of characteristic 0, the Hochschild-to-cyclic spectral sequence degenerates at first term.
Dates: 7, 14 and 21 of May 2021 (Fridays) at 14:00-15:00 (GMT time).
Link: To appear soon. This event is open to all and will be streamed on Microsoft Teams.
Historical context (written by the organizer): In the eighties, Deligne and Illusie (following earlier work of Faltings), gave a purely algebraic proof of the degeneration of the classical Hodge-to-de Rham spectral sequence. Later, in the 00’s, Kontsevich and Soibelman conjectured that the noncommutative Hodge-to-de Rham spectral sequence (where smooth proper algebraic varieties are replaced by smooth proper dg algebras) should also degenerate. Recently, Kaledin proved Kontsevich-Soibelman’s conjecture. Kaledin’s proof makes use of several different tools from algebra, algebraic geometry, representation theory and algebraic topology. Therefore, I believe this short course to be of interest to a broad mathematical audience.
### Rational Points and Galois Representations
nt.number-theory
2021-05-10 through 2021-05-12
University of Pittsburgh
Pittsburgh, PA; USA
Meeting Type: workshop
Contact: Carl Wang-Erickson
### Description
This three-day workshop is meant to bring together mathematicians with interest in rational points on algebraic curves, Galois representations, and related topics. Due to Covid-19, it will be occurring online via Zoom.
We are delighted to feature a colloquium presentation by Prof. Kirsten Wickelgren, seven research talks, one coding demo session, and one problem discussion session led by Prof. David Zureick-Brown.
### Leuca2021 Celebrating Claude Levesque's and Michel Waldschmidt's 75th birthdays
nt.number-theory
2021-05-10 through 2021-05-14
Marina di San Gregorio, Patù (Lecce), Italy; Italy
Meeting Type: conference
Contact: Valerio Talamanca
none
### Special Month On Singularities & K-Stability
nt.number-theory ag.algebraic-geometry ac.commutative-algebra
2021-05-17 through 2021-06-11
University of Utah
Salt Lake City, Utah; USA
Meeting Type: Special Month
Contact: Christopher Hacon, Karl Schwede, Chenyang Xu
### Description
This is a special month on the Singularities & K-Stability. There will be lecture series on Mixed characteristic vanishing theorems and applications and on Recent progress in K-stability of Fano varieties There will also be seminar-style talks and social activities (at most one talk per day).
### Cornell Topology Festival
gr.group-theory gt.geometric-topology
2021-05-20 through 2021-05-21
Cornell University
Ithaca, NY; USA
Meeting Type: conference
Contact: [email protected]
### Description
You are cordially invited to the 56th annual Cornell Topology Festival, which will be held online (for the first and hopefully only time ever!) this May. The main festival takes place Thursday May 20th – Friday, May 21st, 2021, with pre-festival talks on the 4th, 11th, and 18th, and a special colloquium on the 13th. More information about Zoom links, online social events, and discussion will follow for registered participants. We hope you will join us!
Speakers
Pre-Festival
Steve Trettel, Stanford University, 4 May 1pm; Lvzhou Chen, University of Texas Austin, 11 May 1pm; Sarah Koch, University of Michigan, Special Colloquium, 13 May 4pm; Jeremy Hahn, Massachusetts Institute of Technology, 18 May 1pm
20–21 May
Agnes Beaudry, University of Colorado Boulder; Sebastian Hensel, University of Munich; Kate Ponto, University of Kentucky; Manuel Rivera, Purdue University; Federico Rodriguez Hertz, Pennsylvania State University; Nick Salter, Columbia University; Richard Schwartz, Brown University; Matthew Stover, Temple University.
### QUANTUM CHAOS AND NUMBER THEORY A conference in honor of Zeev Rudnick's 60 birthday
nt.number-theory
2021-05-23 through 2021-05-27
Tel Aviv University
Tel Aviv; Israel
Meeting Type: conference
Contact: Lior Rosenzweig
### Description
The conference “Analytic Number Theory, Quantum Chaos and their Interfaces” aims at gathering distinguished researchers working in either of the disciplines to discuss recent research advances in these fields, and serve as a playground for the exchange of ideas between these, rather diverse, research communities. Another purpose of our conference is to provide a solid educational platform for more junior researchers (PhD students, postdoctoral researchers and early career permanent faculty) who aspire to conduct research in the relevant fields and expose them to some of the outstanding results and open problems as well as to meet other researchers with similar academic interests, with high potential to start new research collaborations. Being a home to Professor Zeev Rudnick, who has contributed greatly to unifying the main subjects of the conference and established some relevant fundamental results in these fields, Tel Aviv University is a natural place for such a meeting to take place.
### Curves over Finite Fields
ag.algebraic-geometry nt.number-theory
2021-05-24 through 2021-05-28
Benasque; Spain
Meeting Type: conference
Contact: see conference website
### Description
In the fall semester of 1985, Prof. Jean-Pierre Serre taught at Harvard University an extended series of lectures of his course on Rational Points on Curves over Finite Fields , first taught at Collège de France. Fernando Gouvêa's handwritten notes of this course have been spread all around since then. These notes contain the origin and inspiration of most of the works on the topic since 1985: maximal curves, construction of curves from their jacobians, class field towers, asymptotics of the number of points, etc.
At last, these notes have been edited, revised and are going to be published by the Société Mathématique Française in the Documents Mathématiques series. The present workshop will celebrate the publication of these notes. Experts on the topic will explain the main progress since 1985 and will discuss open questions and new techniques on curves over finite fields. Plenary speakers will be asked to write down their talks in order to publish proceedings which will be a natural continuation of Serre’s book.
### Arithmetic, Geometry, Cryptography and Coding Theory
ag.algebraic-geometry nt.number-theory
2021-05-31 through 2021-06-04
CIRM
Luminy; France
Meeting Type: conference
Contact: see conference website
### Description
Our goal is to organise a conference devoted to interactions between pure mathematics (in particular arithmetic and algebraic geometry) and information theory (especially cryptography and coding theory). This conference will be the eighteenth edition, with the first one held in 1987, in a series that has traditionally brought together some of the top specialists in the domains of arithmetic, geometry, and information theory. The corresponding international community is very active and all of the concerned research domains are developing and expanding rapidly.
The conference is therefore also an important occasion for junior mathematicians (graduate students and postdocs) to interact with established researchers in order to exchange ideas. We aim to create an inclusive atmosphere and to encourage forging new connections between researchers of various different backgrounds.
The conference talks will be devoted to recent advances in arithmetic and algebraic geometry and number theory, with a special emphasis on algorithmic and effective results and applications of these fields to information theory.
The conference will last one week and will be organized as follows :
• There will be one or two plenary talks each day, at the start of each session. They will be given by established researchers, some of whom are new to the established AGC2T community; this will allow for the introduction of emerging topics to the community, which may give rise to applications of arithmetic or algebraic geometry to information theory.
• There will be several shorter specialized talks in each session, often delivered by junior mathematicians.
As with the previous editions of the AGC2T, we aim publish conference proceedings as a special volume of the Contemporary Mathematics collection of the AMS.
Conference Topics
• Algebraic and arithmetic geometry over finite fields and global fields.
• Number theory, especially explicit and algorithmic.
• Algebro-geometric codes constructed from curves and higher-dimensional algebraic varieties over finite fields and global fields.
• Arithmetic and geometric aspects of cryptography (symmetric, public key, and post-quantum) and cryptanalysis.
## June 2021
### Ninth Bucharest Number Theory Days - a conference in honor of Alexandru Zaharescu's 60th Birthday
nt.number-theory
2021-06-01 through 2021-06-03
Institute of Mathematics of the Romanian Academy, UIC, UIUC
Bucharest; Romania
Meeting Type: conference
Contact: Alexandru Popa
none
### Young Researchers in Mathematics - 10th edition
gm.general-mathematics
2021-06-07 through 2021-06-09
University of Bristol
Online; UK
Meeting Type: conference
Contact: Nirvana Coppola
### Description
Young Researchers in Mathematics is the conference for all PhD students in the UK. We want to welcome each and every early career mathematician to this conference, where you can meet researchers from all areas in a friendly environment. YRM is the perfect opportunity to give talks about your maths, whether it be introductory or your own results. We also invite you to the plenary talks, which showcase a wide range of mathematics happening in the UK now. The conference will be online.
### Conference on Arithmetic Geometry in honor of Luc Illusie
ag.algebraic-geometry nt.number-theory
2021-06-07 through 2021-06-11
Morningside Center for Mathematics
Beijing; China
Meeting Type: conference
Contact: see conference website
none
### Motives, quadratic forms and arithmetic
ag.algebraic-geometry kt.k-theory-and-homology
2021-06-07 through 2021-06-11
Université d'Artois
Lens; France
Meeting Type: conference
Contact: Jérôme Burési, Baptiste Calmès, Ivo Dell'Ambrogio, Ahmed Laghribi
### Description
To celebrate Bruno Kahn's 63rd birthday
Motives and quadratic forms interact in fruitful ways: motives have successfully been used to understand and compute invariants of quadratic forms, while conversely, quadratic forms play a surprisingly deep and structural role in the motivic stable homotopy category of schemes. Moreover, over arithmetic bases, motives can be used to extract information of number theoretical nature from quadratic forms or other algebraic or geometric objects.
### 38th Annual Workshop in Geometric Topology
gt.geometric-topology at.algebraic-topology
2021-06-15 through 2021-06-17
Online; USA
Meeting Type: conference
Contact: Greg Friedman
### Description
The 38th Annual Workshop in Geometric Topology will be held online June 15-17, 2021. The featured speaker will be Peter Bubenik of the University of Florida, who will give a series of three one-hour lectures on Topological Data Analysis.
Participants are invited to contribute talks of 20 minutes. Contributed talks need not be directly related to the topic of the principal lectures. Title and abstracts must be submitted by May 15. If there are more volunteers to speak than there are time slots, the organizers will choose talks that provide a balanced collection of topics and respect the historical traditions of the workshop. Earlier responses may be given some preference. Applicants will be notified whether their talk has been accepted by June 1.
Full details, including registration and information about submitting titles and abstracts for contributed talks, can be found at the conference web site at http://faculty.tcu.edu/gfriedman/GTW2021
The Workshops in Geometric Topology are a series of informal annual research conferences that have been held since 1984. In non-pandemic years, the workshops currently rotate among Brigham Young University, Calvin College, Colorado College, Texas Christian University, and the University of Wisconsin at Milwaukee. Each workshop features a series of three lectures by one principal speaker, providing a substantial introduction to an area of current research in geometric topology. Participants are invited to contribute short talks on their own research, and there is ample time set aside each day for informal interactions between participants. Funding for the workshop series is currently provided by a grant from the National Science Foundation (DMS-1764311).
Workshop Organizers: Fredric Ancel, University of Wisconsin-Milwaukee Greg Friedman, Texas Christian University Craig Guilbault, University of Wisconsin-Milwaukee Molly Moran, Colorado College Nathan Sunukjian, Calvin College Eric Swenson, Brigham Young University Frederick Tinsley, Colorado College Gerard Venema, Calvin College
### 4th International Conference on Mathematics and Statistics (ICoMS 2021)
na.numerical-analysis pr.probability st.statistics-theory
2021-06-24 through 2021-06-25
Paris; France
Meeting Type: conference
Contact: see conference website
none
### Applied Topology in Bedlewo
gt.geometric-topology at.algebraic-topology
2021-06-27 through 2021-07-03
Banach Center
Bedlewo; Poland
Meeting Type: conference
Contact: Zbigniew Blaszczyk, Pawel Dlotko
### Description
Applied and computational topology, one of the most rapidly growing branches of mathematics, is becoming a key tool in applied sciences. It is making impact not only in mathematics, but on the wide interdisciplinary environment including material and medical sciences, data science, robotics. Building upon successful conferences held in Bedlewo in 2013 and 2017, the next edition of Applied Topology in Bedlewo will take place in 2021. Similarly as before, our aim is to bring together scientists from all over the world working in various fields of applied topology. This time we will focus on:
• random topology,
• topological methods in combinatorics,
• topological data analysis and shape descriptors,
• topological analysis of time-varying data in biology, engineering and finance,
• topological and geometrical descriptors of porous materials.
### Masterclass: High dimensional cohomology of moduli spaces
at.algebraic-topology gt.geometric-topology
2021-06-28 through 2021-07-02
University of Copenhagen, Centre for Geometry and Topology
Copenhagen; Denmark
Meeting Type: Masterclass
Contact: Peter Patzt
### Description
In this Masterclass, we will learn about the high dimensional cohomology of moduli spaces such as the moduli space of curves, graphs, and lattices. These moduli spaces are classifying spaces of groups such as mapping class groups, automorphism groups of free groups, and arithmetic groups. We will learn about duality groups whose high dimensional cohomology of these moduli spaces is related to the low degree homology groups with twisted coefficients. We will also discuss graph homology and tropical curves.
The masterclass is aimed at advanced graduate students and postdocs with an interest in algebraic topology and geometric group theory. Connections with algebraic geometry and number theory will be mentioned but this is not the primary focus.
## July 2021
### Fundamental Groups and their Representations in Arithmetic Geometry
nt.number-theory ag.algebraic-geometry
2021-07-04 through 2021-07-09
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
In arithmetic geometry, one studies solutions to polynomial equations defined with arithmetically interesting coefficients, such as integers or rational numbers. One way to study such objects, which has seen tremendous success in the last several decades, is by investigating their symmetries. Quite surprisingly, in several interesting situations, many of the geometric and arithmetic properties of the objects in question are actually controlled by the object’s symmetries.
Unfortunately, it is usually impossible to study these symmetries directly with current technology. To get around this, mathematicians working in this area often study simplified (often linearized) versions of the symmetries in question, which still capture a significant amount of information about the given object. This workshop will bring together both senior and junior researchers, including graduate students, postdocs, and leading experts, who study objects of geometric and arithmetic origin from the point of view of their symmetries and their linearized variants.
### Geometry via Arithmetic
nt.number-theory ag.algebraic-geometry
2021-07-11 through 2021-07-16
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
There is an age-old relationship between arithmetic and geometry, going back at least to Euclid's Elements. Historically, it has usually been geometry that has been used to enrich our understanding of arithmetic, but the purpose of this workshop is to study the flow of information in the other direction. Namely, how can arithmetic enhance our understanding of geometry? This meeting will bring together researchers from both sides of the partnership, to explore ways to bind the two fields ever closer together.
### Arithmetic Aspects of Deformation Theory
ag.algebraic-geometry nt.number-theory rt.representation-theory
2021-07-18 through 2021-07-23
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
One focus of modern number theory is to study symmetries of numbers that are roots of polynomial equations. Collections of such symmetries are called Galois groups, and they often encode interesting arithmetical information. The theory of Galois representations provides a way to understand these Galois groups and in particular, how they interact with other areas of mathematics. This workshop will investigate how these Galois representations can be put together into families, and search for new arithmetic applications of these families.
### Explicit Methods in Number Theory
nt.number-theory
2021-07-18 through 2021-07-24
Mathematisches Forschungsinstitut Oberwolfach
Oberwolfach; Germany
Meeting Type: workshop
Contact: Karim Belabas, Bjorn Poonen, Fernando Villegas
### Description
The workshop will bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and L-functions and their special values, and modular forms and functions. Considerable attention is paid to computational issues, but the emphasis is on aspects that are of interest to the pure mathematician.
### A Pair of Automorphic Workshops 2021
nt.number-theory
2021-07-18 through 2021-07-31
University of Oregon
Eugene, OR; USA
Meeting Type: graduate instructional workshop (week 1) and collaborative research workshop (week 2)
Contact: Ellen Eischen
### Description
During the last two weeks of July 2021, the University of Oregon will host a graduate instructional workshop followed by a collaborative research workshop to promote diverse collaborations.
While both of these workshops focus on algebraic and p-adic aspects of automorphic forms, L-functions, and related topics, the two workshops are independent. Applicants are encouraged to apply for one or both of the workshops, and funding decisions will be determined separately for each.
## August 2021
### Diophantine Methods in Algebraic Dynamics
ag.algebraic-geometry ds.dynamical-systems nt.number-theory
2021-08-01 through 2021-08-06
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
Algebraic dynamics is the study of discrete dynamical systems on algebraic varieties. It has its origins in complex dynamics, where one studies self-maps of complex varieties, and now encompasses dynamical systems defined over global fields.
In recent years, researchers have fruitfully investigated the latter by applying number-theoretic techniques, particularly those of Diophantine approximation and geometry, subfields which study the metric and geometric behavior of rational or algebraic points of a variety. The depth of this connection has allowed the mathematical arrow between the two fields to point in both directions; in particular, arithmetic dynamics is providing new approaches to deep classical Diophantine questions involving the arithmetic of abelian varieties. This workshop will focus on communicating and expanding upon the connections between algebraic dynamics and Diophantine geometry. It will bring together leading researchers in both fields, with an aim toward synthesizing recent advances and exploring future directions and applications.
### 9th International Conference on Research and Education in Mathematics (ICREM9)
ac.commutative-algebra fa.functional-analysis mp.mathematical-physics na.numerical-analysis nt.number-theory st.statistics-theory
2021-08-02 through 2021-08-06
Institute for Mathematical Research, Universiti Putra Malaysia
Langkawi, Kedah; Malaysia
Meeting Type: conference
Contact: Nor Azlida
### Description
The ICREM is a biennial conference series organised by the Institute for Mathematical Research (INSPEM), Univeristi Putra Malaysia, since its inaugural in 2003. With the joint efforts from our distinguished partners, the ICREM5 and ICREM8 were successfully organised by the Institut Teknologi Bandung (ITB), Indonesia in 2011 and 2017 respectively. The ICREM6 was successfully organized by the Institute of Mathematics, Vietnam Academy of Science & Technology (IMVAST), Vietnam in 2013.
This year, INSPEM with ITB, IMVAST, Thai Nguyen University of Education, Vietnam (TNUE), and Universität der Bundeswehr München, Germany (UniBw) are delighted to announce that the 9th International Conference on Research and Education in Mathematics (ICREM9) will be held in the beautiful island of Langkawi, MALAYSIA, from 02 - 06 August 2021.
The ICREM9 will have five Satellite Conferences:
• Satellite Conference on Computational Fluid Dynamics (CFD2021)
• Satellite Conference on Artificial Intelligence & Data-Driven Innovations (AIDDI2021)
• Satellite Conference on Scientific Computing, Simulation, and Quantitative Instrumentation (SCSQI2021)
• Satellite Conference on Structural and Analytical Mathematics with Cryptology (SAMC2021)
• Satellite Conference on Literacy (LE2021)
The Organising Committee is looking forward to welcoming the delegates to the ICREM9 held for the first time in the beautiful island known as The Jewel of Kedah (Langkawi Permata Kedah). We are working towards preparing an attractive scientific programme with diverse topics to create a conducive environment suitable for encouraging and facilitating our delegates to share their knowledge and exchange ideas in all aspects of mathematics and its application.
The venue of the conference is the perfect setting to complement the ICREM9. Langkawi is an archipelago made up of 99 islands in the Malacca Strait some 30 km off the mainland coast of northwestern Malaysia. The islands are a part of the state of Kedah, which is adjacent to the Thai border. Surrounded by the turquoise sea, the interior of the main island is a mixture of picturesque paddy fields and jungle-clad hills. The main island spans about 25 km from north to south and slightly more for east and west. The coastal areas consist of flat, alluvial plains punctuated with limestone ridges. Two-thirds of the island is dominated by forest-covered mountains, hills, and natural vegetation. The island's oldest geological formation, the Machinchang Formation, was the first part of Southeast Asia to rise from the seabed in the Cambrian more than half a billion years ago.
We invite you to discover for yourself our little treasures. We are honoured to host the ICREM9 here in Langkawi, Malaysia. It is our hope that you will join us and learn, enjoy, and most importantly be a part of the ICREM9 community. Organise a session, give a talk, and remember to experience this great island.
The Organising Committee is looking forward to playing host to all members of the mathematical community at the ICREM9 Conference.
See you in Langkawi, MALAYSIA for ICREM9!
### Perspectives on quantum link homology theories
gt.geometric-topology qa.quantum-algebra
2021-08-09 through 2021-08-15
Regensburg; Germany
Meeting Type: combined student workshop & research conference
Contact: Claudius Zibrowius, Lukas Lewark
### Description
Details see conference website.
### Géométrie algébrique, Théorie des nombres et Applications (GTA)
ag.algebraic-geometry nt.number-theory
2021-08-16 through 2021-08-20
University of French Polynesia
Tahiti; French Polynesia
Meeting Type: conference
Contact: Gaetan Bisson
### Description
The GTA 2021 conference will bring together world class researchers in mathematics. Its main objectives are to discuss recent advances in the fields of algebraic geometry, number theory and their applications, as well as to foster international collaborations on connected topics.
Although contributions from all related areas of mathematics are welcome, particular emphasis will be placed on research interests of our late colleague Alexey Zykin, namely: zeta-functions and L-functions, algebraic geometry over finite fields, families of fields and varieties, abelian varieties and elliptic curves.
### SIAM Conference on Applied Algebraic Geometry (AG21)
ag.algebraic-geometry
2021-08-16 through 2021-08-20
College Station, TX; USA
Meeting Type: conference
Contact: see conference website
### Description
This is the meeting of the SIAM Activity Group on Algebraic Geometry.
The purpose of the SIAM Activity Group on Algebraic Geometry is to bring together researchers who use algebraic geometry in industrial and applied mathematics. "Algebraic geometry" is interpreted broadly to include at least: algebraic geometry, commutative algebra, noncommutative algebra, symbolic and numeric computation, algebraic and geometric combinatorics, representation theory, and algebraic topology. These methods have already seen applications in: biology, coding theory, cryptography, combustion, computational geometry, computer graphics, quantum computing, control theory, geometric design, complexity theory, machine learning, nonlinear partial differential equations, optimization, robotics, and statistics. We welcome participation from both theoretical mathematical areas and application areas not on this list which fall under this broadly interpreted notion of algebraic geometry and its applications.
### Young Researchers in Algebraic Number Theory
nt.number-theory
2021-08-18 through 2021-08-20
University of Bristol
Bristol; UK
Meeting Type: conference
Contact: Nirvana Coppola
### Description
Y-RANT is a (relatively) new conference aimed at postgraduate students and early career researchers in algebraic number theory, promoting discussion and sharing of ideas between members of the community. Participants are strongly encouraged to register to give short talks on their research or other topics of interest. We will do our best to accommodate all participant's talks, however this might not be possible due to time constraints. In addition there will be three keynote talks given by senior researchers in the field.
### Supersingular Isogeny Graphs in Cryptography
ag.algebraic-geometry nt.number-theory
2021-08-22 through 2021-08-27
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
Despite the enormous commercial potential that quantum computing presents, the existence of large-scale quantum computers also has the potential to destroy current security infrastructures. Post-quantum cryptography aims to develop new security protocols that will remain secure even after powerful quantum computers are built. This workshop focuses on isogeny-based cryptography, one of the most promising areas in post-quantum cryptography. In particular, we will examine the security, feasibility and development of new protocols in isogeny-based cryptography, as well as the intricate and beautiful pure mathematics of the related isogeny graphs and elliptic curve endomorphism rings. To address the goals of both training and research, the program will be comprised of keynote speakers and working group sessions.
### Automorphic Forms, Geometry and Arithmetic
rt.representation-theory nt.number-theory ag.algebraic-geometry
2021-08-22 through 2021-08-28
Mathematisches Forschungsinstitut Oberwolfach
Oberwolfach; Germany
Meeting Type: invitational workshop
Contact: see conference website
none
### Georgian Mathematical Union XI Annual International Conference
ag.algebraic-geometry ap.analysis-of-pdes at.algebraic-topology ca.classical-analysis-and-odes co.combinatorics cv.complex-variables dg.differential-geometry fa.functional-analysis gm.general-mathematics gn.general-topology gt.geometric-topology ho.history-and-overview kt.k-theory-and-homology lo.logic mp.mathematical-physics na.numerical-analysis nt.number-theory pr.probability ra.rings-and-algebras st.statistics-theory
2021-08-23 through 2021-08-28
Georgian Mathematical Union, Shota Rustaveli Batumi State University
Batumi, Georgia; Georgia
Meeting Type: International Conference
Contact: Alexander Meskhi, David Natroshvili, Tinatin Davitashvili
none
## September 2021
### Arithmetic Geometry - Takeshi 60
ag.algebraic-geometry nt.number-theory
2021-09-06 through 2021-09-10
Graduate School of Mathematical Sciences, The University of Tokyo
Tokyo; Japan
Meeting Type: conference
Contact: Ahmed Abbes, Kenichi Bannai, Naoki Imai, Tadashi Ochiai, Atsushi Shiho
### Description
A conference on the occasion of Takeshi Saito's 60th birthday
### Arakelov Geometry
ag.algebraic-geometry nt.number-theory
2021-09-06 through 2021-09-10
Universität Regensburg
Regensburg; Germany
Meeting Type: conference (in virtual or hybrid mode)
Contact: Klaus Künnemann
### Description
The conference Arakelov Geometry is organized by José Ignacio Burgos Gil, Walter Gubler, and Klaus Künnemann. This conference constitutes the eleventh session of the Intercity Seminar on Arakelov Theory organized by José Ignacio Burgos Gil, Vincent Maillot and Atsushi Moriwaki with previous sessions in Barcelona, Beijing, Copenhagen, Kyoto, Paris, Regensburg, and Rome.
### 2nd IMA Conference on Mathematics of Robotics
ag.algebraic-geometry at.algebraic-topology gm.general-mathematics gt.geometric-topology
2021-09-08 through 2021-09-10
Manchester Metropolitan University
Manchester; UK
Meeting Type: conference
Contact: Pam Bye
### Description
This Conference has been organised in cooperation with the Society for Industrial and Applied Mathematics (SIAM).
Areas of interest include, but are not limited to: Topology. Kinematics. Algebraic topology of con?guration spaces of robot mechanisms. Topological aspects of path planning and sensor networks. Differential topology and singularity theory of robot mechanism and moduli spaces. Algebraic Geometry. Varieties generated by linkages and constraints. Geometry of stiffness and inertia matrices. Rigid-body motions. Computational approaches to algebraic geometry. Dynamical Systems and Control. Dynamics of robots and mechanisms. Simulation of multi-body systems, e.g. swarm robots. Geometric control of robots. Optimal control and other optimisation problems. Combinatorial and Stochastic Methods. Rigidity of structures. Path planning algorithms. Modular robots. Statistics. Stochastic control. Localisation. Navigation with uncertainty. Statistical learning theory. Cognitive Robotics. Mathematical aspects of Artificial Intelligence, Developmental Robotics and other Neuroscience based approaches.
Invited speakers: Dr Mini Saag – University of Surrey, UK Prof Frank Sottile - Texas A&M University, USA Prof Stefano Stramigioli - University of Twente, The Netherlands
ag.algebraic-geometry nt.number-theory
2021-09-20 through 2021-09-24
Meeting Type: conference
Contact: see conference website
none
## October 2021
### Lattices and Cohomology of Arithmetic Groups: Geometric and Computational Viewpoints
gr.group-theory gt.geometric-topology nt.number-theory
2021-10-03 through 2021-10-08
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
A lattice is a discrete collection of regularly ordered points in space. Lattices are everywhere around us, from the patterned stacked arrangements of fruits and vegetables at the grocery to the regular networks of atoms in crystalline compounds. Today lattices find applications throughout mathematics and the sciences, applications ranging from chemistry to cryptography and Wi-Fi networks.
The focus of this meeting is the connections between lattices and number theory and geometry. Number theory, one of the oldest branches of pure mathematics, is devoted to the study properties of the integers and more sophisticated number systems. Lattices and number theory have many deep connections. For instance using number theory it was recently demonstrated that certain packings of balls in high dimensions are optimally efficient. Lattices also appear naturally when one studies certain spaces that play an important role in number theory; one of the main focuses of this meeting is to investigate computational and theoretical methods to understand such spaces and to expand the frontier of our algorithmic knowledge in working with them.
### Cohomology of Arithmetic Groups: Duality, Stability, and Computations
gr.group-theory gt.geometric-topology nt.number-theory
2021-10-10 through 2021-10-15
Banff International Research Station
Meeting Type: conference
Contact: see conference website
### Description
The cohomology of arithmetic groups is the study of the properties of holes'' in geometric spaces that contain information about number theory. The workshop will bring together mathematicians with expertise in number theory, topology, and geometric group theory to tackle these problems and explore recent developments.
## January 2022
### Higher Algebraic Structures In Algebra, Topology And Geometry
ag.algebraic-geometry at.algebraic-topology gt.geometric-topology kt.k-theory-and-homology sg.symplectic-geometry
2022-01-10 through 2022-04-29
Institute Mittag-Leffler
Djursholm; Sweden
Meeting Type: research program
Contact: Gregory Arone, Tilman Bauer, Alexander Berglund, Søren Galatius, Jesper Grodal, Thomas Kragh
### Description
We are happy to announce that the Institut Mittag-Leffler will be hosting a research program entitled
"HIGHER ALGEBRAIC STRUCTURES IN ALGEBRA, TOPOLOGY AND GEOMETRY”
from January 10, 2022 to April 29, 2022.
Junior researchers (advanced PhD students or young postdocs) can apply for a fellowship to attend the program, covering all expenses (deadline: December 31, 2020). For all others, the program is by invitation only.
Institut Mittag-Leffler in Danderyd, just north of Stockholm, Sweden, is an international centre for research and postdoctoral training in mathematical sciences. The oldest mathematical research institute in the world, it was founded in 1916 by Professor Gösta Mittag-Leffler and his wife Signe, who donated their magnificent villa, with its first-class library, for the purpose of creating the institute that bears their name.
Junior research fellowships: http://www.mittag-leffler.se/research-programs/junior-fellowship-program
The organizers
Gregory Arone ([email protected])
Tilman Bauer ([email protected])
Alexander Berglund ([email protected])
Søren Galatius ([email protected])
Jesper Grodal ([email protected])
Thomas Kragh ([email protected])
## March 2022
### Rational Points 2022
ag.algebraic-geometry nt.number-theory
2022-03-27 through 2022-04-02
Schney/Lichtenfels, Bavaria; Germany
Meeting Type: workshop
Contact: Michael Stoll
### Description
This workshop aims at bringing together the leading experts in the field, covering a broad spectrum reaching from the more theoretically-oriented over the explicit to the algorithmic aspects. The fundamental problem motivating the workshop asks for a description of the set of rational points X(Q) for a given algebraic variety X defined over Q. When X is a curve, the structure of this set is known, and the most interesting question is how to determine it explicitly for a given curve. When X is higher-dimensional, much less is known about the structure of X(Q), even when X is a surface. So here the open questions are much more basic for our understanding of the situation, and on the algorithmic side, the focus is on trying to decide if a given variety does have any rational point at all.
This is a workshop with about 50 participants. Participation is by invitation. Every participant is expected to contribute actively to the success of the event, by giving talks and/or by taking part in the discussions.
## May 2022
### Franco-Asian Summer School on Arithmetic Geometry in Luminy
nt.number-theory ag.algebraic-geometry
2022-05-30 through 2022-06-03
Centre International de Rencontres Mathématiques (CIRM)
Marseille; France
Meeting Type: conference
Contact: Ahmed Abbes
### Description
Arithmetic geometry is a broad and central area of Mathematics. The summer school will focus on a number of areas of important progress over the last three or four years, notably: p-adic Hodge theory, Ramification of étale l-adic sheaves, Homotopical algebra techniques and applications to motives and epsilon factors. The summer school will consist of 4 mini-courses supplemented by ten one-hour lectures.
## June 2022
### 7th IMA Conference on Numerical Linear Algebra and Optimization
ag.algebraic-geometry gm.general-mathematics
2022-06-29 through 2022-07-01
University of Birmingham
Birmingham; UK
Meeting Type: conference
Contact: Pam Bye
### Description
Early Bird Conference Fees IMA/SIAM Member - £395.00 Non IMA/SIAM Member - £450.00 IMA/SIAM Student - £215.00 Non IMA/SIAM Student - £225.00 Conference Fees will increase by £20 on 22 May 2022 Day Delegate rate: A Day Delegate rate is also available for this Conference if you would like to attend one of the scheduled Conference days. If you would like to find out more information about our Day Delegate rate, please contact us at [email protected]
Accommodation The IMA have booked accommodation at Edgbaston Park Hotel on hold for delegates on a first-come, first-serve basis. The room is £90 Single occupancy, B&B which will be available to book until 16/05/2022. If you are interested in booking at this rate, please contact the Conferences Department for the booking code.
Organising Committee Michal Kocvara, University of Birmingham (co-chair) Daniel Loghin, University of Birmingham (co-chair) Coralia Cartis, University of Oxford Nick Gould, Rutherford Appleton Laboratory Philip Knight, University of Strathclyde Jennifer Scott, Rutherford Appleton Laboratory Valeria Simoncini, University of Bologna Contact information For general conference queries please contact the Conferences Department, Institute of Mathematics and its Applications, Catherine Richards House, 16 Nelson Street, Southend-on-Sea, Essex, SS1 1EF, UK. E-mail: [email protected] Tel: +44 (0) 1702 354 020
## July 2022
### Spec$(\overline{Q})$
ac.commutative-algebra ag.algebraic-geometry nt.number-theory
2022-07-06 through 2022-07-08
Fields Institute
Meeting Type: conference
Contact: see conference website
### Description
Spec(Q¯¯¯¯) is the first conference to celebrate and promote research advances of LGBT2Q mathematicians specialising in algebraic geometry, arithmetic geometry, commutative algebra, and number theory. This conference capitalises on recent thematic program successes in algebraic geometry at Fields, the Thematic Program on Combinatorial Algebraic Geometry (July 1 - December 31, 2016) and the Thematic Program on Homological Algebra of Mirror Symmetry (July 1 - December 31, 2019). Spec(Q¯¯¯¯) will create an empowering and engaging environment which provides LGBT2Q visibility in algebraic geometry, will support junior LGBT2Q academics, and will crystallise new collaborative networks for participants. Algebraic geometry, classically, is the study of the geometry of solutions of polynomial equations; through modern advances it has become an intersectional mathematical field, drawing from various aspects of algebra, number theory, geometry, combinatorics and even mathematical physics. This conference aims to highlight strong mathematical research in a wide array of algebraic geometry, broadly defined. The conference will feature some plenary talks by world-leading researchers from a range of areas of algebraic geometry. To facilitate new connections across the various threads of algebraic geometry, plenary talks at Spec(Q¯¯¯¯) will be aimed a general algebro-geometric audience.
This activity will bring together mathematicians spanning all academic ranks to create ideal networking and mentorship for LGBT2Q academics while disseminating key achievements of trans and queer algebraic geometers. Queer and trans academics often have a diffcult experience developing key collaborations and networks of trusted colleagues. Each research connection, grant, and application involves a conscious decision of how much of one’s queer/trans identity to disclose. This conference provides a safe space to develop ones network while removing these barriers. In such spaces, one can discuss mathematics with new colleagues while unbridled with many societal challenges that they face in mathematical communities. When a mathematician feels free to be themselves in all ways, they are able to immerse themselves in creative mathematical thought.
### Park City Mathematics Institute: Number theory informed by computation
nt.number-theory
2022-07-17 through 2022-08-06
IAS/PCMI
Park City, UT; USA
Meeting Type: conference and summer school
Contact: Bjorn Poonen
none
## August 2022
### Women in Numbers Europe - 4
nt.number-theory
2022-08-29 through 2022-09-02
Utrecht University
Utrecht; Netherlands
Meeting Type: workshop
Contact: Ramla Abdellatif, Valentijn Karemaker, Ariane Mézard
### Description
This is a workshop that aims to support new collaborations for women in number theory. Prior to the conference, the project leaders will design projects and provide background reading and references for their groups. Each participant will be assigned to a working group according to her research interests. During the workshop there will be some talks and career-development activities, but there will also be ample time dedicated to working in the working groups.
## January 2023
### Algebraic Cycles, L-Values, and Euler Systems
nt.number-theory ag.algebraic-geometry
2023-01-17 through 2023-05-26
MSRI
Berkeley, CA; USA
Meeting Type: conference
Contact: see conference website
### Description
The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of L-functions and their leading terms to cycles and Galois cohomology groups.
The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.
The goal of this semester is to bring together researchers working on different aspects of this young but fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles.
### Diophantine Geometry
ag.algebraic-geometry nt.number-theory
2023-01-17 through 2023-05-26
MSRI
Berkeley, CA; USA
Meeting Type: thematic program
Contact: see conference website
### Description
Number Theory concerns the study of properties of the integers, rational numbers, and other structures that share similar features. It is a central branch of mathematics with a well-known feature: it is often the case that easy-to-state problems in number theory turn out to be exceedingly difficult (e.g. Fermat’s Last Theorem), and their study leads to groundbreaking discoveries in other fields of mathematics.
A fundamental theme in number theory concerns the study of integer and rational solutions to Diophantine equations. This topic originated at least 3,700 years ago (as documented in babylonian clay tablets) and it has evolved into the highly sophisticated field of Diophantine Geometry. There are deep and fruitful interactions between Diophantine Geometry and seemingly distant fields such as representation theory, algebraic geometry, topology, complex analysis, and mathematical logic, to mention a few. In recent years, these connections have led to a large number of new results and, specially, to the partial or complete resolution of important conjectures in the field.
While the study of rational solutions of diophantine equations initiated thousands of years ago, our knowledge on this subject has dramatically improved in recent years. Especially, we have witnessed spectacular progress in aspects such as height formulas and height bounds for algebraic points, automorphic methods, unlikely intersection problems, and non-abelian and p-adic approaches to algebraic degeneracy of rational points. All these groundbreaking advances in the study of rational and algebraic points in varieties will be the central theme of the semester program “Diophantine Geometry” at MSRI. The main purpose of this program is to bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to update on recent breakthroughs, and to further advance the field by making progress on fundamental open problems and by developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field, and we strongly encourage their participation.
## April 2025
### test 2
ho.history-and-overview
2025-04-12 through 2025-04-14
test; Armenia
Meeting Type: conference
Contact: see conference website
test 2
|
# OperationsWithSets
## Geometric operations with convex sets
### Methods for determining basic properties of sets
Consider the following example where the set is built from the following constraints
x = sdpvar(2, 1);
constraints = [ 0 <= x <= 5; 4*x(1)^2 - 2*x(2) <= 0.4; sqrt( x(1)^2 + 0.6*x(2)^2 ) <= 1.3 ];
S = YSet(x, constraints);
It is very often of interest to check wheter the set is empty or not. In MPT there exist an isEmptySet method that checks whether the set is empty
S.isEmptySet()
Another useful property is boundedness which can be checked using the isBounded method, i.e.
S.isBounded()
These properties can be veriefied by plotting the set using plot method
S.plot('color', 'lightblue', 'linewidth', 2, 'linestyle', '--')
Note that plotting of general convex sets could become time consuming because the set is sampled with an uniform grid. The value of the grid can be changed using the grid option of the plot method, for details type
help ConvexSet/plot
### Working with multiple sets
Consider following two sets created with the help of YALMIP
z = sdpvar;
t = sdpvar;
constraints1 = [ z^2 - 5*t <= 1 ; 0 <= t <= 1 ];
S = YSet( [z; t], constraints1);
constraints2 = [ z^2 + 5*t <= 1 ; 0 <= t <= 1 ];
R = YSet( [z; t], constraints2);
which are plotted in different colors
plot(S, 'color', 'lightgreen', R, 'color', 'yellow')
The above sets can be concatenated into an array using overloaded [ ] operators. The column concatenation can be done using brackets or vertcat method which are equivalent:
column_array1 = [S; R]
column_array2 = vertcat(S, R)
The row concatenation using brackets or horzcat method
row_array1 = [S, R]
row_array2 = horzcat(S, R)
If the sets are stored in an array, some methods can operate on the array, for instance
row_array1.isBounded()
column_array2.isEmptySet()
If the method is not applicable on the array, it can be invoked for each element in the array using forEach method:
statement = row_array1.forEach(@isBounded)
The forEach method is a fast replacement of for-loop and it can be useful for user-specific operations over an array of convex sets.
To create a new copy of the ConvexSet, the copy must be employed, otherwise the new object points to the same data stored in the original object
Snew = S.copy()
### Geometric methods
The ConvexSet class offers a couple of methods for use in computational geometry. Consider a following set which is created as an intersection of quadratic and linear constraints
x = sdpvar(2, 1);
A = [-0.46 -0.03; 0.08 -1.23; -0.92 -1.9; -1.92 2.37];
b = [1.72; 3.84; 3.05; 0.03];
constraints = [ 0.2*x'*x-[2.1 0.8]*x<=2; A*x<=b ];
S = YSet(x, constraints);
which is plotted in salmon color
S.plot('color', 'salmon')
This set will be used next to demonstrate some methods that can be applied to YSet objects.
#### Set containment test
To check if the point is contained in the set or not, there exist a method contains. For instance, the point x1=[8; 0] lies in the set
x1 = [8; 0];
S.contains(x1)
ans =
1
but the point x2 = [15; 0] lies not:
x2 = [15; 0];
S.contains( x2 )
ans =
0
#### Distance to a point
Computing distance from the set to a given point is achieved using distance method. For instance, the distance to the point x2 that lies outside of the set S can be computed as follows
data = S.distance(x2)
ans =
exitflag: 1
dist: 0.5932
x: [2x1 double]
y: [2x1 double]
The output from the method is a structure with four fields indicating the result of the computing the distance. The field exitflag indicates the exit status from the optimization solver, the actual distance is available in dist field and fields x, y indicate the coordinates of the two points for which the distance has been computed. It can be extracted from the field y which point is the closest to the point x2 and plotted.
data.y
ans =
11.5654
0.7044
#### Point projection onto a set
Computation of the distance is achieved also in the project method which computes the closest point to the set. For the point x2 the projection operation results in a structure with four fields
res = S.project(x2)
res =
exitflag: 1
how: 'Successfully solved (SeDuMi-1.3)'
x: [2x1 double]
dist: 3.5061
The field exitflag informs about the status returned from the optimization solver which can be found also in the how field. The closed point is computed in x field and the value about the distance can be found in dist field. One can verify that the computed point by the projection operation is the same as by distance operation
res.x
ans =
11.5654
0.7044
#### Separation hyperplane from a point
The YSet object implements the separate method that computes a hyperplane that separates the set from a given point. As an example, consider computation of a separating hyperplane from the set S and the point x2:
He = S.separate(x2)
He =
-3.4346 0.7044 -45.3730
which returns a data corresponding to a hyperplane equation { x | He*[x; -1] = 0 }. To plot the computed hyperplane, a Polyhedron object is constructed
P = Polyhedron('He', He)
and plotted.
#### Extreme point in a given direction
Computation of extreme points for a convex set is implemented in the extreme method. The method accepts a point as an argument that specifies the direction in which to compute the extreme point. For instance, for the point x2, the extreme method results in
v1 = S.extreme( x2 )
v1 =
exitflag: 1
how: 'Successfully solved (SeDuMi-1.3)'
x: [2x1 double]
supp: 175.4535
The output variable v1 contains the status returned from the optimization solver in exitflag and how fields. The extreme point is stored in x variable and the field supp corresponds to a support of the set given as { max x'*y s.t. y in Set }. The extreme point in the direction x3 = [0; 5] is given as
x3 = [0; 5];
v2 = S.extreme( x3 )
v2 =
exitflag: 1
how: 'Successfully solved (SeDuMi-1.3)'
x: [2x1 double]
supp: 36.3241
One can visually inspect the location of the computed extreme points v1.x and v2.x in the figure below:
#### Computing a support of the set in a given direction
For a given point x, the support of a set is given as { max x'*y s.t. y in Set }. This feature is implemented in the support method. The syntax of the support method requires a point x which determines the direction and the value of the maximum over the set is returned.
S.support(x3)
ans =
36.3241
Note that computation of the support is available also in the extreme method.
#### Maximum over the set in a given direction - ray shooting
The ray-shooting problem is given by { max alpha s.t. alpha*x in Set } and is available in the shoot method. As an example, consider computation of the maximum of the set in the direction of the point x2 = [15; 0] which gives the value of alpha
alpha = S.shoot( x2 )
alpha =
0.7586
Multiplying this value with the point x2 one obtains the point v = alpha * x2 that lies on the boundary of the set
v = alpha * x2
v =
11.3788
0
#### Outer approximation
Computation of the bounding box over the set is implemented in outerApprox method:
B = S.outerApprox()
The method returns a polyhedron with lower and upper bounds that are stored internally and can be accessed by referring to Internal property
B.Internal.lb
ans =
-0.6775
-2.9647
B.Internal.ub
ans =
11.6969
7.2648
Comparison of the bounding box B wih the orignal set S can be found on the figure below which was generated with the following command:
S.plot('color', 'salmon');
hold on
B.plot('wire', true, 'linestyle', '--', 'linewidth', 2)
hold off
Back to Computational Geometry overview.
|
# Calories 2
Ben eats approximately 2400 calories per day. His wife Sarah eats 5/8 as much. How many calories does Sarah eat per day?
Result
S = 1500 cal/d
#### Solution:
$B = 2400 \ cal/d \ \\ \ \\ S = \dfrac{ 5 }{ 8 } \cdot \ B = \dfrac{ 5 }{ 8 } \cdot \ 2400 = 1500 = 1500 \ \text{ cal/d }$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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|
# Syntax tree
A syntax , derivation or parse tree is a term from theoretical computer science and linguistics . It describes a hierarchical representation of the breakdown of a text. Syntax trees are used both as an aid for graphical visualization of the breakdown and, in the form of a data structure , to display this breakdown for machine processing, e.g. B. in a compiler or translator .
The various terms are not used uniformly in the literature. Only the term derivation tree , which is based on the term derivation, is formally precisely defined . Other names for different types of trees can then be technically defined in more detail, as described below, if necessary.
In contrast to computer science, in which languages can also be defined according to the technical possibilities, linguistics finds more difficult conditions when dealing with natural languages , mainly because the order of the components in a sentence can vary.
## introduction
Representation of a derivation tree
In the (mechanical) analysis of natural language sentences or formal texts (e.g. computer programs), directly after the lexical analysis (the breakdown into tokens or symbols ), the symbols are often hierarchically combined to form related parts of sentences ( constituents ) or sections of the formal Text instead. Conversely, this can also be seen as a dissection of the text. The result is a tree like the one shown on the right. In addition to the graphic form, bracketed representations are also used for syntax trees:${\ displaystyle [_ {S} \ [_ {\ mathit {NP}} \ John] \ [_ {\ mathit {VP}} \ [_ {V} \ hit] \ [_ {\ mathit {NP}} \ [_ {\ mathit {Det}} \ the] \ [_ {N} \ ball]]]]].}$
Technically, the tree on the left is also called a concrete derivation tree , as it shows the resulting structure exactly based on the concrete text. In linguistics, however, models are also common that provide several layers of representation (e.g. surface and deep structure ).
The nodes of the tree are often enriched with attributes (in linguistics these are mainly morphological categories ). This gives you an attributed syntax tree with the associated attributed grammar . While a context-free grammar is used in the first two tree representations , the context dependency comes into play in the latter . These differences are reflected in the Chomsky hierarchy . In such cases, the term semantic analysis is used in compiler construction .
## Derivation trees
Consider a context-free grammar . A derivation tree for this is a tree whose nodes are labeled with symbols from (i.e. terminal and non-terminal symbols and the empty word ). The tree is ordered , i.e. H. the children of each node have a fixed order, and the following applies to the labeling: ${\ displaystyle G = \ left (N, \ Sigma, P, S \ right)}$${\ displaystyle \ Sigma \ cup N \ cup \ {\ varepsilon \}}$
• The root is labeled with the start symbol . This property is sometimes not required. A tree that satisfies them is called a complete derivation tree .${\ displaystyle S}$
• If the children of an inner node labeled with are labeled with the symbols (in that order), the grammar must contain the rule .${\ displaystyle A}$${\ displaystyle z_ {1}, \ ldots, z_ {m}}$${\ displaystyle A \ to z_ {1} \ ldots z_ {m}}$
• The leaves of the tree are labeled with symbols .${\ displaystyle \ Sigma \ cup \ {\ varepsilon \}}$
• If a leaf is marked with , it is the only successor to its predecessor node.${\ displaystyle \ varepsilon}$
Only non-terminal symbols can be used as inner nodes, and only the terminal symbols or the empty word for the leaves.
### Construction of derivation trees
Possible syntax trees / diagrams can often be easily created for short texts by following the production rules. There are many mechanical methods available for longer texts .
For example, the syntax diagram shown in the introduction u. a. the following rules apply:
${\ displaystyle {\ begin {array} {lll} S & \ rightarrow & NP \ VP \\ NP & \ rightarrow & {\ mathtt {John}} \\ NP & \ rightarrow & Det \ N \\ VP & \ rightarrow & V \ NP \\\ end {array}} \ quad \ quad {\ begin {array} {lll} V & \ rightarrow & {\ mathtt {hit}} \\ Det & \ rightarrow & {\ mathtt {the}} \\ N & \ rightarrow & {\ mathtt {ball}} \ end {array}}}$
In order to generate a derivation tree, the rules can be applied step-by-step from the root by systematically replacing one nonterminal on the left side of the rule with the symbols on the right until only terminals are left:
${\ displaystyle S \ \ Rightarrow \ NP \ VP \ \ Rightarrow \ {\ mathtt {John}} \ VP \ \ Rightarrow \ {\ mathtt {John}} \ V \ NP \ \ Rightarrow \ {\ mathtt {John \ hit }} \ NP \ \ Rightarrow \ {\ mathtt {John \ hit}} \ Det \ N \ Rightarrow \ {\ mathtt {John \ hit \ the}} \ N \ Rightarrow \ {\ mathtt {John \ hit \ the \ ball}}}$
With each of the steps you draw a piece of the syntax tree from top to bottom. But you can also apply the rules the other way around and start with the written sentence and build up the tree step by step from bottom to top.
### Derivation trees for unambiguous and ambiguous grammars
If there is more than one derivation tree for a word in the language of a grammar, one speaks of an ambiguous grammar , otherwise of an unambiguous one. For example, the following grammar is ambiguous
${\ displaystyle {\ begin {array} {lll} S & \ rightarrow & S \ S \\ S & \ rightarrow & a \ end {array}}}$
because you can divide "aa a" into two different ways: "[aa] a" and "a [aa]". However, only one possible classification allows the following grammar
${\ displaystyle {\ begin {array} {lll} S & \ rightarrow & a \ S \\ S & \ rightarrow & a \ end {array}}}$
In the case of ambiguous grammars, the number of possible derivation trees for one and the same word can increase sharply with the length of the word. In this case, derivation trees are no longer a suitable representation for the totality of possible derivations. In the case of formal languages, the concrete (surface) grammar is usually formulated clearly. Abstract grammars, on the other hand, are often ambiguous, whereby the uniqueness of the abstract derivation tree then results from the concrete through the course of the analysis.
## Abstract syntax trees
For the representation of syntax trees as a data structure in a computer, the term abstract syntax tree (AST) is now used quite uniformly, although the terminology here also fluctuates and z. B. also of abstract derivation trees , operator trees or the like. can be talked about. An exact connection between abstract syntax tree and concrete derivation tree is shown in the literature e.g. T. indicated. However, in addition to a coarsening of the derivation tree, requirements for further processing are also included in the structure, so that a direct formal derivation from the surface grammar is usually unsatisfactory as a result.
The context-free surface grammar is then opposed to an abstract grammar , which in the narrower sense is mostly an algebraic data type . The syntax trees are then technically represented as versatile terms . The analysis is in the transition between grammatical and algebraic-logical terms, so that one can speak fluently here of non-terminals and types or of trees and terms.
### example
Concrete (left) and abstract (right) syntax tree for the expression ${\ displaystyle a \ times (b + 3)}$
The illustration opposite shows concrete and abstract syntax trees for the following grammars.
concrete grammar abstract grammar algebraic type
E :: E "+" T - expression
:: T
T :: T "*" F - term
:: F
F :: V factor
:: N
:: "(" E ")"
V - variable
N - number
E :: E "+" E
:: E "*" E
:: V
:: N
type E = add (E, E);
mul (E, E);
var (V);
num (N)
The concrete grammar in this example must regulate the order in which the operators are applied to the (partial) expressions - that is, dot before dash and the partial expressions of the same priority are to be grouped together from left to right. The possibility of creating a different summary is also offered with expressions in brackets. Together with certain terminals (here "(", ")", "+", "*") these are merely properties of the syntactic surface that no longer play a role in later analysis and further processing. In particular, the distinction between different types of expressions (here E, T and F) and the key words can be completely dispensed with, as can be seen from the abstract syntax tree, which is also much closer to the "content" of the expression. Furthermore, due to these surface details, concrete derivation trees not only quickly become confusing, but also take up more storage space than necessary as a data structure in the computer due to their details. This is also reflected in the runtime and complexity of the programs that are later to process the derivation tree. For technical reasons, the breakdown of a source text is therefore usually not represented by a specific derivation tree.
### Representation of abstract syntax trees
In addition to the graphical representation as (operator) tree shown in the example, abstract syntax trees are also technically noted as terms , e.g. B .: mul(var('a'), add(var('b'), num(3))).
### Abstract grammar
While abstract syntax trees are data structures and algebraic types take on the role of grammar in them, in the literature, especially in connection with calculi, often only a coarse, ambiguous grammar is given, which, as shown in the example above, has the same structure as the terms but still contain keywords. This form enables a pleasant writing of abstract syntax trees, which is often very close to the actual source. Usually it is pointed out in the introduction that brackets may be used to disambiguate. An abstract syntax tree for the above example would then actually be a * (b + 3)written down as. In the context of this literature, however, the focus is always on the term. As mentioned, the boundaries between grammar and algebra are blurred by playing with form.
A typical example are the expressions in the lambda calculus , the abstract grammar of which is often only just written down as. The same technique is also used for extensive grammars. ${\ displaystyle E: = \ lambda VE \, | \, EE \, | \, V}$
## literature
• Ingo Wegener : Theoretical Computer Science . An algorithm-oriented introduction. BG Teubner, Stuttgart, ISBN 3-519-02123-4 , 6.1 Examples of context-free languages and syntax trees, p. 147-148 .
• Uwe Schöning : Theoretical Computer Science - in a nutshell . 5th edition. Spectrum Akademischer Verlag, Heidelberg, ISBN 978-3-8274-1824-1 , 1.1.4 Syntax trees, p. 15-17 .
• Juraj Hromkovič : Theoretical Computer Science . Formal languages, predictability, complexity theory, algorithms, communication and cryptography. 3. Edition. BG Teubner Verlag, Heidelberg, ISBN 978-3-8351-0043-5 , 10.4 Context-free grammars and push-down automata, p. 378 .
• Hans Zima: Compiler I . Analysis. Bibliographisches Institut, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01644-2 , 4.3 Abstract trees and their attribution, p. 216-229 .
• Stefan Müller : Grammatical Theory. From transformational grammar to constraint-based approaches. 2nd Edition. Language Science Press, Berlin 2018, ISBN 978-3-96110-074-3 , chap. 2 ( langsci-press.org ).
## Individual evidence
1. Müller (2018), p. 59f.
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# Math Help - Why does this not work?
1. ## Why does this not work?
we have the inequality x/(6x-9) <= 1/x , if we get rid of the denominators we get
x^2 <= 6x-9 , x!= 3/2 and 0
x^2-6x+9<=0
(x-3)^2 <=0
So the solution would be only at x=3. However the real solution includes 0<x<1.5 , but this does not work here? Now if we just subtract 1/x from x/(6x-9), it will give us the correct answer.
What did I do wrong, am I missing something in my approach?
2. ## Re: Why does this not work?
You need to remember that when you multiply or divide by negative numbers, the inequality sign changes its direction. So you will need to consider several different cases.
3. ## Re: Why does this not work?
Originally Posted by Prove It
You need to remember that when you multiply or divide by negative numbers, the inequality sign changes its direction. So you will need to consider several different cases.
but I did not multiply or divide by a negative number. So what gives?
4. ## Re: Why does this not work?
x is a VARIABLE, so it can take DIFFERENT values. There are some values of x where the denominator on the left (6x - 9) is negative, and obviously some values of x where the denominator on the right (x) is negative...
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Database
Not applicable
## Sum of Column_B for each distinct Column_A
How do I take a sum of one column (exposures) if and only if another column (ID) is distinct? Note that Exposure is a calculated field and will always be the same for a single ID.
I have a table that looks like this:
ID Exposure Claims
A 1 50
A 1 100
A 1 5
B .25 1000
C 1 10
C 1 100
C 1 500
C 1 252
Output: 2.25
Tags (1)
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## Introduction
In order to estimate behavioral states from telemetry (and biologging data) using this non-parametric Bayesian framework, the data must be formatted in a certain way to run properly. This especially applies to the initial analysis of the raw data by the segmentation model. This tutorial will walk through the different steps of preparing the raw telemetry data for analysis by the models within bayesmove.
## But first…
Before we begin in earnest, the practitioner must make sure the data have been cleaned. This includes the removal of duplicate observations and sorting the observations in consecutive order per animal ID. At a minimum, an object of class data.frame with columns for animal ID, date, x coordinate (e.g., longitude, Easting), and y coordinate (e.g., latitude, Northing) must be included.
## Calculating step lengths, turning angles, and time intervals
In many cases, step lengths and turning angles are used to estimate latent behavioral states from animal movement. Since these metrics are only directly comparable if measured on the same time interval, it is also important to calculate the time interval between successive observations since only those at the primary time interval of interest will be retained for further analysis.
First, let’s take a look at what the data should look like before calculating these data streams:
library(bayesmove)
library(dplyr)
#> Warning: package 'dplyr' was built under R version 4.0.5
library(ggplot2)
library(purrr)
library(tidyr)
#> Warning: package 'tidyr' was built under R version 4.0.5
library(lubridate)
data(tracks)
# Check data structure
#> id date x y
#> 1 id1 2020-07-02 11:59:41 0.00000 0.0000000
#> 2 id1 2020-07-02 12:58:26 10.56492 -1.6654990
#> 3 id1 2020-07-02 13:59:31 25.50174 -0.6096675
#> 4 id1 2020-07-02 15:01:27 31.22014 9.5438464
#> 5 id1 2020-07-02 15:59:56 36.15821 19.8737009
#> 6 id1 2020-07-02 16:58:38 39.06810 26.4996352
str(tracks)
#> 'data.frame': 15003 obs. of 4 variables:
#> $id : chr "id1" "id1" "id1" "id1" ... #>$ date: POSIXct, format: "2020-07-02 11:59:41" "2020-07-02 12:58:26" ...
#> $x : num 0 10.6 25.5 31.2 36.2 ... #>$ y : num 0 -1.67 -0.61 9.54 19.87 ...
We can see that ‘date’ is in a POSIXct format and that the x and y coordinates are stored as numeric variables. Technically, the coordinates can be in a lat-lon format, but this will make interpretation of step lengths more difficult since they will be recorded in map units. Therefore, it is suggested that x and y coordinates are in a UTM projection where the unit of measure is meters. The ‘id’ column can be stored either as character or factor.
Now, let’s calculate step length, turning angle, and time interval:
tracks<- prep_data(dat = tracks, coord.names = c("x","y"), id = "id")
#> id date x y step angle NSD dt
#> 1 id1 2020-07-02 11:59:41 0.00000 0.0000000 10.695 NA 0.000 3526
#> 2 id1 2020-07-02 12:58:26 10.56492 -1.6654990 14.974 0.227 114.392 3664
#> 3 id1 2020-07-02 13:59:31 25.50174 -0.6096675 11.653 0.987 650.710 3716
#> 4 id1 2020-07-02 15:01:27 31.22014 9.5438464 11.449 0.067 1065.782 3509
#> 5 id1 2020-07-02 15:59:56 36.15821 19.8737009 7.237 0.032 1702.380 3522
#> 6 id1 2020-07-02 16:58:38 39.06810 26.4996352 0.119 -2.804 2228.547 3738
The new tracks data frame has three new columns (‘step’, ‘angle’, and ‘dt’), which store the data for step lengths, turning angles, and time intervals, respectively. Since this example uses coordinates that are considered to be in a UTM projection, step lengths are reported in meters. Turning angles are reported in radians and time intervals are reported in seconds. Alternatively, these measures can also be calculated using functions from other R packages, such as adehabitatLT.
## Round times and filter observations
Next, we want to filter the data so that we only retain data for a given time interval (or step). Let’s look at a distribution of the time intervals that were just calculated:
Based on this histogram, it appears that 3600 s (1 hour) is likely the primary time interval, where some observations slightly deviate from this exact interval. I will now round all time intervals (dt) and dates to reflect this rounding of times within a given tolerance window. In this example, I will choose 3 minutes (180 s) as the tolerance on which to round observations close to the primary time interval (3600 s).
tracks<- round_track_time(dat = tracks, id = "id", int = 3600, tol = 180, time.zone = "UTC",
units = "secs")
#> id date x y step angle NSD dt
#> 1 id1 2020-07-02 11:59:41 0.00000 0.0000000 10.695 NA 0.000 3600
#> 2 id1 2020-07-02 12:59:41 10.56492 -1.6654990 14.974 0.227 114.392 3600
#> 3 id1 2020-07-02 13:59:41 25.50174 -0.6096675 11.653 0.987 650.710 3600
#> 4 id1 2020-07-02 14:59:41 31.22014 9.5438464 11.449 0.067 1065.782 3600
#> 5 id1 2020-07-02 15:59:41 36.15821 19.8737009 7.237 0.032 1702.380 3600
#> 6 id1 2020-07-02 16:59:41 39.06810 26.4996352 0.119 -2.804 2228.547 3600
# How many different time intervals?
n_distinct(tracks$dt) #> [1] 112 # How many observations of each time interval? hist(tracks$dt, main = "Rounded Time Intervals (s)")
It looks like nearly all observations had a time interval within the tolerance limit. Now the dataset needs to be filtered to only include observations where dt == 3600.
# Create list from data frame
tracks.list<- df_to_list(dat = tracks, ind = "id")
# Filter observations
tracks_filt.list<- filter_time(dat.list = tracks.list, int = 3600)
# View sample of results
#> id date x y step angle NSD dt
#> 1 id3 2020-07-02 12:01:26 0.000000e+00 0.000000e+00 0.000 NA 0.000 3600
#> 2 id3 2020-07-02 13:01:26 -6.164902e-05 -6.576507e-05 0.145 2.728 0.000 3600
#> 3 id3 2020-07-02 14:01:26 1.336034e-01 5.716719e-02 0.001 -2.109 0.021 3600
#> 4 id3 2020-07-02 15:01:26 1.335014e-01 5.640661e-02 3.321 -0.842 0.021 3600
#> 5 id3 2020-07-02 17:04:44 -2.601031e+00 -1.807549e+00 11.272 0.120 10.033 3600
#> 6 id3 2020-07-02 18:04:44 8.594375e+00 -4.939411e-01 0.000 2.125 74.107 3600
#> obs time1
#> 1 1 1
#> 2 2 2
#> 3 3 3
#> 4 4 4
#> 5 6 5
#> 6 7 6
# Check that only observations at 1 hour time intervals are retained per ID
purrr::map(tracks_filt.list, ~n_distinct(.$dt)) #>$id1
#> [1] 1
#>
#> $id2 #> [1] 1 #> #>$id3
#> [1] 1
There are also two new columns that have been added to the data frame of each ID: ‘obs’ and ‘time1’. The ‘obs’ column holds the number of the observation before the data were filtered, whereas ‘time1’ stores the number of the observation after filtering. This is important since the ‘obs’ column will allow the merging of results from this model with that of the original data and the ‘time1’ column will be used to segment the tracks.
## Discretize data streams
The unique feature of this modeling framework is that it does not rely upon standard parametric density functions that are used in nearly every other model that estimates behavior. This is expected to reduce any constraints posed by the selection of a given parametric density function and allow for greater model flexibility. However, this does require the user to define the number of bins for each variable and how they will be discretized.
Let’s first take a look at how step lengths and turning angles are distributed:
We can see that step lengths are highly right-skewed whereas turning angles are a little more balanced despite having peaks at $$-\pi, 0$$, and $$\pi$$ radians. These distributions will inform how we discretize these variables.
An example is included below for the discretization of step lengths and turning angles, but this can be performed in many different ways.
# Define bin number and limits for turning angles
angle.bin.lims=seq(from=-pi, to=pi, by=pi/4) #8 bins
# Define bin number and limits for step lengths
dist.bin.lims=quantile(tracks[tracks$dt == 3600,]$step,
c(0,0.25,0.50,0.75,0.90,1), na.rm=T) #5 bins
angle.bin.lims
#> [1] -3.1415927 -2.3561945 -1.5707963 -0.7853982 0.0000000 0.7853982 1.5707963
#> [8] 2.3561945 3.1415927
dist.bin.lims
#> 0% 25% 50% 75% 90% 100%
#> 0.00000 0.10700 1.27900 5.73825 10.75250 25.25200
Bins were defined in different ways for each data stream due to their inherent properties. Step lengths were broken into 5 bins (6 limits) using quantiles, which assisted in creating a more balanced distribution of bins since step lengths are typically right-skewed. Only step lengths that were observed at the primary time interval (3600 s; 1 h) were used to calculate quantiles. Since turning angles are already relatively balanced, these were separated into 8 bins (9 limits) of equal width centered at 0 radians (from $$-\pi$$ to $$\pi$$). The following code shows how to use these limits to discretize the data:
# Assign bins to observations
tracks_disc.list<- map(tracks_filt.list,
discrete_move_var,
lims = list(dist.bin.lims, angle.bin.lims),
varIn = c("step", "angle"),
varOut = c("SL", "TA"))
The plots below show the limits used to define the bins for each continuous variable, as well as what these distributions look like after discretization. First, step lengths will be shown:
#> summarise() has grouped output by 'key'. You can override using the .groups
#> argument.
And next the plots for turning angles:
The data are now in the proper format to be analyzed by the segmentation model within bayesmove. While these bin limits are suggested as default settings, they are by no means the only way to discretize the data. This may require trial and error after viewing results from the segmentation model.
|
# Why does the midi system have the widest range of frequencies
###### Question:
Why does the midi system have the widest range of frequencies
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# If the sum of 3, 7, and x is 18, then the average (arithmet
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If the sum of 3, 7, and x is 18, then the average (arithmet [#permalink] 11 Jun 2019, 02:37
Expert's post
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Question Stats:
80% (00:15) correct 20% (00:52) wrong based on 15 sessions
If the sum of 3, 7, and x is 18, then the average (arithmetic mean) of 3, 7, and x is
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
[Reveal] Spoiler: OA
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Re: If the sum of 3, 7, and x is 18, then the average (arithmet [#permalink] 11 Jun 2019, 08:11
Expert's post
Carcass wrote:
If the sum of 3, 7, and x is 18, then the average (arithmetic mean) of 3, 7, and x is
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
Average $$= \frac{3+7+x}{3}$$
Since we're told that the sum of 3, 7, and x is 18, we can replace the numerator with 18 to get:
Average $$= \frac{18}{3}=6$$
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Re: If the sum of 3, 7, and x is 18, then the average (arithmet [#permalink] 11 Jun 2019, 08:11
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Automakers develop consensus list of priority locations for next 19 H2 fueling stations in California
16 June 2015
The automaker OEM Advisory Group (OEM AG) of the California Fuel Cell Partnership (CaFCP) has developed a consensus list of recommended station priority locations for the next 19 hydrogen stations to be built in California. The OEM AG members are American Honda, General Motors, Hyundai, Mercedes-Benz, Nissan, Toyota and Volkswagen.
The priority locations represent general geographic areas that the OEM AG suggests be considered by the California Air Resources Board (ARB), station developers and the California Energy Commission (CEC) in planning the next phase of hydrogen station network development in California. (Earlier post.) The automakers consider their recommendations as preliminary and expect to refine them further through subsequent analysis and further consultation with stakeholders prior to future solicitations.
To prepare the recommendations, the automakers worked individually to ascertain station deployment for their own market needs, then shared the data independently in a double-blind process. The data was then compiled into an aggregate list. The automakers then collaboratively reviewed the data to refine the cluster and regional infrastructure needs.
The automakers worked to ensure:
1. customer travel-time to the nearest hydrogen station is minimized within a regional market;
2. network coverage is sufficiently robust for inter-market travel;
3. increased network capacity; and
4. creation of redundancy in the network.
The recommendations focus on building hydrogen fueling network coverage and redundant capacity throughout the Northern California, Southern California and Central Valley regions.
OEM AG recommendations
(In alphabetical order, not priority ranked)
Primary Priority Secondary Priority
Berkeley/Richmond/Oakland
Beverly Hills/Westwood
Fremont
Lebec*
Manhattan Beach
Sacramento
San Diego #2
San Diego #3
San Francisco
Thousand Oaks/Agoura Hills
Torrance/Palos Verdes
Culver City
Dublin/Pleasanton
Encino/Sherman Oaks/Van Nuys
Irvine South
Los Banos*
Palm Springs
Ventura/Oxnard
* These two locations will further strengthen the I-5 corridor.
In 2012, CaFCP published A California Road Map: Bringing Hydrogen Fuel Cell Vehicles to the Golden State (earlier post) that concluded that California would need 68 hydrogen fueling stations in five geographic clusters in which most early adopters are expected to support the roll-out of fuel cell vehicles. These cluster communities are:
• Berkeley
• South San Francisco Bay Area
• Santa Monica and West LA
• Torrance and nearby coastal communities
• Irvine and southern Orange County
An update in 2014, Hydrogen Progress, Priorities and Opportunities (HyPPO), estimated California will reach 100 stations in 2021. HyPPO also discussed quantifiable improvements in cost reductions, investment strategies and station technology since 2012, while outlining a set of specific actions in six areas to further realize the establishment of the infrastructure required. (Earlier post.)
Good going (as usual with California). This a bare minimum good start. Some 150+ H2 stations would eventually be required to satify growing demands.
Wonder if they (H2 stations) could be linked with H2 pipelines or fed with H2 tank trucks, or with H2 made and compressed locally or a mix of all three?
IMHO this is a waste of time and thus money; but then, I'm biased toward BEVs and believe hydrogen is a scheme to continue the current system of Oil Companies controlling the U.S. energy market. What do I know?
Let's just require that we use renewable hydrogen or we are really just running on dirty fracked gas coming from oil companies.
We have no Oil or NG (yet) but a huge surplus of clean electricity (95% hydro and 5% wind)....CPPs, NPPs and NGPPs were all turn off and decommissioned. A lot more can be developed.
There are enough local objections against Oil and NG shale operations to keep them underground for a long time.
The ideal for us would be a mix of BEVs + FCEVs for extended range during our long cold winters and as a mean to store surplus REs.
It's likely wasted money but unless we give H2 FCEVs a chance to prove themselves in the real world we're going to have to listen to fans talk about how wonderful they would be if we would only give them a chance.
Toyota's going to sell a few hundred FCEVs at a $50k loss and give away fuel for the first two years. (A good way to disguise the actual operating cost.) Let's see how they sell. Greenwashing scam The final solutions may very be with 5-5-5- batteries (in about 2025 or so) and even better with 10-10-10 batteries (in about 2035 or so. Meanwhile, interim solutions such as (1) HEVs have already prospered, (2) PHEVs also, (3) improved ICEVs seems to have a second chance. (4) short range BEVs are being rolled out. (5) a few real expensive extended range BEVs (TESLA's) are coming out. 6) very few extended range PCEVs are also coming out. Finally, only 5) and 6) may compete for the ground vehicles mass market? Deck chairs...Titanic. Hey, maybe if we not only rearrange them but also add a couple more! :) They can produce bio-propane and can use it in a SOFC, a 30KWH battery combined with a 30KW fuel cell is a vehicle most could live with - a 100mile battery range and if you have to go father hit a button and start your fuel cell 30 KW will push a midsize at a steady 70mph. Propane is pretty clean and Bio-propane would be carbon neutral, propane is energy dense and doesn't need an exotic high pressure tank or brand new infrastructure. We do need a few advances in SOFCs working temperatures have been lowered from 900C to 600C but still needs to get down to around 350C. The usual stuff in the comments which assume that it is possible to run a society with a lot of renewables without a huge input from storage. There are AFAIK absolutely no detailed plans which show that anything like that is possible. All of those involve huge need for storage, with hydrogen the biggest one. Here is a recent Finnish study: http://phys.org/news/2015-06-fully-renewable-energy-economically-viable.html Not liking hydrogen does not excuse simply ignoring what is possible, and what isn't. If it were EP posting, it is a different matter, as he advocates building lots of nuclear, which would mean not needing the hydrogen pathway to any large extent. They aren't building them though, and there is no prospect of their doing so to any large extent in the West. So I favour what can actually be made to work. Others seem to prefer ignoring the reality that lots and lots of renewables can't be done without hydrogen. As an aside, these are some of the stations to fuel the cars both of which opponents declared confidently would never be built. Well, they are building them, which is a win for reality. In addition to ignoring winter, died in the wool opponents of fuel cells and hydrogen contrive to ignore the 50% of people with nowhere to plug a car in. They are not going to accept greatly reduced convenience and fool around as a BEV enthusiast might to get a charge. One universal solution does not work. Davemart, "the 50% of people with nowhere to plug a car in" I'm not convinced that 50% of car owners live off the grid. The lack of charging stations is a short-term problem. It can easily be remedied if there is sufficient demand. Yes Davemart...over 50% of potential BEV owners have moved (or are moving) to cities where easy access to Level II charging energy level is not readily available. Even in our 100% electric large condo building, it would cost about$5.5K each to have the internal electrical system modified and the main 25,000 VAC step down transformer may have to be changed. That could cost another $2K+ each. Secondly, there are very few public quick charge and/or level II stations in operation. Since we have huge surpluses of clean (hydro + wind) energy, H2 stations may be an interesting complement/alternative, specially for our long cold winters when wind production picks up. The total cost would certainly not be more than gasoline and ICEVs at$1.35L let along the environmental/pollution/health cost.
Public H2 stations could get electricity 24/7 at $0.03/kWh and for less off peak demand hours. The$50,000,000 or more that it will take to build these 19 H2 fuel stations would build over 300 Tesla Supercharging stations. With the 200 mile Chevrolet Bolt and Tesla Model 3 coming out within a few years, it will be increasingly difficult to make a rational case for creating an entirely new fuel infrastructure with fuel costs 10x and dispensing station costs 16x that of the competition.
When the number of cars and stations are very small, it's easy to obfuscate the real cost with non disclosure agreements on the fuel cost. But as more stations and cars are built, the slight of hand gets noticed. Consumers are smarter than that.
ECIC...with very low cost excess/surplus clean electricity (which we have), H2 will eventually be produced as cheap, if not cheaper, that dirty imported diesel/gasoline.
By adding the very high environmental/health cost to fossil (and bio) fuels, clean H2 may quickly become cheaper and would fit well with our cold winters and long travel distances.
I admit that conditions are different in NYC and many other US cities where dirty electricity goes for $0.30/kWh instead of$0.03/kWh. That is not our case and will not be for years.
We would need 5-5-5 or even 10-10-10 batteries to manage our very cold winter days.
HD> H2 will eventually be produced as cheap...
I'd love to see the whole system costs beat conventional liquid fuels, and if that happened from renewable sources I'd be a supporter. But the likelihood of that happening before someone solves the thermal management problem for batteries in very cold climates seems low.
Is the problem really much different than block heaters?
Bernard:
The 50% of people without easy access to charging refers to those, even in the US, who park their cars beside the road not in a garage.
Worldwide the percentage would be far higher.
If you think providing access for them is trivial, you have not studied the matter.
And if you think that most people who do not have easy access to at home charging are going to fool around to an unlimited extent to save the few dollars a week that fuel costs ex tax you are also mistaken.
Whatever BEV enthusiasts think, cars are a convenience, and people are not going to sacrifice convenience to use batteries.
eci said:
'When the number of cars and stations are very small, it's easy to obfuscate the real cost with non disclosure agreements on the fuel cost. But as more stations and cars are built, the slight of hand gets noticed. Consumers are smarter than that. '
Well, in your case hopefully you are now able to distinguish between prices pre and post tax, as you had based your notions on confusing the two.
You ignore that many pathways may lead to hydrogen from renewables being competitive with petrol per mile driven.
You also ignore that if there is a problem on hydrogen costs PHEV configurations such as the prototype Audi have made could also be used.
Most people are smart enough to know that the major car companies probably know more about how to build cars which work and can be economically fuelled than some guy who won't put his own name to his comments and who runs a blog apparently devoted to the uncritical promotion of Tesla.
Hmmm, well you guys say that we need H2 for energy storage and you might be right.
But that is very different than trying to use H2 for a major part of our transportation infrastructure. For light duty vehicles, at best it's misguided and at worst, it's a scam propagated by special interests, mainly oil and gas industry who are trying to keep us buying transportation fuel from them.
Second, if H2 is really the right storage medium to help us through our colder winter days then that will become viable. But as much as you guys disparage batteries for that use, there are a lot of projects underway in the commercial and utility sector where they're using batteries because...they think it's the right solution. Not because they are getting subsidized to do it.
If H2 is really cheaper and more efficient, it will win plenty of those bids as well.
@DaveD..I have nothing against BEVs and I would gladly buy one or two when they become a practical transportation solution in our very cold area.
Right now they are better suited as a summer short range vehicle.
We will need BEVs with affordable 120+ kWh battery pack and road side quick charge stations to operate safely in winter times.
It is probably coming but it is not really here yet.
Nothing makes it more clear that you've run out of convincing arguments, Davemart, than when you launch into personal attacks. That seems to be a majority of the time with Hydrogen. I guess it's hard to think of something interesting to say given the scorecard of FCVs vs BEVs and PHEVs after several decades of R&D.
If H2 is a commercially viable fuel, why are dispensers asked to sign non-disclosure agreements about the hydrogen fuel cost? How is hiding the price from your end user customer helpful?
@ Bernard (and backing up Dave M)....One paper showing ~50% of US can plug in a PEV says: "study 1 estimates that about half of new car-buying US households park at least one vehicle within 25 ft of a Level 1 (110/120 V) electrical outlet at home"
Axsen, J. and K.S. Kurani (2012). Who can recharge a plug-in electric vehicle at home? Transportation Research Part D, 17(5), 349-353. Available at: http://www.rem.sfu.ca/people/faculty/jaxsen/
There are some arguments that haven't been made yet on H2:
1. what are we doing about medium and heavy duty vehicles that use ~50 billion gal/diesel a year and are responsible for a considerable amount of air quality problems (in addition to GHGs)? You can electrify some, improve efficiency, and try to use as much low GHG biofuels but... (a) you can't electrify everything (b) biofuel carbon intensities are increasingly contentious and won't be settled any time soon and (c) efficiency+electrification won't get us to an 80% reduction in GHGs like the IPCC says we need to stave off a 2 deg C temp change.
2. Even a high range 200-mile Bolt (with an effective range of ~150 miles) will not satisfy range needs of something like 30-40% of drivers if you assume people are willing to charge once per day and are willing to be "inconvenienced" by lack of range a couple times a year. Battery swapping and inductive roadway charging are all nice options but show me they work and are cheaper than a H2 infrastructure (estimated at ~$40-50 billion net present value in the 2013 NAS transitions study). What to do? I don't like building a whole new infrastructure either, but seems our only choice. If cities can install street lamps and parking meters, that are capable of installing chargers wherever cars are parked. With inductive charging, there are not even any exposed wires. The idea that a new refueling infrastructure that costs an order of magnitude more than electric, with fuel that is also enormously more expensive and can not be dispensed automatically overnight or while at work, from rooftop or solar canopy arrays, is more desirable, is a puzzler only for people without economic ties to the fossil fuel / H2 industry. US Department of Energy estimates$500 billion to \$1 trillion for a national H2 infrastructure.
The comments to this entry are closed.
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## mvn: Automagically create a Docker image
Having a Docker image of your software projects may make things easier for you, but will for sure lower the barrier for users to use your tools — at least in many cases ;-)
I am developing many tools in Java using Maven to manage dependencies. Thus, I’ve been looking for means to generate corresponding Docker files using the very same build management. There are already a few approaches to build Docker images through Maven, e.g. alexec’s docker-maven-plugin, and fabric8io’s docker-maven-pluginand so on — just to name a few. However, all theses solutions seem super-heavy and they require learning new syntax and stuff, while it is so easy and doesn’t require any third party plugins.
## Build Docker images using maven-antrun
Maven’s antrun plugin allows for execution of external commands. That means, you just need to maintain a proper Dockerfile along with your sources and after building the tool with maven you can call the docker-build command to create a Docker image of the current version of your tool.
I did that for a Java web application. The Dockerfile is stored as a resource in src/main/docker/Dockerfile is actually very simple:
### Build a Docker image
Using Maven’s antrun-plugin we can call the docker tool:
This executes a command like
after the deploy phase. Thus, it builds a docker image tagged with the current version of your tool. The build’s context is target, so it will use the target/Dockerfile which COPYs the new version of your tool into the image.
### Automatically build images using a Maven profile
I created a docker profile in Maven’s configuration file that is active per default if there is a src/main/docker/Dockerfile in your repository:
### Bonus: Also push the new image to the Docker Hub
To also push the image you need to execute the push command:
And due to the latest-confusion of Docker you also should create the latest-alias and also push that:
However, both is easy. Just append a few more exec calls in the antrun-plugin! The final pom.xml snippet can be found on GitHub.
## Create an Unscanable Letter
Some time ago I’ve heard about the EURion constellation. Never heard about it? Has nothing to do with stars or astrology. It’s the thing on your money! :)
Take a closer look at your bills and you’ll discover plenty of EURions, as shown in the picture on the right. Just a few inconspicuous dots. So what’s it all about? The EURion constellation is a pattern to be recognized by imaging software, so that it can recognize banknotes. It was invented to prevent people from copying money :)
But I don’t know of any law that prohibits using that EURion, so I’ve been playing around with it. Took me some trials to find the optimal size, but I was able to create a $$LaTeX$$ document that includes the EURion. That’s the essential tex code:
The whole $$LaTeX$$ environment can be found on GitHub, together with the EURion image and stuff. I also provide the resulting letter.
Of course I immediately asked some friends to try to scan the letter, but it turns out, that not all scanners/printers are aware of the EURion… So it’s a bit disappointing, but I learned another thing. Good good. And to be honest, I do not have a good use case. Why should I prevent someone from printing my letters? Maybe photographers can use the EURion in their images. Copyright bullshit or something…
## Monitoring of XOS devices
This week I developed some plugins for Nagios/Icinga to monitor network devices of the vendor Extreme Networks. All these plugins receive status information of, eg. switches, via SNMP.
## The Basic: Check Mem, CPU, and Fans
Checking for available memory, for the device’s temperature, for the power supplies, and for fan states is quite straight forward. You just ask the switch for the values of a few OIDs, evaluate the answer, and tell Nagios/Icinga what to do.
The Simple Network Management Protocol (SNMP) is actually a very easy to use protocol. There is an SNMP server, such as a router or a switch, which exposes management data through the SNMP protocol. To access these data you just send an object identify (OID) to an SNMP server and receive the corresponding value. So called management information bases (MIB) can tell you what a certain OID stands for.
On the command line, for example, you could use snmpwalk to iterate over an OID subtree to, e.g., obtain information about the memory on a device:
usr@srv $snmpwalk -v 2c -c publicCommunityString switch.address.com 1.3.6.1.4.1.1916.1.32.2.2.1 1.3.6.1.4.1.1916.1.32.2.2.1.1.1 = Gauge32: 1 1.3.6.1.4.1.1916.1.32.2.2.1.2.1 = STRING: "262144" 1.3.6.1.4.1.1916.1.32.2.2.1.3.1 = STRING: "116268" 1.3.6.1.4.1.1916.1.32.2.2.1.4.1 = STRING: "7504" 1.3.6.1.4.1.1916.1.32.2.2.1.5.1 = STRING: "138372" The OID 1.3.6.1.4.1.1916.1.32.2.2.1 addresses the memory information table of the SNMP provider at switch.address.com. The value at *.2.1 shows how much memory is installed, *.3.1 shows how much memory is free, *.4.1 shows how much is consumed by the system, and *.5.1 shows how much is consumed by user processes. Basic calculations tell us there are 262144/1024 = 256KB in total and 100*116268/262144 = 44.35% is free. A bit more logic for a warning/critical switch and the plugin is done. ## The Feature: Monitoring of the FDB But I would probably not write about that basic stuff if there was not an extra feature! I implemented a script to also monitor the FDB. FDB is and abbreviation for forwarding databases: The switch maintains a forwarding database (FDB) of all MAC addresses received on all of its ports. It, for example, uses the information in this database to decide whether a frame should be forwarded or filtered. Each entry consists of • the MAC address of the device behind the port • the associated VLAN • the age of the entry – depending on the configuration the entries age out of the table • some flags – e.g. is the entry dynamic or static • the port The table may look like the following: > show fdb Mac Vlan Age Flags Port / Virtual Port List ------------------------------------------------------------------------------ 01:23:45:67:89:ab worknet(0060) 0056 n m 9 01:23:42:67:89:ab mobnet(0040) 0001 n m 21 Flags : d - Dynamic, s - Static, p - Permanent, n - NetLogin, m - MAC, i - IP, x - IPX, l - lockdown MAC, L - lockdown-timeout MAC, M- Mirror, B - Egress Blackhole, b - Ingress Blackhole, v - MAC-Based VLAN, P - Private VLAN, T - VLAN translation, D - drop packet, h - Hardware Aging, o - IEEE 802.1ah Backbone MAC, S - Software Controlled Deletion, r - MSRP As soon as the switch gets a frame on one port it learns the corresponding MAC address, port number, etc. into this table. So if a frame for this MAC address arrives it know where to send it to. However, that content of a networking class. All we need to know is that a switch can tell you which device which MAC address is is connected to which port. And that’s the idea of check_extreme_fdb.pl! It compares the entries of the FDB with some expected entries in an CSV file. The CSV is supposed to contain three coloumns: mac,port,vlan If a MAC address in the FDB matches the MAC address in the CSV file it checks the ports and vlans. If those do not match, it will raise an error. For the CSV: Feel free to leave port or vlan empty if you do not care about this detail. That means, if you just want to make sure that the device with the MAC 01:23:45:67:89:ab is in vlan worknet you add an entry such as: 01:23:45:67:89:ab,,worknet Use -e <FILE> to pass the CSV file containing expected entry to the program and call it like beckham: perl -w check_extreme_fdb.pl -s <SWITCH> -C <COMMUNITY-STRING> -e <EXPECTED> Here, SWITCH being the switch’s address and COMMUNITY-STRING beeing the SNMP “passphrase”. You may also want to add -w to raise a warning if one of the entries in the CSV file wasn’t found in the FDB. To create a sample CSV file that matches the current FDB you can call it with --print. To get the script have a look at the check_extreme_fdb.pl software page. ## More Extreme Stuff In addition there are some other scripts to monitor Extreme Networks devices: ## Do I have a CD-RW? You don’t know whether the CD drive on your machine is able to burn CDs? And too lazy to go off with your head under your table? Or you’re remote on the machine? Then that’s your command line: $ cat /proc/sys/dev/cdrom/info
CD-ROM information, Id: cdrom.c 3.20 2003/12/17
drive name: sr0
drive speed: 32
drive # of slots: 1
Can close tray: 1
Can open tray: 1
Can lock tray: 1
Can change speed: 1
Can select disk: 0
Reports media changed: 1
Can play audio: 1
Can write CD-R: 1
Can write CD-RW: 1
Can write DVD-R: 1
Can write DVD-RAM: 1
Can write MRW: 1
Can write RAM: 1
Works on Debian based systems :)
## Docker Jail for Skype
As I’m now permanently installed at our University (yeah!) I probably need to use skype more often than desired. However, I still try to avoid proprietary software, and skype is the worst of all. Skype is an
obfuscated malicious binary blob with network capabilities
as jvoisin beautifully put into words. I came in contact with skype multiple times and it was always a mess. Ok, but what are the options if I need skype? So far I’ve been using a virtual box if I needed to call somebody who insisted on using skype, but now that I’ll be using skype more often I need an alternative to running a second OS on my machine. My friend Tom meant to make a joke about using Docker and … TA-DAH! … Turns out it’s actually possible to jail a usable skype inside a Docker container! Guided by jvoisin’s article Running Skype in docker I created my own setup:
# The Dockerfile
The Dockerfile is available from the skype-on-docker project on GitHub. Just clone the project and change into the directory:
$git clone https://github.com/binfalse/skype-on-docker.git$ cd skype-on-docker
$ls -l total 12 -rw-r--r-- 1 martin martin 32 Jan 4 17:26 authorized_keys -rw-r--r-- 1 martin martin 1144 Jan 4 17:26 Dockerfile -rw-r--r-- 1 martin martin 729 Jan 4 17:26 README.md The Docker image is based on a Debian:stable. It will install an OpenSSH server (it exposes 22) and download the skype binaries. It will also install the authorized_keys file in the home directories of root and the unprivileged user. Thus, to be able to connect to the container you need to copy your public SSH key into that file: $ cat ~/.ssh/id_rsa.pub >> authorized_keys
Good so far? Ok, then go for it! Build a docker image:
$docker build -t binfalse/skype . This might take a while. Docker will execute the commands given in the Dockerfile and create a new Docker image with the name binfalse/skype. Feel free to choose a different name.. As soon as that’s finished you can instantiate and run a new container using: $ docker run -d -p 127.0.0.1:55757:22 --name skype_container binfalse/skype
This will start the container as a daemon (-d) with the name skype_container (--name skype_container) and the host’s port 55757 mapped to the container’s port 22 (-p 127.0.0.1:55757:22). Give it a millisecond to come up and then you should be able to connect to that container via ssh. From that shell you should be able to start an configure skype:
$ssh -X -p 55555 [email protected] The programs included with the Debian GNU/Linux system are free software; the exact distribution terms for each program are described in the individual files in /usr/share/doc/*/copyright. Debian GNU/Linux comes with ABSOLUTELY NO WARRANTY, to the extent permitted by applicable law. Last login: Mon Jan 4 23:07:37 2016 from 172.17.42.1$ skype
You can immediately go and do your chats and stuff, but you can also just configure skype. Do setup everything just like you want to find it when starting skype, for example tick the auto-login button to get rid of the login screen etc. As soon as that’s done, commit the changes to build a new image reflecting your preferences:
$docker commit skype_container binfalse/deb-skype Now you’ll have an image called binfalse/deb-skype that contains a fully configured skype installation. Just kill the other container: $ docker stop skype_container
\$ docker rm skype_container
And now your typical workflow might look like:
docker run -d -p 127.0.0.1:55757:22 --name skype__ binfalse/deb-skype
sleep 1
ssh -X -p 55757 [email protected] skype && docker rm -f skype__
Feel free to cast it in a mould just as I did. The script is also available from my apt repo, it’s name is bf-skype-on-docker:
echo "deb http://apt.binfalse.de binfalse main" > /etc/apt/sources.list.d/binfalse.list
apt-get update && apt-get install bf-skype-on-docker
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# Mass Spectrometer Question!
1. Oct 28, 2005
### 123Sub-Zero
1.Some data obtained from the mass spectrum of a sample of carbon are given below.
Ion 12C+ 13C+
Absolute mass of one ion/g 1.993 x 10-23 2.158 x 10-23
Relative abundance/ % 98.9 1.1
Use the data to calculate the mass of one neutron, the RAM of 13C and the RAM of carbon in the sample.
You may neglect the mass of an electron.
Work out:
1.Mass of one neutron
2.RAM of 13C
3.RAM of carbon in the sample
I know how to work out the RAM form a mass spectrometer graph but cannot work this out. I presume you multiply the mass of one ion by relative abundance to (for the RAM questions) but then dont know the next step.
2. Nov 13, 2005
$$A_{r} of samples:\\ ^{13}C:~~\frac{13 x 1.1}{100} = 1.43 x 10^{-1} = 0.143\\ Carbon:~~\frac{(12 x 98.9)~+~(13 x 1.1)}{(98.9 + 1.1)} = 12.011 = 12$$
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# NAG C Library Function Document
## 1Purpose
nag_ranks_and_scores (g01dhc) computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.
## 2Specification
#include #include
void nag_ranks_and_scores (Nag_Scores scores, Nag_Ties ties, Integer n, const double x[], double r[], NagError *fail)
## 3Description
nag_ranks_and_scores (g01dhc) computes one of the following scores for a sample of observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
1. Rank Scores
The ranks are assigned to the data in ascending order, that is the $i$th observation has score ${s}_{i}=k$ if it is the $k$th smallest observation in the sample.
2. Normal Scores
The Normal scores are the expected values of the Normal order statistics from a sample of size $n$. If ${x}_{i}$ is the $k$th smallest observation in the sample, then the score for that observation, ${s}_{i}$, is $E\left({Z}_{k}\right)$ where ${Z}_{k}$ is the $k$th order statistic in a sample of size $n$ from a standard Normal distribution and $E$ is the expectation operator.
3. Blom, Tukey and van der Waerden Scores
These scores are approximations to the Normal scores. The scores are obtained by evaluating the inverse cumulative Normal distribution function, ${\Phi }^{-1}\left(·\right)$, at the values of the ranks scaled into the interval $\left(0,1\right)$ using different scaling transformations.
The Blom scores use the scaling transformation $\frac{{r}_{i}-\frac{3}{8}}{n+\frac{1}{4}}$ for the rank ${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. Thus the Blom score corresponding to the observation ${x}_{i}$ is
$si = Φ-1 ri - 38 n+14 .$
The Tukey scores use the scaling transformation $\frac{{r}_{i}-\frac{1}{3}}{n+\frac{1}{3}}$; the Tukey score corresponding to the observation ${x}_{i}$ is
$si = Φ-1 ri - 13 n+13 .$
The van der Waerden scores use the scaling transformation $\frac{{r}_{i}}{n+1}$; the van der Waerden score corresponding to the observation ${x}_{i}$ is
$si = Φ-1 ri n+1 .$
The van der Waerden scores may be used to carry out the van der Waerden test for testing for differences between several population distributions, see Conover (1980).
4. Savage Scores
The Savage scores are the expected values of the exponential order statistics from a sample of size $n$. They may be used in a test discussed by Savage (1956) and Lehmann (1975). If ${x}_{i}$ is the $k$th smallest observation in the sample, then the score for that observation is
$si = EYk = 1n + 1n-1 + ⋯ + 1n-k+1 ,$
where ${Y}_{k}$ is the $k$th order statistic in a sample of size $n$ from a standard exponential distribution and $E$ is the expectation operator.
Ties may be handled in one of five ways. Let ${x}_{t\left(\mathit{i}\right)}$, for $\mathit{i}=1,2,\dots ,m$, denote $m$ tied observations, that is ${x}_{t\left(1\right)}={x}_{t\left(2\right)}=\cdots ={x}_{t\left(m\right)}$ with $t\left(1\right). If the rank of ${x}_{t\left(1\right)}$ is $k$, then if ties are ignored the rank of ${x}_{t\left(j\right)}$ will be $k+j-1$. Let the scores ignoring ties be ${s}_{t\left(1\right)}^{*},{s}_{t\left(2\right)}^{*},\dots ,{s}_{t\left(m\right)}^{*}$. Then the scores, ${s}_{t\left(\mathit{i}\right)}$, for $\mathit{i}=1,2,\dots ,m$, may be calculated as follows:
• – if averages are used, then ${s}_{t\left(i\right)}=\sum _{j=1}^{m}{s}_{t\left(j\right)}^{*}/m$;
• – if the lowest score is used, then ${s}_{t\left(i\right)}={s}_{t\left(1\right)}^{*}$;
• – if the highest score is used, then ${s}_{t\left(i\right)}={s}_{t\left(m\right)}^{*}$;
• – if ties are to be broken randomly, then ${s}_{t\left(i\right)}={s}_{t\left(I\right)}^{*}$ where $I\in \left\{\text{random permutation of }1,2,\dots ,m\right\}$;
• – if ties are to be ignored, then ${s}_{t\left(i\right)}={s}_{t\left(i\right)}^{*}$.
## 4References
Blom G (1958) Statistical Estimates and Transformed Beta-variables Wiley
Conover W J (1980) Practical Nonparametric Statistics Wiley
Lehmann E L (1975) Nonparametrics: Statistical Methods Based on Ranks Holden–Day
Savage I R (1956) Contributions to the theory of rank order statistics – the two-sample case Ann. Math. Statist. 27 590–615
Tukey J W (1962) The future of data analysis Ann. Math. Statist. 33 1–67
## 5Arguments
1: $\mathbf{scores}$Nag_ScoresInput
On entry: indicates which of the following scores are required.
${\mathbf{scores}}=\mathrm{Nag_RankScores}$
The ranks.
${\mathbf{scores}}=\mathrm{Nag_NormalScores}$
The Normal scores, that is the expected value of the Normal order statistics.
${\mathbf{scores}}=\mathrm{Nag_BlomScores}$
The Blom version of the Normal scores.
${\mathbf{scores}}=\mathrm{Nag_TukeyScores}$
The Tukey version of the Normal scores.
${\mathbf{scores}}=\mathrm{Nag_WaerdenScores}$
The van der Waerden version of the Normal scores.
${\mathbf{scores}}=\mathrm{Nag_SavageScores}$
The Savage scores, that is the expected value of the exponential order statistics.
Constraint: ${\mathbf{scores}}=\mathrm{Nag_RankScores}$, $\mathrm{Nag_NormalScores}$, $\mathrm{Nag_BlomScores}$, $\mathrm{Nag_TukeyScores}$, $\mathrm{Nag_WaerdenScores}$ or $\mathrm{Nag_SavageScores}$.
2: $\mathbf{ties}$Nag_TiesInput
On entry: indicates which of the following methods is to be used to assign scores to tied observations.
${\mathbf{ties}}=\mathrm{Nag_AverageTies}$
The average of the scores for tied observations is used.
${\mathbf{ties}}=\mathrm{Nag_LowestTies}$
The lowest score in the group of ties is used.
${\mathbf{ties}}=\mathrm{Nag_HighestTies}$
The highest score in the group of ties is used.
${\mathbf{ties}}=\mathrm{Nag_RandomTies}$
The repeatable random number generator is used to randomly untie any group of tied observations.
${\mathbf{ties}}=\mathrm{Nag_IgnoreTies}$
Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.
Constraint: ${\mathbf{ties}}=\mathrm{Nag_AverageTies}$, $\mathrm{Nag_LowestTies}$, $\mathrm{Nag_HighestTies}$, $\mathrm{Nag_RandomTies}$ or $\mathrm{Nag_IgnoreTies}$.
3: $\mathbf{n}$IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the sample of observations, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
5: $\mathbf{r}\left[{\mathbf{n}}\right]$doubleOutput
On exit: contains the scores, ${s}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, as specified by scores.
6: $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
## 7Accuracy
For ${\mathbf{scores}}=\mathrm{Nag_RankScores}$, the results should be accurate to machine precision.
For ${\mathbf{scores}}=\mathrm{Nag_SavageScores}$, the results should be accurate to a small multiple of machine precision.
For ${\mathbf{scores}}=\mathrm{Nag_NormalScores}$, the results should have a relative accuracy of at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(100×\epsilon ,{10}^{-8}\right)$ where $\epsilon$ is the machine precision.
For ${\mathbf{scores}}=\mathrm{Nag_BlomScores}$, $\mathrm{Nag_TukeyScores}$ or $\mathrm{Nag_WaerdenScores}$, the results should have a relative accuracy of at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10×\epsilon ,{10}^{-12}\right)$.
## 8Parallelism and Performance
nag_ranks_and_scores (g01dhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
If more accurate Normal scores are required nag_normal_scores_exact (g01dac) should be used with appropriate settings for the input argument etol.
## 10Example
This example computes and prints the Savage scores for a sample of five observations. The average of the scores of any tied observations is used.
### 10.1Program Text
Program Text (g01dhce.c)
### 10.2Program Data
Program Data (g01dhce.d)
### 10.3Program Results
Program Results (g01dhce.r)
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# 600-cell
600-cell
Schlegel diagram, vertex-centered
(vertices and edges)
Type Convex regular 4-polytope
Schläfli symbol {3,3,5}
Coxeter diagram
Cells 600 (3.3.3)
Faces 1200 {3}
Edges 720
Vertices 120
Vertex figure
icosahedron
Petrie polygon 30-gon
Coxeter group H4, [3,3,5], order 14400
Dual 120-cell
Properties convex, isogonal, isotoxal, isohedral
Uniform index 35
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also called a C600, hexacosichoron and hexacosidedroid.[1]
The 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex (abbreviated from "tetrahedral complex") and polytetrahedron, being bounded by tetrahedral cells.
## Geometry
Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons.
The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = ${\displaystyle \pi /5}$, 60°= ${\displaystyle \pi /3}$, 72° = ${\displaystyle 2\pi /5}$, 90° = ${\displaystyle \pi /2}$, 108° = ${\displaystyle 3\pi /5}$, 120° = ${\displaystyle 2\pi /3}$, 144° = ${\displaystyle 4\pi /5}$, and 180° = ${\displaystyle \pi }$. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V. References: S.L. van Oss (1899); F. Buekenhout and M. Parker (1998).
Its vertex figure is an icosahedron, and its dual polytope is the 120-cell. It has a dihedral angle of cos-1( (2 - 5cos(π/15)) / 3) = ~164.48°.[2]
Each cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
## Coordinates
The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5) /2 is the golden ratio), can be given as follows: 16 vertices of the form:[3]
(±½, ±½, ±½, ±½),
and 8 vertices obtained from
(0, 0, 0, ±1)
by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of
½ (±φ, ±1, ±1/φ, 0).
Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.
When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion -Id. We have the isomorphism RSG ≅ (2IL × 2IR) / {Id, -Id}. The order of RSG equals 120 × 120 / 2 = 7200.
The binary icosahedral group is isomorphic to SL(2,5).
The full symmetry group of the 600-cell is the Weyl group of H4. This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss (1899); see References.
## Visualization
The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron, which with some effort can be seen in most of the below perspective projections.
A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.[4]
## Union of two tori
100 tetrahedra in a 10x10 array forming a clifford torus boundary in the 600 cell.
The 120-cell can be decomposed into two disjoint tori. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to a 10-vertex decagon path in the 600-cell. Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each "saucer" with 10 tetrahedra forming an icosahedron. You can view this as five, vertex stacked, icosahedral pyramids, with the five extra annular ring gaps also filled in. The surface is the same as that of ten stacked pentagonal antiprisms. You now have a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above. These paths spiral around the center core path, but mathematically they are all equivalent. Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a clifford torus. They can be "unrolled" into a square 10x10 array. Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb.
A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen stereographic projection A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection
There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra.
The 600-cell can be further partitioned into 20 disjoint intertwining rings of 30 cells and ten edges long each, forming a discrete Hopf fibration. These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix. Five such helices nest and spiral around each of the 10-vertex decagon paths, forming the initial 150 cell torus mentioned above.
This decomposition of the 600-cell has symmetry [[10,2+,10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of tetrahedra, similar to the belt of an icosahedron with the 5 top and 5 bottom triangles removed (pentagonal antiprism).
## Images
### 2D projections
The H3 decagonal projection shows the plane of the van Oss polygon.
Orthographic projections by Coxeter planes
H4 - F4
[30]
[20]
[12]
H3 A2 / B3 / D4 A3 / B2
[10]
[6]
[4]
### 3D projections
Vertex-first projection
This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
• The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
• The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image.
Cell-first projection
This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
• The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
• The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint have been culled for clarity.
This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
Stereographic projection (on 3-sphere)
Cell-Centered
Simple Rotation
A 3D projection of a 600-cell performing a simple rotation.
Frame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.
## Diminished 600-cells
The snub 24-cell may be obtained from the 600-cell by removing the vertices of an inscribed 24-cell and taking the convex hull of the remaining vertices. This process is a diminishing of the 600-cell.
The grand antiprism may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.
A bi-24-diminished 600-cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell.
## Related complex polygons
The regular complex polytopes 3{5}3, and 5{3}5, , in ${\displaystyle \mathbb {C} ^{2}}$ have a real representation as 600-cell in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group 3[5]3, order 360, and the second has symmetry 5[3]5, order 600.[5]
## Related polytopes and honeycombs
The 600-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:
It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, 16-cell {3,3,4} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.
This 4-polytope is a part of a sequence of 4-polytope and honeycombs with icosahedron vertex figures:
## Notes
1. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
2. ^ Coxeter, Regular polygons, p.293
3. ^
4. ^ Grossman, Wendy A.; Sebline, Edouard, eds. (2015), Man Ray Human Equations: A journey from mathematics to Shakespeare, Hatje Cantz. See in particular mathematical object mo-6.2, p. 58; Antony and Cleopatra, SE-6, p. 59; mathematical object mo-9, p. 64; Merchant of Venice, SE-9, p. 65, and "The Hexacosichoron", Philip Ordning, p. 96.
5. ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). pp.48-49
## References
• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]
• Oss, Salomon Levi van: Das regelmässige 600-Zell und seine selbstdeckenden Bewegungen. Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 Deel 7 Nummer 1 (Afdeeling Natuurkunde). Amsterdam: 1899. Online at URL [3], reachable from the home page of the KNAW Digital Library at URL [4]. REMARK: Van Oss does not mention the arc distances between vertices of the 600-cell.
• F. Buekenhout, M. Parker: The number of nets of the regular convex polytopes in dimension <= 4. Discrete Mathematics, Volume 186, Issues 1-3, 15 May 1998, Pages 69-94. REMARK: The authors do mention the arc distances between vertices of the 600-cell.
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# Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Proposition 7.4.3.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Then there exists a pullback diagram
$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, }$
where $\overline{U}$ is a cocartesian fibration and a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ which exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$.
Proof. Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. Applying Proposition 6.3.2.1, we deduce that there exists an $\infty$-category $\operatorname{\mathcal{E}}[W^{-1}]$ and a diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{E}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. In particular, the diagram $\mathrm{Rf}$ carries each $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ to an isomorphism in $\operatorname{\mathcal{E}}[W^{-1}]$. Let $\overline{\operatorname{\mathcal{E}}}$ denote the relative join $\operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}]$ (Construction 5.2.3.1). Applying Lemma 5.2.3.17 to the commutative diagram
$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \mathrm{Rf}} \ar [d]^{U} & \operatorname{\mathcal{E}}[W^{-1} ] \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \Delta ^0, }$
we deduce that vertical maps induce a cocartesian fibration
$\overline{U}: \overline{\operatorname{\mathcal{E}}} = \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] \rightarrow \operatorname{\mathcal{C}}\star _{\Delta ^0} \Delta ^0 \simeq \operatorname{\mathcal{C}}^{\triangleright }.$
By construction, we have a pullback diagram of simplicial sets
$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, }$
and the fiber of $\overline{U}$ over the cone point ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ can be identified with the $\infty$-category $\operatorname{\mathcal{E}}[W^{-1}]$. Moreover, $\mathrm{Rf}$ induces a morphism of simplicial sets
$H: \Delta ^1 \times \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] = \overline{\operatorname{\mathcal{E}}}$
for which $H|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$ is the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$, and $H|_{ \{ 1\} \times \operatorname{\mathcal{E}}}$ is the diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$. For every vertex $X \in \operatorname{\mathcal{E}}$, the criterion of Lemma 5.2.3.17 guarantees that $H|_{ \Delta ^1 \times \{ X\} }$ is a $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$, so that $H$ exhibits $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ as a covariant refraction diagram. $\square$
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# Speed of light and Time Travel
1. Mar 3, 2004
Well, We all know about Einstein Theory of Relativity.
The speed of light is constant.
And if someone travels close to the speed of light; time goes slower than someone walking on earth etc.
Ok, this is my question. (Involves Pilots)
Ok, since pilots travel at amazing speeds of 1,000 - 3,000 miles or higher in Military Jet planes and commercial planes and they experience 3-6 times the G force of earth. Let's say a pilot is flying around the world for 24 Hours straight let's assume.
So, in those 24 hours traveling at those fast speeds. Does the pilot age a bit slower than normal human aging during that 24 hour period?
Explain.
2. Mar 3, 2004
### Wooh
I believe that the answer is yes, but only very very very very very very slightly. I mean, how does 3000 compare to 300000000? And then, in many cases, 90000000000000000? Not much at all.
3. Mar 3, 2004
The pilots i seen so young and they are very old.
Specially this 79 year old men looking like in his 40's year old.
But, the others around 50's
He must of pulled a lot of G's in his flights. ;)
hehe.
4. Mar 3, 2004
### franznietzsche
$$t = \tau\gamma$$
$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$v = 14432 \frac{meters}{sec}$$
$$t = \frac{\tau}{\sqrt{1-\frac{14432^2}{299792456^2}}} = \frac{\tau}{\sqrt{1-2.317x10^{-9}}}$$
$$= \frac{\tau}{0.99999999884127} = 1.0000000011587\tau$$
given that the g forces are only experienced during acceleration and never exceed ten gs (for medical reasons) we can calculate:
$$ds^2 = 1-\frac{\frac{2GM}{c^2}}{R}dt^2$$
$$ds = \sqrt{1-\frac{8.86112x10^{-3}}{6x10^24}}dt$$
$$ds = \sqrt{1-1.47685x10^{-27}}dt$$
that is for 1 g, which the coefficient is obviously insinificant. If we make the earth ten times more massive(10 g basically) we get:
$$ds^2 = 1-\frac{\frac{2GM}{c^2}}{R}dt^2$$
$$ds = \sqrt{1-\frac{8.86112x10^{-2}}{6x10^24}}dt$$
$$ds = \sqrt{1-1.47685x10^{-26}}dt$$
Which remains insignificant. Any added g forces will have essentially nil effect, the special relativisitc effects of the high velocity however are measurable by atomic clocks, but not significant in terms of a human lifetime or any amount of time recognizable by humans. For example if you lived a hundred years by your watch at that velocity some remote observer would measure 100.00000011587 years, which amounts to a net difference of 3 seconds over 100 years. Not much.
Last edited: Mar 3, 2004
5. Mar 3, 2004
HEY franznietzsche!
I did not typed that second post.
I was away from the computer and my little immature brother sat down and post that.. i think he deleted some of my files..
kids.. bah...
6. Mar 3, 2004
### franznietzsche
meh, either way, was good excersize for me.
But yeah the pilot does age more slowly, just not much. Its very insignificant as the numers show.
7. Mar 4, 2004
### HallsofIvy
Staff Emeritus
Hey! I'm going to use that the next time I suddenly realize that I had just said something stupid!
8. Mar 4, 2004
### ZapperZ
Staff Emeritus
Hehe... GREAT idea! I'll use that too, except I'll blame it on my evil twin Skippy! :)
Zz.
9. Mar 4, 2004
hmm..
HallsofENVY you think you're some kind of physic?
Ook , ms cleooo^^^
10. Mar 4, 2004
### pallidin
Damn. So, one would have to travel at a velocity 10.5 million times that of a jet for 100 years to "gain" a single year due to time dilation? That's if I even have my math right.
In any event, I can see that we don't fly jets for longevity.
11. Jul 8, 2010
### dave_baksh
From Franz's post above - am I right in thinking that the centrapeital force experienced from loop de looping in a plane (or in any other way) contributes towards time dilation in the same fashion as being in a gravitational field?
Because I've never heard that before (my knowledge of special relativity is reasonable and of general, minimal.)
12. Sep 7, 2010
### SANDU
Sir
if we travel around earth in opposite motion for 20 yrs then could we can go to past
13. Sep 7, 2010
### HallsofIvy
Staff Emeritus
A very strange assertion. Do you have any evidence to support it?
14. Sep 7, 2010
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New Conduction and Reversible Memory Phenomena in Thin Insulating Films
J. G. Simmons, R. R. Verderber
Abstract
It has been observed that when thin (200 to 3000 $\overset{\circ} {\mathrm A}$) film insulators have been formed by the electrolytic introduction of gold ions from one of the electrodes they can draw appreciable currents. They then show temperature-independent conductivity, voltage-controlled negative resistance and reversible voltage and thermal-voltage memory effects. It is postulated that the injected ions introduce a broad band of localized impurity levels within the normally forbidden band of the insulator. The electrons are assumed to move through the insulator by tunnelling between adjacent sites within the impurity band; it is also assumed that the electrons can, under certain conditions, be trapped within the impurity band. A model based on these ideas accounts in a self-consistent manner for all the experimental observations, and calculations of current-voltage characteristics based on the model are in fact in agreement with them.
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# Charged Black Holes: The Reissner-Nordström Geometry
Back to Collapse to a Black Hole
Forward to The Extremal Reissner-Nordström Geometry
index | movies | approach | orbit | singularity | dive | Schwarzschild | wormhole | collapse | Reissner-Nordström | extremal RN | Hawking | quiz | home | links
Reissner-Nordström geometry The Reissner-Nordström geometry describes the geometry of empty space surrounding a charged black hole. If the charge of the black hole is less than its mass (measured in geometric units $$G = c = 1$$), then the geometry contains two horizons, an outer horizon and an inner horizon. Between the two horizons space is like a waterfall, falling faster than the speed of light, carrying everything with it. Upstream and downstream of the waterfall, space moves slower than the speed of light, and relative calm prevails. Fundamental charged particles like electrons and quarks are not black holes: their charge is much greater than their mass, and they do not contain horizons. If the geometry is continued all the way to the centre of the black hole, then there is a gravitationally repulsive, negative-mass singularity there. Uncharged persons who fall into the charged black hole are repelled by the singularity, and do not fall into it. The diagram at left is an embedding diagram of the Reissner-Nordström geometry, a 2-dimensional representation of the 3-dimensional spatial geometry at an instant of Reissner-Nordström time. Between the horizons, radial lines at fixed Reissner-Nordström time are time-like rather than space-like, which is to say that they are possible wordlines of radially infalling (albeit not freely falling) observers. The animated dashes follow the positions of such infalling observers as a function of their own proper time.
Caveats The Universe at large appears to be electrically neutral, or close to it. Thus real black holes are unlikely to be charged. If a black hole did somehow become charged, it would quickly neutralize itself by accreting charge of the opposite sign. It is not clear how a gravitationally repulsive, negative-mass singularity could form. If it did, it is likely that the singularity would spontaneously destroy itself by popping charged particle-antiparticle pairs out of the vacuum inside the inner horizon. By swallowing particles of charge opposite to itself, the singularity would tend to neutralize both its charge and its negative mass, redistributing the charge over space inside the inner horizon. In these pages I have somewhat arbitrarily replaced the Reissner-Nordström geometry near the singularity with flat space. Specifically, the inward rush of space into the black hole slows to a halt at the turnaround point $$r_0$$ inside the inner horizon (see the discussion in the section below on the Free-fall spacetime diagram), and I have replaced the space interior to $$r_0$$ with flat space. This is equivalent to concentrating all the charge of the black hole into a thin shell at the turnaround point $$r_0$$.
Reissner-Nordström metric The Reissner-Nordström metric is $d s^2 = - \, B(r) d t^2 + {d r^2 \over B(r)} + r^2 d o^2$ where the metric coefficient B($$r$$) is $B(r) = 1 - {2 M \over r} + {Q^2 \over r^2} \ .$ This expression is in geometric units, works also for the Reissner-Nordström geometry. The mass $$M(r)$$ at radial position $$r$$ is the effective mass interior to $$r$$ which is the total mass $$M$$ at infinity, less the mass $$Q^2 / (2r)$$ contained in the electromagnetic field outside $$r$$: $M(r) = M - {Q^2 \over 2r} \ .$ The electromagnetic mass $$Q^2 / (2r)$$ is the mass outside $$r$$ associated with the energy density $$E / (8\pi)$$ of the electric field $$E = Q / r^2$$ surrounding a charge $$Q$$. The infall velocity $$v$$ of space passes the speed of light $$c$$ at the outer horizon $$r_+$$, but slows back down to less than the speed of light at the inner horizon $$r_-$$. The velocity slows all the way to zero at the turnaround point $$r_0$$ inside the inner horizon, $r_0 = {Q^2 \over 2M} \ .$ The free-fall metric for the Reissner-Nordström geometry takes the same form as for Schwarzschild, $d s^2 = - d t_\textrm{ff} + (d r - v \, d t_\textrm{ff})^2 + r^2 d o^2 \ ,$ with free-fall velocity $v = - \sqrt{2 M(r) \over r} \ .$ The free-fall time coordinate $$t_\textrm{ff}$$ is the proper time experienced by persons who free-fall at velocity $$d r / d t_\textrm{ff} = v$$ from zero velocity at infinity: $t_\textrm{ff} = t + \sqrt{2 M} \left[ 2 \sqrt{x} - {r_+ \sqrt{x_+} \over r_+ - r_-} \ln \left( {\sqrt{x} + \sqrt{x_+} \over \sqrt{x} - \sqrt{x_+}} \right) + {r_- \sqrt{x_-} \over r_+ - r_-} \ln \left( {\sqrt{x} + \sqrt{x_-} \over \sqrt{x} - \sqrt{x_-}} \right) \right] \ ,$ where the coordinate $$x$$ is the radial position relative to the turnaround point $$r_0$$, $x \equiv r - r_0 \ ,$ and $$x_\pm \equiv r_\pm - r_0$$ are the values of $$x$$ at the horizons $$r_\pm$$. The free-fall metric shows that the spatial geometry is flat, having spatial metric $$d r^2 + r^2 d o^2$$, on hypersurfaces of fixed free-fall time, $$d t_\textrm{ff} = 0$$. The colouring of lines in the free-fall spacetime diagram is as in the Reissner-Nordström spacetime diagram, with the addition of green lines which are worldlines of observers who free fall radially from zero velocity at infinity, and horizontal dark green lines which are lines of constant free-fall time $$t_\textrm{ff}$$.
Finkelstein spacetime diagram of the Reissner-Nordström geometry As usual, the Finkelstein radial coordinate $$r$$ is the circumferential radius, defined so that the proper circumference of a sphere at radius $$r$$ is $$2\pi r$$, while the Finkelstein time coordinate is defined so that radially infalling light rays (yellow lines) move at $$45^\circ$$ in the spacetime diagram. Finkelstein time $$t_\textrm{F}$$ is related to Reissner-Nordström time $$r$$ by $t_\textrm{F} = t + {1 \over 2 g_+} \ln \left( {r - r_+ \over r_0 - r_+} \right) + {1 \over 2 g_-} \ln \left( {r - r_- \over r_0 - r_-} \right) \ ,$ where $$g_\pm \equiv g(r_\pm)$$ are the surface gravities at the two horizons $g_\pm = \pm {r_+ - r_- \over 2 r_\pm^2} \ .$ The gravity $$g(r)$$ at radial position $$r$$ is the inward acceleration $g(r) = {d v \over d t_\textrm{ff}} = - v {d v \over d r} = {1 \over 2} {d B \over d r} \ .$ The colouring of lines is as in the Schwarzschild case: the red line is the horizon, the cyan line at zero radius is the singularity, yellow and ochre lines are respectively the wordlines of radially infalling and outgoing light rays, while dark purple and blue lines are respectively lines of constant Schwarzschild time and constant circumferential radius.
Penrose diagram of the Reissner-Nordström geometry Coordinates of Penrose diagram constructed so that the metric is well-behaved across both outer and inner horizons. Given this restriction, it's impossible to make the zero radius part vertical.
Penrose diagram of the complete Reissner-Nordström geometry Suppose that you fall into a charged black hole. At the moment that you cross the inner horizon, you see an infinitely blueshifted point of light appear directly ahead, in the direction of the black hole. This infinitely blueshifted point of light is a record of the entire past history of the Universe, condensed into an instant. Inside the inner horizon, the gravitational repulsion of the central singularity slows you down and turns you around, accelerating you back out through the inner horizon of a white hole. As you approach the inner horizon of the white hole, this time looking outward directly away from the black hole, part of the image of the outside Universe seems to break away from the rest. As you pass through the inner horizon this breakaway image concentrates into another infinitely blueshifted point of light, which disappears in a blazing flash. This time the infinitely blueshifted point of light contains the entire future of the Universe, condensed into an instant. The white hole spews you out into a new Universe. Since light cannot fall into the white hole from the new Universe, you do not see the new Universe until you pass through the outer horizon of the white hole. At the instant you pass through the outer horizon, you witness once again an infinitely blueshifted point of light appear directly ahead, away from the white hole. The infinitely blueshifted point of light contains the entire past of the new Universe concentrated into an instant. The point of light opens up to reveal the new Universe, which you join. Looking back into the white hole, you can see the Universe from which you came, but to which you cannot return.
Back to Collapse to a Black Hole
Forward to The Extremal Reissner-Nordström Geometry
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## km meaning in text
Posted by on December 22, 2020 / Posted in Uncategorized
The total length of the public road's network is 29151 km, out of which 1243 km are motorways, 6810 km of national roads and 21098 km of regional and local roads. Note: We have 131 other definitions for KMS in our Acronym Attic. KMS means “Kill Myself” on the internet and in real life, though like many abbreviations and some slang terms, it’s typed more often than it’s spoken. Wat does :3 mean, I hav abunch of girls sayin dat. KM means Keep Mum. An abbreviation that is widely used in texting and chat, and on Facebook and elsewhere on the internet, but what does KMT mean in slang? Source(s): myself. Elise Moreau is a writer that has covered social media, texting, messaging, and streaming for Lifewire. It is conjugated as a regular verb. Lo encontrarás en al menos una de las líneas abajo. I don’t know all the symbols here. Jai September 19, 2012 at 12:22 pm. by. Meaning of KM. Information and translations of KM in the most comprehensive dictionary definitions resource on the web. Definition *required. Km definition, kilometer; kilometers. Submitted by Walter Rader (Editor) from Sacramento, CA, USA on Sep 09 2011. We hope you have found this useful. Dictionary entry overview: What does km mean? what does km mean in texting? Reply. This page explains what the abbreviation "kms" means. Ten miles, flat out, as fast as you can go. Browse our Scrabble Word Finder, Words With Friends cheat dictionary, and WordHub word solver to find words starting with km. written abbreviation – plural km or kms – kilometre (s). Parents aren’t always on top of the latest hip teen linguistics, but they should know that if their teen is texting their friends about getting “lit” or “ turnt up,” it means that your teen is partying with drugs and/or alcohol. Change your default dictionary to American English. Submit a new or better definition for KMT. The general thought is that the more surface area (wrinkles, creases, etc.) What Does LMK Mean in Texting? If I am writing "I ran X distance", should I leave a space between the value and abbreviated distance, or is it more appropriate to bunch them together: a) "I ran 5km" b) "I ran 5 km" 4 Answers. Reply. Judges don't usually have the time to look through your text messages or emails unless your case goes to trial. what does uefa mean in texting. Another verb derived from English is chatear, to chat. km abbreviation (plural km, kms) jump to other results (in writing) kilometre(s) Topics Maths and measurement a2; See km in the Oxford Advanced American Dictionary. Most Common KMT Meaning KMT stands for Kiss My Teeth. Freelance Contributor. km definition: 1. written abbreviation for kilometre UK 2. written abbreviation for kilometer 3. abbreviation for…. showing only Slang/Internet Slang definitions (show all 32 definitions). Used in a Sentence: [recaptcha recaptcha-385] Thus concludes our slang archive for KMT. The definition, example, and related terms listed above have been written and compiled by the Slangit team. Enter search text. {\displaystyle k_ {\text {cat}}/K_ {\mathrm {M} }} (catalytic efficiency) is a measure of how efficiently an enzyme converts a substrate into product. This acronym represents a phrase you hear in everyday life all the time. Although it's frowned on by purists and isn't in most dictionaries, the verb textear is often used as the equivalent of "to text." Relevance. Favourite answer. Definition of KM in the Definitions.net dictionary. The noun form is a cognate, texto. Last edited on Apr 14 2013. + KMS meaning as it’s used in various places around the web, in real life (IRL), and everywhere in between. MEANINGS. new search; suggest new definition; Search for KMS in Online Dictionary Encyclopedia Pao. kilometer, kilometre, km, klick (noun) a metric unit of length equal to 1000 meters (or 0.621371 miles) • KM (noun) The noun KM has 1 sense:. If you have any additional definitions of KM that should be on this list, or know of any slang terms that we haven't already published, click here to let us know! We hope you have found this useful. Her work has appeared on Techvibes, SlashGear, Lifehack and others. See more words with the same meaning: to disagree, disapprove, or doubt. Abbreviation: km See more. a brain has, the smarter the person is.Conversely, a person with a smooth brain (no wrinkles) has less surface area and would therefore be … Thanks for the share. 0 0. mono_loco. Used in a Sentence: [recaptcha recaptcha-385] Thus concludes our slang archive for KM. Definition of km abbreviation from the Oxford Advanced Learner's Dictionary. | Meaning, pronunciation, translations and examples Elise Moreau. A stupid person; it refers to the lack of surface area on an individual's brain. What does KM mean? Definition and synonyms of km from the online English dictionary from Macmillan Education. km es un término alternativo para kilometer. It is in one or more of the lines below. Submit a new or better definition for KM. If you’re looking to find what KMS means, you’ve come to the right place! source: ASECAP. at.srichinmoyraces.org After 41 hours and 244 km Gerhard Burster was well on the way to break the world record for the men's category M 65, which was 255,505 km at that time. What does KMA stand for in Text Messaging? Text Messaging KMA abbreviation meaning defined here. This is the British English definition of km.View American English definition of km. DICTIONARY.COM; THESAURUS.COM; MEANINGS. Puede estar recorriendo 500 km diario. kms definition. We are constantly updating our database with new slang terms, acronyms, and abbreviations. km translations: “kilometer”의 약어. 0.3 km 9.0 km 88.9 km 0.1 km send inquiry Hotel Kopanice - Special offers and discounts Do not miss the romantic horse-drawn carriage ride. Emoji; Slang; Acronyms; Pop Culture; Memes; Gender and Sexuality; Mixed-up Meanings ... (Commonwealth) extended its hold on second place after they also dug deep and covered 24 km. Kilometer definition, a unit of length, the common measure of distances equal to 1,000 meters, and equivalent to 3280.8 feet or 0.621 mile. Gerhard Burster war nach 41 Stunden mit 244 km voll auf Kurs in der Altersklasse M 65 den Weltrekord zu pulverisieren, der bis dahin bei 255,505 km stand. Denia se encuentra a 7 km, Pedreguer a 2 km y la playa a 3 km. Correrán 16 km a velocidad máxima. km - Translation to Spanish, pronunciation, and forum discussions. KMS definition: kill myself: used esp in text messaging and social media | Meaning, pronunciation, translations and examples 1 people chose this as the best definition of km: Kilometer.... See the dictionary meaning, pronunciation, and sentence examples. Yeah September 21, 2012 at 9:24 pm. Answer Save. I’m gonna try this text. Definition *required. 1 decade ago. Km definition: km is a written abbreviation for → kilometre . Learn more in the Cambridge English-Korean Dictionary. However, I saw a sentence where “out of which” was used in a phrase, such as “out of which 1000 km is motorway”. Get the top KMA abbreviation related to Text Messaging. 'km' is an alternate term for 'kilometer'. 0.3 km 9.0 km 88.9 km 0.1 km Anfrage absenden Hotel Kopanice - Spezialangebote und Ermässingungen Verpassen Sie nicht die romantische Pferdeschlitten-Kutschfahrt. Or use our Unscramble word solver to find your best possible play! since she is welcoming you in her world, KM means Kiss Me! This page explains how KM is used on Snapchat, Whatsapp, Facebook, Twitter, and Instagram as well as in texts and chat forums such as Teams. It is situated 7 km from Denia, 2 km from Pedreguer and 4 km from the sandy beach Las Marinas. 1 km = 1000 m; 1 m = 100 cm = 1000 mm = 10^6 um = 10^9 nm Metric Unit multiplied by = English Unit: millimeters centimeters centimeters meters meters meters kilometers kilometers: 0.0394 0.394 0.0328 39.4 3.28 1.1 3,281 0.621: inches (in) inches feet (ft) inches … Other results I live 5 km from the airport; a 5 km drive. See more. stands for: kill my self. Diffusion limited enzymes, such as fumarase, work at the theoretical upper limit of 108 – 1010 M−1s−1, limited by diffusion of substrate into the active site. See more words with the same meaning: Internet, texting, SMS, email, chat acronyms (list of). 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This is “Analyzing Cash Flow Information”, section 12.5 from the book Accounting for Managers (v. 1.0). For details on it (including licensing), click here.
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12.5 Analyzing Cash Flow Information
Learning Objective
1. Analyze cash flow information.
Question: Companies and analysts tend to use income statement and balance sheet information to evaluate financial performance. In fact, financial results presented to the investing public typically focus on earnings per share (Chapter 13 "How Do Managers Use Financial and Nonfinancial Performance Measures?" discusses earnings per share in detail). However, analysis of cash flow information is becoming increasingly important to managers, auditors, and outside analysts. What measures are commonly used to evaluate performance related to cash flows?
Answer: Three common cash flow measures used to evaluate organizations are (1) operating cash flow ratio, (2) capital expenditure ratio, and (3) free cash flow. (Further coverage of these measures can be found in the following article: John R. Mills and Jeanne H. Yamamura, “The Power of Cash Flow Ratios,” Journal of Accountancy, October 1998.) We will use two large home improvement retail companies, The Home Depot, Inc., and Lowe’s Companies, Inc., to illustrate these measures.
Operating Cash Flow Ratio
Question: The operating cash flow ratioA cash flow performance measure calculated as cash provided by operating activities divided by current liabilities. is cash provided by operating activities divided by current liabilities. What does this ratio tell us, and how is it calculated?
Answer: This ratio measures the company’s ability to generate enough cash from daily operations over the course of a year to cover current obligations. Although similar to the commonly used current ratio, this ratio replaces current assets in the numerator with cash provided by operating activities. The operating cash flow ratio is as follows:
Key Equation
The numerator, cash provided by operating activities, comes from the bottom of the operating activities section of the statement of cash flows. The denominator, current liabilities, comes from the liabilities section of the balance sheet. (Note that if current liabilities vary significantly from one period to the next, some analysts prefer to use average current liabilities. We will use ending current liabilities unless noted otherwise.)
As with most financial measures, the resulting ratio must be compared to similar companies in the industry to determine whether the ratio is reasonable. Some industries have a large operating cash flow relative to current liabilities (e.g., mature computer chip makers, such as Intel Corporation), while others do not (e.g., startup medical device companies).
The operating cash flow ratio is calculated for Home Depot and Lowe’s in the following using information from each company’s balance sheet and statement of cash flows.
Home Depot and Lowe’s are in the same industry and have comparable ratios, which is what we would expect for similar companies.
Capital Expenditure Ratio
Question: The capital expenditure ratioA cash flow performance measure calculated as cash provided by operating activities divided by capital expenditures. is cash provided by operating activities divided by capital expenditures. What does this ratio tell us, and how is it calculated?
Answer: This ratio measures the company’s ability to generate enough cash from daily operations to cover capital expenditures. A ratio in excess of 1.0, for example, indicates the company was able to generate enough operating cash to cover investments in property, plant, and equipment. The capital expenditure ratio is as follows:
Key Equation
The numerator, cash provided by operating activities, comes from the bottom of the operating activities section of the statement of cash flows. The denominator, capital expenditures, comes from information within the investing activities section of the statement of cash flows.
The capital expenditure ratio is calculated for Home Depot and Lowe’s in the following using information from each company’s statement of cash flows.
Since the capital expenditure ratio for each company is above 1.0, both companies were able to generate enough cash from operating activities to cover investments in property, plant, and equipment (also called fixed assets).
Free Cash Flow
Question: Another measure used to evaluate organizations, called free cash flow, is simply a variation of the capital expenditure ratio described previously. What does this measure tell us, and how is it calculated?
Answer: Rather than using a ratio to determine whether the company generates enough cash from daily operations to cover capital expenditures, free cash flow is measured in dollars. Free cash flowA cash flow performance measure calculated as cash provided by operating activities minus capital expenditures. is cash provided by operating activities minus capital expenditures. The idea is that companies must continue to invest in fixed assets to remain competitive. Free cash flow provides information regarding how much cash generated from daily operations is left over after investing in fixed assets. Many organizations, such as Amazon.com, consider this measure to be one of the most important in evaluating financial performance (see Note 12.34 "Business in Action 12.5"). The free cash flow formula is as follows:
Key Equation
Free cash flow = Cash provided by operating activities − Capital expenditures
The cash provided by operating activities comes from the bottom of the operating activities section of the statement of cash flows. The capital expenditures amount comes from information within the investing activities section of the statement of cash flows.
The free cash flow amount is calculated for Home Depot and Lowe’s as follows using information from each company’s statement of cash flows.
Because free cash flow for each company is above zero, both companies were able to generate enough cash from operating activities to cover investments in fixed assets and have some left over to invest elsewhere. This conclusion is consistent with the capital expenditure ratio analysis, which uses the same information to assess the company’s ability to cover fixed asset expenditures.
Formulas for the cash flow performance measures presented in this chapter are summarized in Table 12.1 "Summary of Cash Flow Performance Measures".
Table 12.1 Summary of Cash Flow Performance Measures
Free Cash Flow at Amazon.com
Amazon.com is an online retailer that began selling books in 1996 and has since expanded into other areas of retail sales. The founder and CEO (Jeff Bezos) believes free cash flow is so important, the annual report included a letter from Mr. Bezos to the shareholders, which began with this statement, “Our ultimate financial measure, and the one we want to drive over the long-term, is free cash flow per share.”
The company justifies this focus on free cash flow by making the point that earnings presented on the income statement do not translate into cash flows, and shares are valued based on the present value of future cash flows. This implies shareholders should be most interested in free cash flow per share rather than earnings per share. Mr. Bezos goes on to state, “Cash flow statements often don’t receive as much attention as they deserve. Discerning investors don’t stop with the income statement.”
Amazon.com’s free cash flow for 2010 totaled $2,164,000,000, compared to$2,880,000,000 in 2009. Net income for 2010 totaled $1,152,000,000, compared to$902,000,000 in 2009. It is interesting to note that free cash flow is significantly higher than net income for 2010 and 2009.
Key Takeaway
• Three measures are often used to evaluate cash flow. The operating cash flow ratio measures the company’s ability to generate enough cash from daily operations over the course of a year to cover current obligations. The formula is as follows:
The capital expenditure ratio measures the company’s ability to generate enough cash from daily operations to cover capital expenditures. The formula is as follows:
Free cash flow measures the company’s ability to generate enough cash from daily operations to cover capital expenditures and determines how much cash is remaining to invest elsewhere in the company. The formula is as follows:
Free cash flow = Cash provided by operating activities − Capital expenditures
Review Problem 12.8
The following financial information is for PepsiCo Inc. and Coca-Cola Company for fiscal year 2010.
For PepsiCo and Coca-Cola, calculate the following measures and comment on your results:
1. Operating cash flow ratio
2. Capital expenditure ratio (Hint: fixed asset expenditures are the same as capital expenditures.)
3. Free cash flow
Solution to Review Problem 12.8
All dollar amounts are in millions.
1. The formula for calculating the operating cash flow ratio is as follows:
PepsiCo generated slightly more cash from operating activities to cover current liabilities than Coca-Cola.
2. The formula for calculating the capital expenditure ratio is as follows:
Both companies generated more than enough cash from operating activities to cover capital expenditures.
3. The formula to calculate free cash flow is as follows:
Free cash flow = Cash provided by operating activities − Capital expenditures
The conclusion reached in requirement two is confirmed here. Both companies generated more than enough cash from operating activities to cover capital expenditures. In fact, PepsiCo had $5,195,000,000 remaining from operating activities after investing in fixed assets, and Coca-Cola had$7,317,000,000 remaining.
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# wx.Gauge¶
A gauge is a horizontal or vertical bar which shows a quantity (often time).
wx.Gauge supports two working modes: determinate and indeterminate progress.
The first is the usual working mode (see SetValue and SetRange) while the second can be used when the program is doing some processing but you don’t know how much progress is being done. In this case, you can periodically call the Pulse function to make the progress bar switch to indeterminate mode (graphically it’s usually a set of blocks which move or bounce in the bar control).
wx.Gauge supports dynamic switch between these two work modes.
There are no user commands for the gauge.
## Window Styles¶
This class supports the following styles:
• wx.GA_HORIZONTAL: Creates a horizontal gauge.
• wx.GA_VERTICAL: Creates a vertical gauge.
• wx.GA_SMOOTH: Creates smooth progress bar with one pixel wide update step (not supported by all platforms).
## Class Hierarchy¶
Inheritance diagram for class Gauge:
wxMSW
wxMAC
wxGTK
## Methods Summary¶
__init__ Default constructor. Create Creates the gauge for two-step construction. GetBezelFace Returns the width of the 3D bezel face. GetClassDefaultAttributes GetRange Returns the maximum position of the gauge. GetShadowWidth Returns the 3D shadow margin width. GetValue Returns the current position of the gauge. IsVertical Returns True if the gauge is vertical (has GA_VERTICAL style) and False otherwise. Pulse Switch the gauge to indeterminate mode (if required) and makes the gauge move a bit to indicate the user that some progress has been made. SetBezelFace Sets the 3D bezel face width. SetRange Sets the range (maximum value) of the gauge. SetShadowWidth Sets the 3D shadow width. SetValue Sets the position of the gauge.
## Class API¶
class wx.Gauge(Control)
Possible constructors:
Gauge()
Gauge(parent, id=ID_ANY, range=100, pos=DefaultPosition,
size=DefaultSize, style=GA_HORIZONTAL, validator=DefaultValidator,
name=GaugeNameStr)
A gauge is a horizontal or vertical bar which shows a quantity (often time).
### Methods¶
__init__(self, *args, **kw)
__init__ (self)
Default constructor.
__init__ (self, parent, id=ID_ANY, range=100, pos=DefaultPosition, size=DefaultSize, style=GA_HORIZONTAL, validator=DefaultValidator, name=GaugeNameStr)
Constructor, creating and showing a gauge.
Parameters
• parent (wx.Window) – Window parent.
• id (wx.WindowID) – Window identifier.
• range (int) – Integer range (maximum value) of the gauge. See SetRange for more details about the meaning of this value when using the gauge in indeterminate mode.
• pos (wx.Point) – Window position.
• size (wx.Size) – Window size.
• style (long) – Gauge style.
• validator (wx.Validator) – Window validator.
• name (string) – Window name.
Create(self, parent, id=ID_ANY, range=100, pos=DefaultPosition, size=DefaultSize, style=GA_HORIZONTAL, validator=DefaultValidator, name=GaugeNameStr)
Creates the gauge for two-step construction.
See wx.Gauge for further details.
Parameters
Return type
bool
GetBezelFace(self)
Returns the width of the 3D bezel face.
Return type
int
Note
This method is not implemented (returns 0) for most platforms.
static GetClassDefaultAttributes(variant=WINDOW_VARIANT_NORMAL)
Parameters
variant (WindowVariant) –
Return type
wx.VisualAttributes
GetRange(self)
Returns the maximum position of the gauge.
Return type
int
GetShadowWidth(self)
Returns the 3D shadow margin width.
Return type
int
Note
This method is not implemented (returns 0) for most platforms.
GetValue(self)
Returns the current position of the gauge.
Return type
int
IsVertical(self)
Returns True if the gauge is vertical (has GA_VERTICAL style) and False otherwise.
Return type
bool
Pulse(self)
Switch the gauge to indeterminate mode (if required) and makes the gauge move a bit to indicate the user that some progress has been made.
Note
After calling this function the value returned by GetValue is undefined and thus you need to explicitly call SetValue if you want to restore the determinate mode.
SetBezelFace(self, width)
Sets the 3D bezel face width.
Parameters
width (int) –
Note
This method is not implemented (doesn’t do anything) for most platforms.
SetRange(self, range)
Sets the range (maximum value) of the gauge.
This function makes the gauge switch to determinate mode, if it’s not already.
When the gauge is in indeterminate mode, under wxMSW the gauge repeatedly goes from zero to range and back; under other ports when in indeterminate mode, the range setting is ignored.
Parameters
range (int) –
SetShadowWidth(self, width)
Parameters
width (int) –
Note
This method is not implemented (doesn’t do anything) for most platforms.
SetValue(self, pos)
Sets the position of the gauge.
The pos must be between 0 and the gauge range as returned by GetRange , inclusive.
This function makes the gauge switch to determinate mode, if it was in indeterminate mode before.
Parameters
pos (int) – Position for the gauge level.
### Properties¶
BezelFace
Range
ShadowWidth
Value
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# How are pulse trains generated in a radio transmitter?
When simulating a PSK signal in MATLAB or Python, I often create an impulse train where each impulse is modulated with a symbol. This modulated impulse train then passes through a pulse shape filter to yield the PSK signal.
However I am wondering if this is indeed how PSK signals are generated in real radio electronics hardware. Is there an electronics module which generates modulated impulses which passes through an analog filter? If so, how narrow are such pulses typically?
I know the block diagram representation of PSK implementation in hardware often shows dataflow split into two paths, one for In-phase and one for Quadrature-Phase. Each path passes through a Root Raised Cosine filter before combining back again. However it is not clear to me how this RRC module is implemented in hardware.
If it's 1950 through about 1980 or 1990, then the all-analog way to do it would be to generate the carrier and a $$90^\circ$$ shifted version of it, then switch in the carrier, the carrier shifted $$180^\circ$$ (which is trivial), the $$90^\circ$$ carrier, or its $$180^\circ$$-shifted version. (Alternately, you'd always have the carrier or it's $$180^\circ$$-shifted version added to the $$90^\circ$$ shifted or $$-90^\circ$$ shifted version -- it may be easier to realize in circuitry, and you'd still get phase shifts in $$90^\circ$$ increments.)
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# Math Help - Probability Distributions
1. ## Probability Distributions
1) Verify f(x) = (2x)/(k(k+1)) for x = 1,2,3,...,k can serve as the probability distribution function of a random variable with the given range.
I know this has to satisfy two parts. I got the first part, where for each value in the domain, f(x) >= 0. I am having trouble with the part where the sum has to equal 1. How do I show this?
2. Hello,
Originally Posted by noles2188
1) Verify f(x) = (2x)/(k(k+1)) for x = 1,2,3,...,k can serve as the probability distribution function of a random variable with the given range.
I know this has to satisfy two parts. I got the first part, where for each value in the domain, f(x) >= 0. I am having trouble with the part where the sum has to equal 1. How do I show this?
$\sum_{x=1}^k f(x)=\sum_{x=1}^k \frac{2x}{k(k+1)}=\frac{2}{k(k+1)}\sum_{x=1}^k x$
and the sum of the first k integers is $\frac{k(k+1)}{2}$
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## College Algebra (6th Edition)
By the text of the exercise: $a_1=4$ $a_2=2a_1+3=2\cdot4+3=11$ $a_3=2a_2+3=2\cdot11+3=25$ $a_4=2a_3+3=2\cdot25+3=53$
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# Problem 1
Chemistry Level 2
When $$8.8\text{ g}$$ of an organic acid, with molar mass $$88 \text{ gmol}^{-1}$$, is burnt in excess oxygen, $$17.6\text{ g}$$ of carbon dioxide and $$7.2\text{ g}$$ of water are produced. Calculate its empirical formula.
×
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# Markov chain
## Primary tabs
\documentclass{article}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\md}{d}
\newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector
\newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}}
\newcommand{\mvi}[1]{\mv{#1}^{-1}}
\newcommand{\mderiv}[1]{\frac{\md}{\md {#1}}} %d/dx
\newcommand{\mnthderiv}[2]{\frac{\md^{#2}}{\md {#1}^{#2}}} %d^n/dx
\newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} %partial d^n/dx
\newcommand{\mnthpderiv}[2]{\frac{\partial^{#2}}{\partial {#1}^{#2}}} %partial d^n/dx
\newcommand{\borel}{\mathfrak{B}}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\rationals}{\mathbb{Q}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\complexes}{\mathbb{C}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\defined}{:=}
\newcommand{\var}{\mathrm{var}}
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\corr}{\mathrm{corr}}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\powerset}[1]{\mathcal{P}(#1)}
\newcommand{\bra}[1]{\langle#1 \vert}
\newcommand{\ket}[1]{\vert \hspace{1pt}#1\rangle}
\newcommand{\braket}[2]{\langle #1 \ket{#2}}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\norm}[1]{\left|\left|#1\right|\right|}
\newcommand{\esssup}{\mathrm{ess\ sup}}
\newcommand{\Lspace}[1]{L^{#1}}
\newcommand{\Lone}{\Lspace{1}}
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\newcommand{\Lp}{\Lspace{p}}
\newcommand{\Lq}{\Lspace{q}}
\newcommand{\Linf}{\Lspace{\infty}}
\newcommand{\sequence}[1]{\{#1\}}
\begin{document}
\paragraph{Definition}
We begin with a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $I$ be a countable set, $(X_n: n \in \integers)$ be a collection of random variables taking values in $I$, $\mv{T} = (t_{ij}: i,j \in I)$ be a stochastic matrix, and $\mv{\lambda}$ be a distribution. We call $(X_n)_{n\ge 0}$ a \emph{Markov chain} with initial distribution $\mv{\lambda}$ and \emph{transition matrix} $\mv{T}$ if:
\begin{enumerate}
\item{$X_0$ has distribution $\mv{\lambda}$.}
\item{For $n \ge 0$, $\mathbb{P}(X_{n+1}=i_{n+1} | X_0 = i_0, \ldots, X_n = i_n) = t_{i_n i_{n+1}}$.}
\end{enumerate}
That is, the next value of the chain depends only on the current value, not any previous values. This is often summed up in the pithy phrase, Markov chains have no memory.''
As a special case of (2) we have that $\mathbb{P}(X_{n+1} = i | X_n = j) = t_{ij}$ whenever $i,j \in I$. The values $t_{ij}$ are therefore called
\emph{transition probabilities} for the Markov chain.
\paragraph{Discussion}
Markov chains are arguably the simplest examples of random processes. They come in discrete and continuous versions; the discrete version is presented above.
%%%%%
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nd{document}
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# AllerCaps
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Number of Capsules 90 Caps Aller Assist Blend™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 mg, Aquamin® F Mineralized Red Algae, Reishi, Organic Milk Thistle (seed) (Silybum marianum), Fermented Cordyceps (mycelia) Extract (Cordyceps sinensis), Organic Turmeric, Bromelain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 mg, Calcium (as Calcium Carbonate from Aquamin® F Algae) . . . . . . . . . . . . . . . 44 mg, N-Acetyl-L-Cysteine (NAC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 mg, Other Ingredients: Vegetable Cellulose Capsules, Quercetin Dihydrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 mg, Three Vegetable Capsules Provide: Take 3 capsules daily or as directed by a health professional.
## Reviews
There are no reviews yet.
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# zbMATH — the first resource for mathematics
Spanning cycles in regular matroids without $$M^{*}(K_{5})$$ minors. (English) Zbl 1126.05031
Summary: Catlin and Jaeger proved that the cycle matroid of a 4-edge-connected graph has a spanning cycle. This result can not be generalized to regular matroids as there exist infinitely many connected cographic matroids, each of which contains a $$M^{*}(K_{5})$$-minor and has arbitrarily large cogirth, that do not have spanning cycles. In this paper, we proved that if a connected regular matroid without a $$M^{*}(K_{5})$$-minor has cogirth at least 4, then it has a spanning cycle.
##### MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
Full Text:
##### References:
[1] Appel, K.; Haken, W., Every planar map is four colorable, part I: discharging, Illinois J. math., 21, 429-490, (1977) · Zbl 0387.05009 [2] Appel, K.; Haken, W.; Koch, J., Every planar map is four colorable, part II: reducibility, Illinois J. math., 21, 491-567, (1977) · Zbl 0387.05010 [3] Boesch, F.T.; Suffel, C.; Tindell, R., The spanning subgraphs of Eulerian graphs, J. graph theory, 1, 79-84, (1977) · Zbl 0363.05042 [4] Bollobás, B., Graph theory, (1979), Springer-Verlag New York · Zbl 0418.05049 [5] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), American Elsevier New York · Zbl 1134.05001 [6] Catlin, P.A., A reduction method to find spanning Eulerian subgraphs, J. graph theory, 12, 29-44, (1988) · Zbl 0659.05073 [7] P.A. Catlin, H.-J. Lai, Y. Shao, Edge-connectivity and edge-disjoint spanning trees (submitted for publication) [8] Catlin, P.A.; Lai, H.-J., Spanning trails joining two given edges, (), 207-222 · Zbl 0841.05067 [9] Catlin, P.A.; Han, Z.Y.; Lai, H.-J., Graphs without spanning closed trails, Discrete math., 160, 81-91, (1996) · Zbl 0859.05060 [10] Jaeger, F., A note on Subeulerian graphs, J. graph theory, 3, 91-93, (1979) [11] Nash-Williams, C.St.J.A., Edge-disjoint spanning trees of finite graphs, J. London math. soc., 36, 445-450, (1961) · Zbl 0102.38805 [12] Nash-Williams, C.St.J.A., Decomposition of finite graphs into forests, J. London math. soc., 39, 12, (1964) · Zbl 0119.38805 [13] Oxley, J.G., Matroid theory, (1992), Oxford University Press New York · Zbl 0784.05002 [14] Pulleyblank, W.R., A note on graphs spanned by Eulerian graphs, J. graph theory, 3, 309-310, (1979) · Zbl 0414.05040 [15] Robertson, N.; Sanders, D.; Seymour, P.; Thomas, R., The four-color theorem, J. combin. theory ser. B, 70, 2-44, (1997) · Zbl 0883.05056 [16] Seymour, P.D., Decomposition of regular matroids, J. combin. theory ser. B, 28, 305-359, (1980) · Zbl 0443.05027 [17] Seymour, P.D., Matroids and multicommodity flows, European J. combin. theory ser. B., 2, 257-290, (1981) · Zbl 0479.05023 [18] Tutte, W.T., A homotopy theorem for matroids, I, II, Trans. amer. math. soc., 88, 144-174, (1958) · Zbl 0081.17301 [19] Tutte, W.T., On the problem of decomposing a graph into $$n$$ connected factors, J. London math. soc., 36, 80-91, (1961) · Zbl 0096.38001 [20] Veblen, O., An application of modular equations in analysis situs, Ann. of math., 14, 86-94, (1912-1913) · JFM 43.0574.01 [21] Wagner, K., Über eine eigneschaft der ebenen komplexe, Math. ann., 144, 570-590, (1937) · JFM 63.0550.01 [22] Welsh, D.J.A., Matroid theory, (1976), Academic Press London · Zbl 0343.05002 [23] Welsh, D.J.A., Euler and bipartite matroids, J. combin. theory, 6, 313-316, (1969) · Zbl 0167.01704 [24] Zhan, S.M., Hamiltonian connectedness of line graphs, Ars combin., 22, 89-95, (1986) · Zbl 0611.05038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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CGAL 4.4 - 3D Alpha Shapes
User Manual
Assume we are given a set $$S$$ of points in 2D or 3D and we'd like to have something like "the shape formed by these points." This is quite a vague notion and there are probably many possible interpretations, the alpha shape being one of them. Alpha shapes can be used for shape reconstruction from a dense unorganized set of data points. Indeed, an alpha shape is demarcated by a frontier, which is a linear approximation of the original shape [1].
As mentioned in Edelsbrunner's and Mücke's paper [2], one can intuitively think of an alpha shape as the following. Imagine a huge mass of ice-cream making up the space $$\mathbb{R}^3$$ and containing the points as "hard" chocolate pieces. Using one of those sphere-formed ice-cream spoons we carve out all parts of the ice-cream block we can reach without bumping into chocolate pieces, thereby even carving out holes in the inside (e.g. parts not reachable by simply moving the spoon from the outside). We will eventually end up with a (not necessarily convex) object bounded by caps, arcs and points. If we now straighten all "round" faces to triangles and line segments, we have an intuitive description of what is called the alpha shape of $$S$$. Here's an example for this process in 2D (where our ice-cream spoon is simply a circle):
Alpha shapes depend on a parameter $$\alpha$$ from which they are named. What is $$\alpha$$ in the ice-cream game? $$\alpha$$ is the squared radius of the carving spoon. A very small value will allow us to eat up all of the ice-cream except the chocolate points themselves. Thus we already see that the alpha shape degenerates to the point-set $$S$$ for $$\alpha \rightarrow 0$$. On the other hand, a huge value of $$\alpha$$ will prevent us even from moving the spoon between two points since it's way too large. So we will never spoon up ice-cream lying in the inside of the convex hull of $$S$$, and hence the alpha shape for $$\alpha \rightarrow \infty$$ is the convex hull of $$S$$.
# Definitions
More precisely, the definition of alpha shapes is based on an underlying triangulation that may be a Delaunay triangulation in case of basic alpha shapes or a regular triangulation (cf. Section Regular Triangulations) in case of weighted alpha shapes.
Let us consider the basic case with a Delaunay triangulation. We first define the alpha complex of the set of points $$S$$. The alpha complex is a subcomplex of the Delaunay triangulation. For a given value of $$\alpha$$, the alpha complex includes all the simplices in the Delaunay triangulation which have an empty circumscribing sphere with squared radius equal or smaller than $$\alpha$$. Here "empty" means that the open sphere do not include any points of $$S$$. The alpha shape is then simply the domain covered by the simplices of the alpha complex (see [2]).
In general, an alpha complex is a disconnected and non-pure complex: This means in particular that the alpha complex may have singular faces. For $$0 \leq k \leq d-1$$, a $$k$$-simplex of the alpha complex is said to be singular if it is not a facet of a $$(k+1)$$-simplex of the complex. CGAL provides two versions of alpha shapes. In the general mode, the alpha shapes correspond strictly to the above definition. The regularized mode provides a regularized version of the alpha shapes. It corresponds to the domain covered by a regularized version of the alpha complex where singular faces are removed (See Figure 41.1 for an example).
Figure 41.1 Comparison of general and regularized alpha-shape. Left: Some points are taken on the surface of a torus, three points being taken relatively far from the surface of the torus; Middle: The general alpha-shape (for a large enough alpha value) contains the singular triangle facet of the three isolated points; Right: The regularized version (for the same value of alpha) does not contains any singular facet.
The alpha shapes of a set of points $$S$$ form a discrete family, even though they are defined for all real numbers $$\alpha$$. The entire family of alpha shapes can be represented through the underlying triangulation of $$S$$. In this representation each $$k$$-simplex of the underlying triangulation is associated with an interval that specifies for which values of $$\alpha$$ the $$k$$-simplex belongs to the alpha complex. Relying on this fact, the family of alpha shapes can be computed efficiently and relatively easily. Furthermore, we can select the optimal value of $$\alpha$$ to get an alpha shape including all data points and having less than a given number of connected components. Also, the alpha-values allows to define a filtration on the faces of the triangulation of a set of points. In this filtration, the faces of the triangulation are output in increasing order of the alpha value for which they appear in the alpha complex. In case of equal alpha value lower dimensional faces are output first.
The definition is analog in the case of weighted alpha shapes. The input set is now a set of weighted points (which can be regarded as spheres) and the underlying triangulation is the regular triangulation of this set. Two spheres, or two weighted points , with centers $$C_1, C_2$$ and radii $$r_1, r_2$$ are said to be orthogonal iff $$C_1C_2 ^2 = r_1^2 + r_2^2$$ and suborthogonal iff $$C_1C_2 ^2 < r_1^2 + r_2^2$$. For a given value of $$\alpha$$ the weighted alpha complex is formed with the simplices of the regular triangulation triangulation such that there is a sphere orthogonal to the weighted points associated with the vertices of the simplex and suborthogonal to all the other input weighted points. Once again the alpha shape is then defined as the domain covered by a the alpha complex and comes in general and regularized versions.
# Functionality
## Family of Alpha Shapes
The class Alpha_shape_3<Dt,ExactAlphaComparisonTag> represents the whole family of alpha shapes for a given set of points. The class includes the underlying triangulation Dt of the set, and associates to each $$k$$-face of this triangulation an interval specifying for which values of $$\alpha$$ the face belongs to the alpha complex.
The class provides functions to set and get the current $$\alpha$$-value, as well as an iterator that enumerates the $$\alpha$$ values where the alpha shape changes.
Also the class has a filtration member function that, given an output iterator with Object as value type, outputs the faces of the triangulation according to the order of apparition in the alpha complex when alpha increases.
Finally, it provides a function to determine the smallest value $$\alpha$$ such that the alpha shape satisfies the following two properties
(i) all data points are either on the boundary or in the interior of the regularized version of the alpha shape (no singular faces).
(ii) The number of components is equal or less than a given number.
The current implementation is static, that is after its construction points cannot be inserted or removed.
## Alpha Shape for a Fixed Alpha
Given a value of alpha, the class Fixed_alpha_shape_3<Dt> represents one alpha shape for a given set of points. The class includes the underlying triangulation Dt of the set, and associates to each $$k$$-face of this triangulation a classification type. This class is dynamic, that is after its construction points can be inserted or removed.
## Classification and Iterators
Both classes provide member functions to classify for a (given) value of $$\alpha$$ the different faces of the triangulation as EXTERIOR, SINGULAR, REGULAR or INTERIOR with respect to the alpha shape. A $$k$$-face on the boundary of the alpha complex is said to be: REGULAR if it is a subface of the alpha-complex which is a subface of a $$(k+1)$$-face of the alpha complex, and SINGULAR otherwise. A $$k$$-face of the alpha complex which is not on the boundary of the alpha complex is said to be INTERIOR. See Figure 41.2 for a 2D illustration.
Figure 41.2 Classification of simplices, a 2D example. Left: The 2D Delaunay triangulation of a set of points; Right: Classification of simplices for a given alpha (the squared radius of the red circle). INTERIOR, REGULAR and SINGULAR simplices are depicted in black, green and blue respectively. EXTERIOR simplices are not depicted. The vertex s and the edge tu are SINGULAR since all higher dimension simplices they are incident to are EXTERIOR. The facet pqr is EXTERIOR because the squared radius of its circumscribed circle is larger than alpha.
The classes provide also output iterators to get for a given alpha value the vertices, edges, facets and cells of the different types (EXTERIOR, SINGULAR, REGULAR or INTERIOR).
# Concepts and Models
We currently do not specify concepts for the underlying triangulation type. Models that work for a family of alpha-shapes are the instantiations of the classes Delaunay_triangulation_3 and Periodic_3_Delaunay_triangulation_3 (see example Example for Periodic Alpha Shapes). A model that works for a fixed alpha-shape are the instantiations of the class Delaunay_triangulation_3. A model that works for a weighted alpha-shape is the class Regular_triangulation_3. The triangulation needs a geometric traits class and a triangulation data structure as template parameters.
For the class Alpha_shape_3<Dt,ExactAlphaComparisonTag>, the requirements of the traits class are described in the concepts AlphaShapeTraits_3 in the non-weighted case and WeightedAlphaShapeTraits_3 in the weighted case. The CGAL kernels are models in the non-weighted case, and the class Regular_triangulation_euclidean_traits_3 is a model in the weighted case. The triangulation data structure of the triangulation has to be a model of the concept TriangulationDataStructure_3, and it must be parameterized with vertex and cell classes, which are model of the concepts AlphaShapeVertex_3 and AlphaShapeCell_3. The package provides by default the classes Alpha_shape_vertex_base_3<Gt> and Alpha_shape_cell_base_3<Gt>. When using Periodic_3_Delaunay_triangulation_3 as underlying triangulation the vertex and cell classes need to be models to both AlphaShapeVertex_3 and Periodic_3TriangulationDSVertexBase_3 as well as AlphaShapeCell_3 and Periodic_3TriangulationDSCellBase_3 (see example Example for Periodic Alpha Shapes).
For the class Fixed_alpha_shape_3<Dt>, the requirements of the traits class are described in the concepts FixedAlphaShapeTraits_3 in the non-weighted case and FixedWeightedAlphaShapeTraits_3 in the weighted case. The CGAL kernels are models in the non-weighted case, and the class Regular_triangulation_euclidean_traits_3 is a model in the weighted case. The triangulation data structure of the triangulation has to be a model of the concept TriangulationDataStructure_3, and it must be parameterized with vertex and cell classes, which are model of the concepts FixedAlphaShapeVertex_3 and FixedAlphaShapeCell_3. The package provides models Fixed_alpha_shape_vertex_base_3<Gt> and Fixed_alpha_shape_cell_base_3<Gt>, respectively.
# Alpha_shape_3 vs.\ Fixed_alpha_shape_3
The class Alpha_shape_3<Dt,ExactAlphaComparisonTag> represents the whole family of alpha shapes for a given set of points while the class Fixed_alpha_shape_3<Dt> represents only one alpha shape (for a fixed alpha). When using the same kernel, Fixed_alpha_shape_3<Dt> being a lighter version, it is naturally much more efficient when the alpha-shape is needed for a single given value of alpha. In addition note that the class Alpha_shape_3<Dt,ExactAlphaComparisonTag> requires constructions (squared radius of simplices) while the class Fixed_alpha_shape_3<Dt> uses only predicates. This implies that a certified construction of one (several) alpha-shape, using the Alpha_shape_3<Dt,ExactAlphaComparisonTag> requires a kernel with exact predicates and exact constructions (or setting ExactAlphaComparisonTag to Tag_true) while using a kernel with exact predicates is sufficient for the class Fixed_alpha_shape_3<Dt>. This makes the class Fixed_alpha_shape_3<Dt> even more efficient in this setting. In addition, note that the Fixed version is the only of the two that supports incremental insertion and removal of points.
We give the time spent while computing the alpha shape of a protein (considered as a set of weighted points) featuring 4251 atoms (using gcc 4.3 under Linux with -O3 and -DNDEBUG flags, on a 2.27GHz Intel(R) Xeon(R) E5520 CPU): Using Exact_predicates_inexact_constructions_kernel, building the regular triangulation requires 0.09s, then the class Fixed_alpha_shape_3<Dt> required 0.05s while the class Alpha_shape_3<Dt,ExactAlphaComparisonTag> requires 0.35s if ExactAlphaComparisonTag is Tag_false (and 0.70s with Tag_true). Using Exact_predicates_exact_constructions_kernel, building the regular triangulation requires 0.19s and then the class Alpha_shape_3<Dt,ExactAlphaComparisonTag> requires 0.90s.
# Examples
## Example for Basic Alpha-Shapes
This example builds a basic alpha shape using a Delaunay triangulation as underlying triangulation.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Alpha_shape_3.h>
#include <fstream>
#include <list>
#include <cassert>
typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds;
typedef Gt::Point_3 Point;
typedef Alpha_shape_3::Alpha_iterator Alpha_iterator;
int main()
{
std::list<Point> lp;
//read input
std::ifstream is("./data/bunny_1000");
int n;
is >> n;
std::cout << "Reading " << n << " points " << std::endl;
Point p;
for( ; n>0 ; n--) {
is >> p;
lp.push_back(p);
}
// compute alpha shape
Alpha_shape_3 as(lp.begin(),lp.end());
std::cout << "Alpha shape computed in REGULARIZED mode by default"
<< std::endl;
// find optimal alpha value
Alpha_iterator opt = as.find_optimal_alpha(1);
std::cout << "Optimal alpha value to get one connected component is "
<< *opt << std::endl;
as.set_alpha(*opt);
assert(as.number_of_solid_components() == 1);
return 0;
}
## Building Basic Alpha Shapes for Many Points
When many points are input in the alpha shape, say more than 10 000, it may pay off to use a Delaunay triangulation with Fast_location policy as underlying triangulation in order to speed up point location queries (cf. Section The Location Policy Parameter).
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Alpha_shape_3.h>
#include <fstream>
#include <list>
#include <cassert>
typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds;
typedef CGAL::Alpha_shape_3<Delaunay> Alpha_shape_3;
typedef K::Point_3 Point;
typedef Alpha_shape_3::Alpha_iterator Alpha_iterator;
typedef Alpha_shape_3::NT NT;
int main()
{
Delaunay dt;
std::ifstream is("./data/bunny_1000");
int n;
is >> n;
Point p;
std::cout << n << " points read" << std::endl;
for( ; n>0 ; n--) {
is >> p;
dt.insert(p);
}
std::cout << "Delaunay computed." << std::endl;
// compute alpha shape
Alpha_shape_3 as(dt);
std::cout << "Alpha shape computed in REGULARIZED mode by defaut."
<< std::endl;
// find optimal alpha values
Alpha_shape_3::NT alpha_solid = as.find_alpha_solid();
Alpha_iterator opt = as.find_optimal_alpha(1);
std::cout << "Smallest alpha value to get a solid through data points is "
<< alpha_solid << std::endl;
std::cout << "Optimal alpha value to get one connected component is "
<< *opt << std::endl;
as.set_alpha(*opt);
assert(as.number_of_solid_components() == 1);
return 0;
}
## Example for Weighted Alpha-Shapes
The following examples build a weighted alpha shape requiring a regular triangulation as underlying triangulation. The alpha shape is built in GENERAL mode.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Regular_triangulation_euclidean_traits_3.h>
#include <CGAL/Regular_triangulation_3.h>
#include <CGAL/Alpha_shape_3.h>
#include <list>
typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds;
typedef CGAL::Regular_triangulation_3<Gt,Tds> Triangulation_3;
typedef Alpha_shape_3::Cell_handle Cell_handle;
typedef Alpha_shape_3::Vertex_handle Vertex_handle;
typedef Alpha_shape_3::Facet Facet;
typedef Alpha_shape_3::Edge Edge;
typedef Gt::Weighted_point Weighted_point;
typedef Gt::Bare_point Bare_point;
int main()
{
std::list<Weighted_point> lwp;
//input : a small molecule
lwp.push_back(Weighted_point(Bare_point( 1, -1, -1), 4));
lwp.push_back(Weighted_point(Bare_point(-1, 1, -1), 4));
lwp.push_back(Weighted_point(Bare_point(-1, -1, 1), 4));
lwp.push_back(Weighted_point(Bare_point( 1, 1, 1), 4));
lwp.push_back(Weighted_point(Bare_point( 2, 2, 2), 1));
//build alpha_shape in GENERAL mode and set alpha=0
Alpha_shape_3 as(lwp.begin(), lwp.end(), 0, Alpha_shape_3::GENERAL);
//explore the 0-shape - It is dual to the boundary of the union.
std::list<Cell_handle> cells;
std::list<Facet> facets;
std::list<Edge> edges;
as.get_alpha_shape_cells(std::back_inserter(cells),
as.get_alpha_shape_facets(std::back_inserter(facets),
as.get_alpha_shape_facets(std::back_inserter(facets),
as.get_alpha_shape_edges(std::back_inserter(edges),
std::cout << " The 0-shape has : " << std::endl;
std::cout << cells.size() << " interior tetrahedra" << std::endl;
std::cout << facets.size() << " boundary facets" << std::endl;
std::cout << edges.size() << " singular edges" << std::endl;
return 0;
}
## Example for Fixed Weighted Alpha-Shapes
Same example as previous but using a fixed value of alpha.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Regular_triangulation_3.h>
#include <CGAL/Regular_triangulation_euclidean_traits_3.h>
#include <CGAL/Fixed_alpha_shape_3.h>
#include <CGAL/Fixed_alpha_shape_vertex_base_3.h>
#include <CGAL/Fixed_alpha_shape_cell_base_3.h>
#include <list>
typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds;
typedef CGAL::Regular_triangulation_3<Gt,Tds> Triangulation_3;
typedef CGAL::Fixed_alpha_shape_3<Triangulation_3> Fixed_alpha_shape_3;
typedef Fixed_alpha_shape_3::Cell_handle Cell_handle;
typedef Fixed_alpha_shape_3::Vertex_handle Vertex_handle;
typedef Fixed_alpha_shape_3::Facet Facet;
typedef Fixed_alpha_shape_3::Edge Edge;
typedef Gt::Weighted_point Weighted_point;
typedef Gt::Bare_point Bare_point;
int main()
{
std::list<Weighted_point> lwp;
//input : a small molecule
lwp.push_back(Weighted_point(Bare_point( 1, -1, -1), 4));
lwp.push_back(Weighted_point(Bare_point(-1, 1, -1), 4));
lwp.push_back(Weighted_point(Bare_point(-1, -1, 1), 4));
lwp.push_back(Weighted_point(Bare_point( 1, 1, 1), 4));
lwp.push_back(Weighted_point(Bare_point( 2, 2, 2), 1));
//build one alpha_shape with alpha=0
Fixed_alpha_shape_3 as(lwp.begin(), lwp.end(), 0);
//explore the 0-shape - It is dual to the boundary of the union.
std::list<Cell_handle> cells;
std::list<Facet> facets;
std::list<Edge> edges;
as.get_alpha_shape_cells(std::back_inserter(cells),
as.get_alpha_shape_facets(std::back_inserter(facets),
as.get_alpha_shape_facets(std::back_inserter(facets),
as.get_alpha_shape_edges(std::back_inserter(edges),
std::cout << " The 0-shape has : " << std::endl;
std::cout << cells.size() << " interior tetrahedra" << std::endl;
std::cout << facets.size() << " boundary facets" << std::endl;
std::cout << edges.size() << " singular edges" << std::endl;
return 0;
}
## Building an Alpha Shapes with Exact Comparisons of Alpha Values
On some platforms, the alpha shapes of the set of points of this example cannot be computed when using a traits with inexact constructions. To be able to compute them with a traits with inexact constructions, the tag ExactAlphaComparisonTag is set to Tag_true.
File Alpha_shapes_3/ex_alpha_shapes_exact_alpha.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Alpha_shape_3.h>
#include <fstream>
#include <list>
#include <cassert>
typedef CGAL::Tag_true Alpha_cmp_tag;
//We use CGAL::Default to skip the complete declaration of base classes
typedef CGAL::Triangulation_data_structure_3<Vb,Fb> Tds;
//Alpha shape with ExactAlphaComparisonTag set to true (note that the tag is also
//set to true for Vb and Fb)
typedef Gt::Point_3 Point;
int main()
{
//Set of points for which the alpha shapes cannot be computed with
//a floating point alpha value (on certain platforms)
std::list<Point> lp;
lp.push_back(Point(49.2559,29.1767,6.7723));
lp.push_back(Point(49.3696,31.4775,5.33777));
lp.push_back(Point(49.4264,32.6279,4.6205));
lp.push_back(Point(49.3127,30.3271,6.05503));
// compute alpha shape
Alpha_shape_3 as(lp.begin(),lp.end(),0,Alpha_shape_3::GENERAL);
return 0;
}
## Example for Periodic Alpha Shapes
The following example shows how to use the periodic Delaunay triangulation (Chapter 3D Periodic Triangulations) as underlying triangulation for the alpha shape computation.
In order to define the original domain and to benefit from the built-in heuristic optimizations of the periodic Delaunay triangulation computation it is recommended to first construct the triangulation and then construct the alpha shape from it. The alpha shape constructor that takes a point range can be used as well but in this case the original domain cannot be specified and the default unit cube will be chosen and no optimizations will be used.
It is also recommended to switch the triangulation to 1-sheeted covering if possible. Note that a periodic triangulation in 27-sheeted covering space is degenerate. In this case an exact constructions kernel needs to be used to compute the alpha shapes. Otherwise the results will suffer from round-off problems.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_3_triangulation_traits_3.h>
#include <CGAL/Periodic_3_Delaunay_triangulation_3.h>
#include <CGAL/Alpha_shape_3.h>
#include <CGAL/Random.h>
#include <CGAL/point_generators_3.h>
// Traits
// Vertex type
// Cell type
typedef CGAL::Triangulation_data_structure_3<AsVb,AsCb> Tds;
typedef CGAL::Alpha_shape_3<P3DT3> Alpha_shape_3;
typedef PK::Point_3 Point;
int main()
{
CGAL::Random random(7);
CGAL::Random_points_in_cube_3<Point, Creator> in_cube(1, random);
std::vector<Point> pts;
// Generating 1000 random points
for (int i=0 ; i < 1000 ; i++) {
Point p = *in_cube++;
pts.push_back(p);
}
// Define the periodic cube
P3DT3 pdt(PK::Iso_cuboid_3(-1,-1,-1,1,1,1));
// Heuristic for inserting large point sets (if pts is reasonably large)
pdt.insert(pts.begin(), pts.end(), true);
// As pdt won't be modified anymore switch to 1-sheeted cover if possible
if (pdt.is_triangulation_in_1_sheet()) pdt.convert_to_1_sheeted_covering();
std::cout << "Periodic Delaunay computed." << std::endl;
// compute alpha shape
Alpha_shape_3 as(pdt);
std::cout << "Alpha shape computed in REGULARIZED mode by default."
<< std::endl;
// find optimal alpha values
Alpha_shape_3::NT alpha_solid = as.find_alpha_solid();
Alpha_shape_3::Alpha_iterator opt = as.find_optimal_alpha(1);
std::cout << "Smallest alpha value to get a solid through data points is "
<< alpha_solid << std::endl;
std::cout << "Optimal alpha value to get one connected component is "
<< *opt << std::endl;
as.set_alpha(*opt);
assert(as.number_of_solid_components() == 1);
return 0;
}
|
# Find a Positive Value of X for Which the Given Equation is Satisfied: X 2 − 9 5 + X 2 = − 5 9 - Mathematics
Sum
Find a positive value of x for which the given equation is satisfied:
$\frac{x^2 - 9}{5 + x^2} = - \frac{5}{9}$
#### Solution
$\frac{x^2 - 9}{5 + x^2} = \frac{- 5}{9}$
$\text{ or }9 x^2 - 81 = - 25 - 5 x^2 [\text{ After cross multiplication }]$
$\text{ or }9 x^2 + 5 x^2 = - 25 + 81$
$\text{ or }14 x^2 = 56$
$\text{ or }x^2 = \frac{56}{14}$
$\text{ or }x^2 = 4 = 2^2$
$\text{ or }x = 2$
$\text{ Thus, }x = 2\text{ is the solution of the given equation . }$
$\text{ Check: }$
$\text{ Substituting }x = 2\text{ in the given equation, we get: }$
$\text{ L . H . S . }= \frac{2^2 - 9}{5 + 2^2} = \frac{4 - 9}{5 + 4} = \frac{- 5}{9}$
$\text{ R . H . S . }= \frac{- 5}{9}$
$\therefore\text{ L . H . S . = R . H . S . for }x = 2 .$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 8 Maths
Chapter 9 Linear Equation in One Variable
Exercise 9.3 | Q 24.1 | Page 17
|
The theory determinant is a vast topic, so we won’t dive too much into it here.
The derterminant can be used for quickly testing whether a set of vectors are linearly dependent.
## 1. Calculation
The calculation of determinant of order 1 proceeds as follows :
The calculation of determinant of order 2 proceeds as follows :
The calculation of determinant of order 3 proceeds as follows :
The calculation of determinant of order higher than 3 is beyond the scope of the this course. Please refer here if you want to know more about it.
## 2. Dependence test
The test of linear dependence is done by using the following property of determinants :
The contraposition of $(4)$ gives :
#### Example I
Let’s test the linear dependence of the following vectors : $\left(\begin{smallmatrix} 8 \\ -4 \\ -3 \end{smallmatrix}\right), \left( \begin{smallmatrix} 2 \\ -4 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} -2 \\ 0 \\ 1 \end{smallmatrix} \right)$
According to $(4)$ the vetors are linearly dependent.
## 3. But more concretely ?
How does the determinant help telling about the linear dependance of vectors ?
Figure 7.1 : area of a parallelogramm (wikipedia)
Figure 7.2 : volume of a parallelepiped (wikipedia)
The determinant actually gives the area of the parallelogram that two vectors of $\mathbb{R}^2$ (Figure 7.1) form together, respectively the volume of the parallelepiped that three vectors of $\mathbb{R}^3$ form togheter (Figure 7.2).
Figure 7.3 illustrates the line segment that two linearly dependent vectors of $\mathbb{R}^2$ form together. Since that one has no width, the area is equal to $0$. Figure 7.4 illustrates the same idea in $\mathbb{R}^3$. Without any height the volume equals $0$.
Figure 7.3 : area equals zero
Figure 7.4 : volume equals zero
## Recapitulation
Determinant of vectors can be used for quickly testing whether those vectors are linearly dependent or not.
The determinant gives (in absolute value) the area of the parallelogram that two vectors of $\mathbb{R}^2$ form together, respectively the volume of the parallelepided that three vectors of $\mathbb{R}^3$ form together. More generally, the determinant gives the hypervolume of the hyperparallelepiped that $n$ vectors of $\mathbb{R}^n$ form together.
If a set of vectors are linearly dependent, then their determinant equals $0$.
|
# zbMATH — the first resource for mathematics
A new proof of a theorem concerning the set $$\{ N^-_k (x)\}^n_{k=0}$$. (English) Zbl 0917.41019
The set $$\{N_k^-(x)\}_{k=0}^n$$ is defined on the complex plane $$\mathbb{C}$$ with $$N_k^-(x)=| x |^k$$, if $$k$$ is even and $$N_k^-(x)=x| x |^{k-1}$$, if $$k$$ is odd. Let $$V^-_n(x_1,\dots,x_n)$$ denote the $$n\times n$$ matrix with the $$(i,j)$$ entry $$N^-_{i-1}(x_j)$$. The author earlier [Approximation Theory Appl. 12, No. 2, 45-53 (1996; Zbl 0861.41025)] discussed some approximation properties of the functions $$N^-_k(x)$$. Among others he proved the following theorem: “If $$x_1,\dots,x_n \in \mathbb{C}$$ are distinct and no three of them have the same modulus, then $\det V^-_n(x_1,\dots,x_n)=\prod_{1\leq i<j\leq n}(a_{ji}x_j-a_{ji}x_i)\not =0,$ where $$a_{ji}, a_{ij} \in \mathbb{C}$$ such that $$| a_{ij}| =| a_{ji}|=1$$ for all $$1\leq i < j \leq n$$.” In the paper under review the author gives a new proof to this theorem.
Reviewer: M.Lenard (Kuwait)
##### MSC:
41A50 Best approximation, Chebyshev systems
|
# Question
Formatted question description: https://leetcode.ca/all/1694.html
1694. Reformat Phone Number
You are given a phone number as a string number. number consists of digits, spaces ' ', and/or dashes '-'.
You would like to reformat the phone number in a certain manner.
Firstly, remove all spaces and dashes.
Then, group the digits from left to right into blocks of length 3 until there are 4 or fewer digits.
The final digits are then grouped as follows:
2 digits: A single block of length 2.
3 digits: A single block of length 3.
4 digits: Two blocks of length 2 each.
The blocks are then joined by dashes. Notice that the reformatting process should never produce any blocks of length 1 and produce at most two blocks of length 2.
Return the phone number after formatting.
Example 1:
Input: number = "1-23-45 6"
Output: "123-456"
Explanation: The digits are "123456".
Step 1: There are more than 4 digits, so group the next 3 digits. The 1st block is "123".
Step 2: There are 3 digits remaining, so put them in a single block of length 3. The 2nd block is "456".
Joining the blocks gives "123-456".
Example 2:
Input: number = "123 4-567"
Output: "123-45-67"
Explanation: The digits are "1234567".
Step 1: There are more than 4 digits, so group the next 3 digits. The 1st block is "123".
Step 2: There are 4 digits left, so split them into two blocks of length 2. The blocks are "45" and "67".
Joining the blocks gives "123-45-67".
Example 3:
Input: number = "123 4-5678"
Output: "123-456-78"
Explanation: The digits are "12345678".
Step 1: The 1st block is "123".
Step 2: The 2nd block is "456".
Step 3: There are 2 digits left, so put them in a single block of length 2. The 3rd block is "78".
Joining the blocks gives "123-456-78".
Example 4:
Input: number = "12"
Output: "12"
Example 5:
Input: number = "--17-5 229 35-39475 "
Output: "175-229-353-94-75"
Constraints:
2 <= number.length <= 100
number consists of digits and the characters '-' and ' '.
There are at least two digits in number.
# Algorithm
The idea is to first take out all the numbers in it, and then follow every 3 with a’-‘. Take a final look, if the last character happens to be’-‘, remove it; if the penultimate character is’-‘, then swap the penultimate and third characters
# Code
Java
Java
class Solution {
public String reformatNumber(String number) {
StringBuffer digits = new StringBuffer();
int length = number.length();
for (int i = 0; i < length; i++) {
char c = number.charAt(i);
if (Character.isDigit(c))
digits.append(c);
}
int digitsCount = digits.length();
if (digitsCount <= 3)
return digits.toString();
if (digitsCount == 4)
return digits.substring(0, 2) + "-" + digits.substring(2);
int remainder = digitsCount % 3;
if (remainder == 1)
remainder += 3;
int firstPartLength = digitsCount - remainder;
StringBuffer sb = new StringBuffer();
for (int i = 0; i < firstPartLength; i += 3) {
if (i > 0)
sb.append('-');
sb.append(digits.charAt(i));
sb.append(digits.charAt(i + 1));
sb.append(digits.charAt(i + 2));
}
if (remainder > 0) {
sb.append('-');
if (remainder == 4) {
sb.append(digits.substring(firstPartLength, firstPartLength + 2));
sb.append('-');
sb.append(digits.substring(firstPartLength + 2));
} else
sb.append(digits.substring(firstPartLength));
}
return sb.toString();
}
}
|
python command line version 2048 games
Complete code, please move Python implements command line version 2048.
It is reasonable to see here and write here, you should be considered to have completed the introduction, so next, you should use the knowledge learned by the introduction to write a command-line version of 2048 game.
The logic of 2048 is nothing more than the operation of 4x4 squares. There is a number in each square. We can operate these numbers to move. If two identical numbers collide under our operation, they can be merged.
And this 4x4 square is nothing more than a matrix.
Our operation can be understood as input characters. wsad stands for up, down, left and right, y stands for OK, and n stands for cancel.
python's function to receive characters is input, for example
>>> x = input("input a number")
input a number5
>>> x
'5'
To create a matrix, np.zeros([4,4]).astype(int) can be used to represent the creation of a matrix 4 × 4 4\times4 four × All 0 matrices of 4 and transformed into shaping.
There are only 16 elements in the matrix. Although the cycle efficiency is low, it is enough to meet people's operation speed.
Operation logic
2048 has only four gesture actions, i.e. up, down, left and right. The results caused by these four actions can be attributed to the operation of a single line or column, and then to the operation of a list.
For example, for [0,2,2,0], if you merge to the right, the output is [0,0,0,4], and if you merge to the left, the output is [4,0,0,0].
The implementation method is as follows
def addZeros(lst:list,flag:bool=True):
zeros = [0]*(4-len(lst))
return zeros+lst if flag else lst+zeros
# When flat is true, fill 0 on the left; Otherwise, fill the right
def rmZero(lst, flag=True):
lst = [x for x in lst if x]
end = len(lst)-1
if end < 1:
index = range(end) if flag else range(end,-1,-1)
iter = 1 if flag else -1
for i in index:
if lst[i] == lst[i+iter]:
lst[i] *= 2
lst[i+iter] = 0
lst = [x for x in lst if x]
Then, rmZero is executed for both rows and columns. Where wsad represents up, down, left and right respectively
# w. S, a and D are up, down, left and right respectively
def updateMat(mat,op):
flag = op in "sd"
if op in "ws":
for i in range(4):
mat[:,i] = rmZero(mat[:,i],flag)
else:
for i in range(4):
mat[i,:] = rmZero(mat[i,:],flag)
return mat
These three functions have completed the interaction logic of 2048. The next step is to embed the updateMat function into the game process.
initialization
Before the 2048 game starts, you need to initialize a 4x4 matrix, and then before each operation, you need to randomly generate a number at the position of 0 in the matrix. The value range of randomly generated numbers determines the difficulty of the game, so the generation method is also flexible. A common generation method is given below
def addNew(mat):
i,j = randint(4),randint(4)
while(mat[i,j]!=0):
i,j = randint(4),randint(4)
else:
x = randint(1,100)
x = 7 - np.floor(np.log2(x))
mat[i,j] = int(2**x)
print(mat)
Interactive operation
Then there is interactive operation. asdw means operation and q means exit.
def InputNew(mat):
op = input("input operator:")
if op in "asdw":
newMat = updateMat(mat*1,op)
if np.max(np.abs(mat-newMat))==0:
print("Invalid operation")
return mat,"Error"
else:
return newMat,"True"
if op == 'q':
return mat, "Exit"
print("Invalid Input")
return mat, "Error"
main
if __name__ == "__main__":
while(1):
newMat,flag = InputNew(mat)
while flag=="Error":
newMat,flag = InputNew(mat)
if flag == "Exit":
break
mat = newMat
print(mat)
print('-'*30)
if np.max(mat)==2048: flag == "win"
if np.min(mat)!=0: flag == "lose"
if flag in ["win","lose"]:
if input(f"you {flag}, play again? ")=="y":
mat *= 0
else: break
Posted by haijie1984 at Nov 14, 2021 - 9:03 PM Tag: Python Back-end
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# Quick Start¶
For a more advanced use of PyGMO, please refer to our Tutorials, or Examples.
## On one CPU¶
Let us try to solve the 50-dimensional Schwefel problem.
from PyGMO import *
prob = problem.schwefel(dim = 50)
algo = algorithm.de(gen = 500)
isl = island(algo,prob,20)
print isl.population.champion.f
isl.evolve(10)
print isl.population.champion.f
And it is done!! We have used the algorithm Differential Evolution and we have evolved ten times 500 generations.
(17643.0955597,)
(0.0006364301698340569,)
## On many CPUs ...¶
Let us try to solve, again, the 50-dimensional Schwefel problem.
from PyGMO import *
prob = problem.schwefel(dim = 50)
algo = algorithm.de(gen = 500)
archi = archipelago(algo,prob,8,20)
print min([isl.population.champion.f for isl in archi])
archi.evolve(10)
print min([isl.population.champion.f for isl in archi])
And it is done!! We have launched eight separated threads each one running an instance of Differential Evolution. Each thread evolves separately ten times 500 generations. We then print the best found in the 8 runs.
## ... and migrating solutions ...¶
Let us try to solve, again, the 50-dimensional Schwefel problem.
from PyGMO import *
prob = problem.schwefel(dim = 50)
algo = algorithm.de(gen = 500)
archi = archipelago(algo,prob,8,20, topo = topology.ring())
print min([isl.population.champion.f for isl in archi])
archi.evolve(10)
print min([isl.population.champion.f for isl in archi])
And it is done!! We have launched eight separated threads each one running an instance of Differential Evolution. Each thread evolves for 500 generation its population, then it exchanges solutions according to the defined ring topology. All happens asynchronously in the background We then print the best found in the 8 runs.
## ... and between different algorithms ...¶
Let us try to solve, again, the 50-dimensional Schwefel problem.
from PyGMO import *
prob = problem.schwefel(dim = 50)
algo = []
for i in range(1,9):
algo.append(algorithm.de(gen=500,variant=i))
archi = archipelago(topo=topology.ring())
for i in range(0,8):
archi.push_back(island(algo[i],prob,20))
print min([isl.population.champion.f for isl in archi])
archi.evolve(20)
print min([isl.population.champion.f for isl in archi])
And it is done!! We have instantiated 8 different variants of Differential Evolution and have them cooperate to solve the same optimization problem!
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# “Ethics in Quantitative Finance”
Just before going to the workshop on dependencies in finance and insurance, Tim Johnson (also known as @TCJUK on Twitter), researcher at Heriot-Watt University in Edinburgh and blogger on http://magic-maths-money.blogspot, sent me a copy of his manuscript entitled Ethics in Quantitative Finance: a pragmatic theory of markets. While opening the book, we think of Peter L. Bernstein, his masterpieces Capital Ideas (or the later Capital Ideas Evolving) as well as Against the Gods. But Tim’s book is quite different. This book is not really about finance, but about financial valuation and actuarial science. We can clearly see the deep interactions between financial mathematics and actuarial science. About uncertainy, prices and probabilities. And all those topics are embeded with a philosophical perspective
the argument is presented that financial markets are radically uncertain environments, where correspondence theories of truth are meaningless since there are no matters of fact about an uncertain financial future. In the face of this uncertainty, markets are places where “the opinion which is fated to be ultimately agreed to by all who investigate” is sought and opinions are expressed through asset prices. This implies that markets are centres of communicative action and money is behaving as a language. Using Jürgen Habermas’ analysis, this implies that market prices ‒ statements of opinions ‒ must satisfy objective, subjective and social truth criteria. The argument presented is that reciprocity guarantees the objective truth, sincerity guarantees the subjective truth and charity guarantees the rightness of a price. This explains why reciprocity is embedded in financial mathematics.
The book takes a chronological perspective. We start with a chapter Genesis of money and its impact, folllowed by one on Finance and Ethics in Medieval Europe. I should also mention that the book is full of fascinating anecdotes (in those chapters, but also later on) for instance on gambling
Gambling is often regarded as an illicit activity today and is frequently outlawed, but in ancient societies, gambling was often associated with sacrificial practises (…) Gambling was important in archaic societies because it re-distributed resources in a non-subjective manner and so inhibited the formation of hierarchies.
as described in Altman (1985) in the context of contemporary Australian aboriginal goups. I have to admit that I downloaded a dozen articles mentioned in the book, to read them this summer. I agree with Tim when he mentions Fibonacci as a major reference in financial mathematics (and discounting, as discussed in a previous post on this blog, in French – unfortunately – published recently in Risques, see also Davide Castelvecchi’s recent post).
At the start of the thirteenth century Western Europe was going through a financial revolution and the creation and management of the poena, censii, prestiti, societas and Bills of Exchange in an environment of changing values and prices required complex negotiation and calculation. To cope with the situation, the merchants turned to Leonardo Bonacci, better known as Fibonacci, who would change European culture by changing western mathematics.
There is also a series of very interesting thoughts in those chapters on causality, truth, information, starting with Hume’s is/ought problem, as well as Kant’s perspective. All those ideas are very important when we have in mind that prices in financial markets are related to beliefs and (subjective) information (we’ll get back on that issue later on).
We have then a nice chapter on the Philosophical Basis of Modernity, with Descartes, Spinoza or Locke. This is usually not an important issue is other books on the history of financial mathematics, but it is clearly an important issue (just think of algorithmic trading and ethical questions related to artificial intelligence – those points will be discussed in the last chapter). I was glad to see this chapter here. We finally reach the chapter on The financial revolution of the XVIIth Century, followed by The Enlightenment and l’homme éclairThe starting point is simple,
When trade was carried out over longer distances, more money was needed, the time-scales longer and the risks greater. In these circumstances the societas, partnerships of recognisable individuals, were inadequate and new types of commercial organisation, based on the idea of a corporation, emerged in the Italian city states to enable more people to pool their resources in larger scale commercial operations.
In those chapter, we discover more how modern financial markets emerged a few centuries ago
When the value of an asset was uncertain it would be harder for a broker to find property-owners, who could agree a price. In these situations, liquidity was provided by ‘jobbers’ or ‘market-makers’ who ensured that when a property owner, usually dealing through a broker, wished to trade the market had an opinion as to the price of the asset. Jobbers could form an opinion by trading in blanco, so they did not need the resources to buy the physical assets, and jobbers were associated with people with limited resources (…) The practice emerged of jobbers, today known as ‘dealers’ or ‘market-makers’, being required to simultaneously quote ‘bid’ prices, at which they would buy an asset, and ‘offer’ or ‘ask’ prices, at which they would sell, without knowing if the counter-party is seeking to buy or sell the asset ‒ though the quantity would affect the quoted price.
including some remarks on ‘public opinion’ that might be related to so-called predictive markets
The development of trust between the government and the market did not simply appear but was part of a process that saw power migrate from the aristocratic court to the ‘public opinion’ of the propertied middle classes. London’s coffee-houses became central in the formation of this public opinion (…) Like the Greek agora and the Roman forum, the London coffee-house acted as a focus of market practice and legal theory and the middle-classes, following Locke, came to believe that, like money, the law should be a universal, abstract entity.
The next chapter is Practical Mathematics: the development of probability theory, with some etymological perspective (that I love)
Huygens had to translate his Dutch text into Latin so that it satisfied the requirements of the universities. He struggled to find good Latin words for the terms he was using in Dutch, which originated in gambling. He had used the Dutch word kans (chance) for ‘expectation’, which would usually be translated into Latin as sors, and eventually he, or van Schooten, chose expectatio, giving the English term ‘expectation’. However, Huygens had also considered using the Latin word spes which was the classical term for the virtue ‘Hope’. In French, spes was chosen and today the French use espérance when referring to mathematical expectation, while the Dutch, faithful to Stevin’s precedent, use their own term, verwachting, meaning hope, promise, expectation.
Again, I love in that book the actuarial perspective considered to motivate classical financial theories,
The Huygens brothers did not manage to value annuities. This problem would be solved in 1671 by another student of van Schooten’s Johan de Witt, in a report, Waerdye van Lyf-renten Naer Proportie van Los-Renten (On the Valuation of Annuities in Proportion to Redeemable Loans), for the Dutch government. De Witt used the age-old practice of employing the ‘law of one price’, or arbitrage, and argued that to calculate the expectations, and so value, of annuities he should use the value of equivalent debt contracts
Usually, the ‘law of one price‘ is mentioned with a financial mathematics perspective, and it is nice to have the actuarial one. There is also an interesting discussion (that is also in Ian Hacking’s Emergence of Probability) on connexions between beliefs, prices and so-called “probabilities”
The final section is the most significant but has proved problematic because Bernoulli considered situations where the sum of probabilities could be greater than one. To have probabilities summing to more than one is an issue if you think of chance as being a consequence of relative frequency, as discussed in the first parts of the Ars (…) Bernoulli considered situations where probabilities did not sum to one because he was working at a time when what was important was just treatment in financial contracts. It was not necessary that a probability summed to one, only unjust if it did not. Today, this can be understood in terms of gambling through a third party, where the probabilities, inferred from the cost of a game and the expected winnings are never equal to one, and so indicate a lack of reciprocal justice: the book-maker or casino is taking turpe lucrum..
It is actually a very import point that can be related (more generally) to predictive markets (see e.g. Wolfers & Zitzewitz (2004)),
The attitude that it is illogical for probabilities not to sum to one emerged out of a different conception of probability that was developed in the context of gaming by two Frenchmen, Pierre Rémond de Montmort and Abraham de Moive, and would come to dominate representations of probability in the nineteenth century.
In the middle of the book, we reach the chapter on the ascendency of financial mathematics. This chapter starts with a very interesting between the probability to have 0 are events in a given period of time (from the Poisson distribution, the law of small numbers), i.e. $e^{-rT}$ and the standard discounting factor used in continuous time financial models – also $e^{-rT}$. But then, we have the perspective of economists (rather than philosophers and mathematicians).
Knight felt that economics had split into two strands. There was a mathematical science, which studied closed systems based on distorting assumptions, and a descriptive science, which could deduce nothing. Economics needed to take a middle path that was both realistic and informative (…) At the time, economic theory claimed that markets brought “the value [price] of economic goods to equality with their cost” but this equality, was in fact, only an “occasional accident”. Knight argued that the reason for the theory diverging from the practice arose out of the difference between a ‘known uncertainty’, which he termed a ‘risk’ and an ‘unknown uncertainty’, which he called ‘uncertainty’.
But there are still very interesting points and references on connexions between prices and probabilities
Early in his career, Keynes had written a Treatise on Probability where he had observed that in some cases cardinal probabilities of events could be deduced, in others, ordinal probabilities ‒ one event was more likely than another ‒ could be inferred, but there were a large class of problems that were not reducible to the concept of probability. Keynes’ argument was challenged by a young Cambridge mathematician, Frank Ramsey, who in Truth and Probability (1926) argued that probability relations between a premise and a conclusion could always exist. He defined ‘probability’ as simply ‘a degree of belief’ that could always be decided through a (betting) market. Keynes, a friend and mentor of Ramsey, appears to have been satisfied with the argument and came to believe that the only way to resolve ‘radical uncertainty’ was through discussion. Because Ramsey died young, at the age of 26 in 1930, his approach is more familiar through the Italian, Bruno de Finetti (published 1931) and the American statistician Leonard Savage (published 1954) . Collectively these approaches are considered subjectivist or Bayesian, pointing to their relationship to the eighteenth-century Bayes’ Rule that could be used to update probabilities. De Finetti had enrolled at Milan Polytechnic in 1923 with a view to following in his father’s footsteps into railway engineering but transferred to mathematics and graduated from the University of Milan in 1927. He took a job at the Italian Central Statistical Institute but left to work for an insurance company in Trieste, Assicurazioni Generali, in 1931. He would work as an actuary for the next fifteen years, taking a couple of academic posts along the way. In 1947 he became a full-time academic, finishing his career at La Sapienza University in Rome. De Finetti asserted that “Probability does not exist” because it was merely an expression of an individual’s opinion. He employed the notation ‘Pr’ because it could mean ‘probability’, ‘price’ or ‘prevision’ and could not be tied down. De Finetti argued that in science there were two types of laws: deterministic “necessary and immutable laws; phenomena in nature are determined by their antecedents” and ‘truth-like’ or probable laws that express statistical regularities.
I was glad to see Bruno de Finetti here, because he is a major reference (and he deserve to be more popular in economics). Before reaching the end of the chapter (with a nice connexion between the “representative agent” and Quételet’s “homme moyen“) there is a brief introduction about portfolio selection, with Harry Markowitz and Arthur D. Roy, and I have to admit that I disagree when it says
The question of portfolio choice was one of balancing the risks of disaster against the opportunities for reward, a version of the Scholastic argument that without risk there could be no profit. The question Roy and Markowitz needed to answer was how should risk be measured. Both Markowitz and Roy chose to use the variance, a measure of the average distance of a sample point from the mean, as a proxy for risk. This is not obvious, since risks are colloquially associated with losses, while variance regards high gains as equally unattractive as high losses and reveals that they were thinking about profit being related to uncertainty.
I think that Roy’s Safety First and the Holding of Assets is much more general, and also probably more interesting (from a philosophical perspective, not a pratical one). This safety-first criterion selects a portfolio based on the criterion that the probability of the portfolio’s return falling below a minimum desired threshold is minimized. What I like about this perspective is that it relates risks to quantile levels, and to ruin probabilies used in actuarial mathematics. As mentioned in Safety First and the Holding of Assets, assuming that returns are normally distributed means that the risk of the portfolio is related to the variance, but actually, in the philosophical framework, we are not interested by the “average distance from the mean”, but a quantile level. Anyway, that is just a comment on a brief sentence.
Then we have a detailed chapter on The Fundamental Theorem of Asset Pricing. It starts with a difficult mathematical question, with philosophical implications, related to the concept of measures
This solved the problem of worrying about outcomes but left the issue of identifying the probability of events. The most obvious ‘measure’ of an event is to count its elements or the relative size of different events, but this means you must identify each outcome in an event, which is impossible. In associating a probability with an abstract measure, Kolmogorov had freed it from being tied to concepts rooted in counting elements of event sets. (…) In the classical approach, a probability of zero implies impossibility, whereas a probability of one implies certainty. In Kolmogorov’s conception, this is not so straightforward.
Indeed, it is a rather important and complex question, that cannot be solved without a significant mathematical background. I wanted also to add a brief comment to a sentence that uses a word I do not like…
They made this assessment using a mathematical equation, the Gaussian copula, which had been identified in the 1950s.
This idea started with articles such as Wired or the Financial Times in 2009), based on David Li’s work on Gaussian models for credit risk models (see also the paper by Donald MacKenzie and Taylor Spears entitled ‘The Formula That Killed Wall Street’? The Gaussian Copula and the Material Cultures of Modelling). The idea “Gaussian copula” remained, but actually, the underlying strory is much more simple (actually, in 1998, I was developping a credit risk model for a French firm based on that idea). As in Merton’s model for default, there is a default when the “value of the debt” goes above a threshold. And this non-observable value is supposed to be Gaussian. This can be some sort of a probit model. With several companies, it is rather natural to consider a multivariate joint normal distribution, which yields a multivariate probit model. With that model, the latent unobservable distribution is a multivariate Gaussian, just like in portfolio management. I am no a big fan of the terminology, since copulas became popular in finance in the late 90’s. Actually, this model is rather old, and is only a multivariate probit model. Nothing nerdy here actually… There is then an inspiring paragraph on economics, mathematics, and modeling
For economists, mathematics was “part of the plumbing” that supported economic theory, a view similar to the one they had of money as a neutral tool. Mathematicians are concerned with understanding the relationships between objects and mathematics can reveal connections or differences. Trygve Håvelmo had recognised the problem when he had been awarded the Nobel Prize for Economics in 1989. He reflected that his aspirations for introducing mathematics to economics had not been met. He identified the primary issue as being that the economic models that ‘econometricians’ had been trying to apply to the data were probably wrong. More fundamentally, economics never generated new mathematics ‒ ways of seeing relationships ‒ in the way that the physical sciences had stimulated developments in mathematics. Economists had simply adapted concepts from other fields to their own devices.
Then we reach the chapter (with an odd name) entitled Two Women and a Duck: a Pragmatic Theory of Markets, which starts with an interesting point on models
When an idea is taken to be true by a culture but is in fact an illusion it has become an ideology. An argument that goes back, employed by Marx amongst others, is that ideologies emerge out of an intent to deceive, which implies that there is a Laplacian will capable of persuading a community to accept an ideology. A less intentional explanation is that ideologies are simply convenient models
And then, we get back to our discussion on connexions between probabilities and prices. I leave here the complete page, which give a good overview on the style and the perspective of Tim’s book
In general, the martingale measure specifies where the current price of an asset lies in the distribution of future prices. In being based on observed prices, the martingale measure represents an objective pricing measure that should be used in preference to any subjective measures. This idea that prices give probabilities was in Huygens’ Van Rekeningh of 1655 and was the approach de Witt had taken in pricing annuities in 1671. Probability measures based on historic prices yield subjective measures in that they relate to the past, not the future. Jacob Bernoulli, in Ars Conjectandi, considered situations where probabilities did not sum to 1. These were illogical in a frequentist approach to probability but meaningful in representing unfair arbitrages. The objectivity of probability does not arise from the materiality of counting possible outcomes but in the ethical concept of fairness. In markets, as Aristotle had observed, mathematics establishes the equality necessary for justice in exchange, contributing to social cohesion. BSM (Black-Scholes-Merton) guarantees the coherence of its prices on the basis that a price must preclude arbitrage opportunities. Specifically, if a market-maker offered a price that presented an arbitrage, other traders would exploit the market-maker’s obligation to be sincere in offering both bid and ask prices and bankrupt the market-maker. This practical observation had been shown in Frank Ramsey’s argument that probabilities exist for radically uncertain events. Ramsey noted that a standard way of measuring ‘degrees of belief’, or a probability, is through betting odds and went on to formulate some laws of probability, finishing with the observation that
These are the laws of probability, … If anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event. This is the ‘Dutch Book’ argument and is an alternative to the ‘Golden Rule’ ‒ “Do to others as you would have them do to you” and re-emerges as Kant’s categorical imperative. It is founded on the moral concepts of fairness and reciprocity, not on material acts of dynamic hedging. Ramsey went on to argue that having any definite degree of belief implies a certain measure of consistency, namely willingness to bet on a given proposition at the same odds for any stake, the stakes being measured in terms of ultimate values. Having degrees of belief
obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you.
Then we start a fascinating discussion on truth, and what could be this “true price” given by a mathematical model
The meaning of ‘true’ in relation to prices in markets is unclear because of the uncertainty in finance. The word ‘true’ derives from the Germanic triuwe meaning faithful, reliable or secure and, at its most basic, the truth of a statement rests on whether it corresponds to the facts: it is either true or not that the balance of births and deaths in an English parish in the year 1780 was x. In this conception, a belief is independent of the fact and is true only if it corresponds to the fact. These correspondence theories depend on a statement being verifiable and are central to logical positivism, but are impossible to employ in complex situations or those involving an uncertain future. To deal with this problem of correspondence theories being irrelevant to most human experience, coherence theories emerged out of idealism. For idealists, what was important was that beliefs formed a coherent whole that reflected the unity of knowledge. The problem with this approach is that a perfectly coherent set of beliefs might not correspond to the facts.
To be more specific
In response to the inadequacy of these two approaches to truth, the American philosopher Charles Sanders Peirce proposed a novel definition of truth in the late nineteenth century as “The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth”. This conception of truth rests on the idea of a ‘community’ that stands for the ‘all’ that comes to an agreement. A consequence is that knowledge need not be based on rigorous deductions; Peirce said it should resemble a cable of thin interweaving strands rather than a chain of strong links that is vulnerable to a single link failing. The three conceptions of truth ‒ correspondence, coherence and pragmatic ‒ are relevant to finance where they are characterised by three different types of agents.
We start seing the word “pragmatic” that was in the subtitle of the book actually.
The word ‘pragmatism’ ‒ deriving from the Greek pragmatikos meaning ‘business like’ or effective ‒ emphasised experience and practice over the idealism and theory usually associated with philosophy.
and we finally have the explanation of the chapter’s title
Financial markets, made by market-makers making assertions as to the price of an asset that are challenged by traders, are primarily concerned with a community converging on agreement. The idea of markets as places where an understanding of prices is formed, rather than just a place where goods are exchanged, is captured in a Vietnamese proverb that “two women and a duck make a market”. There is nothing in the proverb that suggests either of the women owns the duck; what is implied is that the women will converse and during that discussion they will come to some agreement as to the value of the duck. This highlights that the value of the duck cannot be established based on either an objective valuation or the subjective belief of a single person but in, at least, a three-way interaction between a speaker, an interpreter and the object under discussion. The truth of an individual’s important beliefs can only be confirmed, or refuted, through discussion with others.
And finally, in a conclusion – Some Implications of a Pragmatic Approach to Finance – we have some thoughts about pragmatism, and connexions with models, and algorithmic implementation
While an algorithm can be objective ‒ and deliver reciprocity ‒ it is not so obvious that an algorithm can be sincere in the way that people understand sincerity. It is even more difficult to think of an algorithm as being capable of charity, the most intangible market norm that is also the most human norm. Consequently, the individual borrower is alienated from the lender and the banker’s role as a mentor of the entrepreneur disappears. The bank’s task of optimising the ‘harvesting’ of loans is a departure from the Quaker principle that asked a borrower how they intended to repay a loan in the time agreed.
which is getting back to initial statements of the book, with a modern perspective.
Faced by radical uncertainty, those involved with finance cannot rely only on models rooted in physica to ensure equality between what is given and received. Rather, they must also conform to norms that ensure that their judgements can be trusted; they must ensure sincerity and charity as well as reciprocity. The fundamental concern of algorithmic trading is that, while it could ensure reciprocity and sincerity, it would be difficult to deliver charity. If prices, financial judgements, are determined by an algorithm, it would not stand for the ascendency of machines to human consciousness, but the descent of man to machine as charity disappears. This decline is avoided through the human sciences. These acknowledge the limitations of human understanding and the need to reinforce norms of behaviour through the repetition of stories, which enable individuals to imagine alternative futures and offer lessons of character to guide action. Since mathematics is neither part of physica nor practica the problem does not lie in the use of mathematics but the motivation behind that use. If the likelihood of financial crises is to be reduced, mathematical approaches to finance must be rooted in the human, not the physical, sciences.
This book is really inspiring, not only in the context of financial valuation, but more generally on ethics of modeling in economics, extracting information about ‘the truth’ and giving a value, a price, to an uncertain random quantity. It is clearly worth reading, thanks Tim.
## One thought on ““Ethics in Quantitative Finance””
1. Mohamed Amr Elgeneidy says:
This book seems interesting. Thanks for the review. I’ll pick it up when it gets published. Also thanks for the link to the blog.
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TheInfoList
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– as supplied by Edexcel Sample Assessment Material. The Proof TEST is the latest in StudyWell’s collection of downloadable resources.
Progressive matrix question - squares, circles, triangles in the corners. This is the new goal, that we split into: $gt(5,y)$ and $gt(y,2)$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The specific system used here is the one found in forall x: Calgary Remix. Is there a formula for absolute magnitude that does not contain an apparent magnitude term? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Proof by Induction.
For more Proof by Exhaustion examples and to test your knowledge of mathematical proof methods take the StudyWell PROOF test: Are you ready to test your Pure Maths knowledge? Why do flight schools refuse to tell the courses price? How is it different from resolution proof? A PowerPoint covering the Proof section of the new A-level (both years). It includes disproof by counterexample, proof by deduction, proof by exhaustion and proof by contradiction, with examples for each. The final slide lists a few suggested sources of further examples and questions on this topic. I have a question: Solve the following by deduction using backward reasoning to prove gt(5,2). If somebody can help me I really appreciate. $\lnot g(5,2)$ has been added to the set of premises and the resolution proof procedure has to be applied. I found from wikipedia that backward reasoning is same as backward chaining. For this reason, the following are very useful to know when trying to prove by deduction: Prove that the difference between the squares of any two consecutive integers is equal to the sum of those integers. Thus, there is no way to derive $gt(y,y)$. Use MathJax to format equations. Linked concepts and Questions: By goal driven search it means that we have to start at the current state. Am I a dual citizen? Hence, we have proved by deduction that the difference between the squares of any two consecutive integers is equal to the sum of those integers. PowerPoint slideshow version also included - suitable for upload to a VLE. Making statements based on opinion; back them up with references or personal experience. An interesting problem with "decomposing" natural numbers. Using again fact 1) with substitution $\{ 5/x, y/z \}$ we get: $gt(5,y) \land gt(y,y) \to gt(5,y)$. Should I try by doing the replacements provided at the end of each level of the tree? Firstly, choose and to be any two consecutive integers. To learn more, see our tips on writing great answers. registered in England (Company No 02017289) with its registered office at 26 Red Lion It follows that, in maths, proof by deduction means that you can prove that something is true by showing that it must be true for all cases that could possibly be considered. Difference or relation between Inference, Reasoning, Deduction, and Induction? But the third fact is: $\forall x \lnot gt(x,x)$. Proof Using Natural Deduction (including '=' rules), Proving this sequent using natural deduction, Proof using natural deduction (Tautology), Proof Disjunctive Syllogism using Natural Deduction, Strategies to work backward and forward when doing natural deduction proof, sed with next line (+N option) and frequency (~N) together. I have a question: Sorry I don't have any idea. This is a demo of a proof checker for Fitch-style natural deduction systems found in many popular introductory logic textbooks. How to notate this two-voice syncopation in 12/8 time? Square Can a small family retire early with 1.2M + a part time job? It follows that, in maths, proof by deduction means that you can prove that something is true by showing that it must be true for all cases that could possibly be considered. Asking for help, clarification, or responding to other answers. Solve the following by deduction using backward reasoning to prove gt(5,2). Is it possible to define an internal model of ZFC which is not set-like and which is not elementary equivalent to any definable set-like model? I found from wikipedia that backward reasoning is … It includes disproof by counterexample, proof by deduction, proof by exhaustion and proof by contradiction, with examples for each. Does the new Netflix series "The Queen's Gambit" resemble any real life chess master? By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy.
11.1 Proof by deduction Proof by deduction is the most commonly used form of proof throughout this book – for example, the proofs of the sine and cosine rules in Chapter 6 Trigonometry. Proof by deduction is a process in maths where a statement is proved to be true based on well-known mathematical principles. Can I go to Japan, where I was born? © Copyright of StudyWell Publications Ltd. 2020, Vector Arithmetic – addition/subtraction and scalar multiplication. Book says that backward chaining is same as goal dependent search. Suppose we are given the following facts: Somebody please guide me. Moreover, the below given Logical Deduction Questions are the best and most commonly collected question need to answer logically. Proof by deduction is the drawing of a conclusion by using the general rules of mathematics and usually involves the use of algebra. The proof by deduction section also includes a few practice questions… There are 12 questions in the Proof TEST (16 including subquestions) covering proof by deduction, proof by exhaustion and disproof by counterexample.
MathJax reference. Often, 2n is used to Conditions. This website and its content is subject to our Terms and How to draw the crossings in a tikz picture? The word deduce means to establish facts through reasoning or make conclusions about a particular instance by referring to a general rule or principle. London WC1R 4HQ. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It only takes a minute to sign up. Furthermore, to attempt any of the competitive exams, there will be Reasoning concept for sure.
Adding together the original two consecutive numbers also gives . The proof by deduction section also includes a few practice questions, with solutions in a separate file. Proof by deduction may require the use of algebraic symbols to represent certain numbers. How it is different from resolution proof? With natural deduction, the proof is quite straightforward: apply and-elimination followed by or-elimination (i.e. Yes it is a Resolution proof: the negation of the statement to be proved, i.e. Kindly check it and explain if possibe. Proof by deduction may require the use of algebraic symbols to represent certain numbers. Created: Mar 30, 2018| Updated: Oct 2, 2020. – Mauro ALLEGRANZA Apr 9 '16 at 11:55 How should I request a professor to restrict communication to email? Furthermore, deduction is the noun associated with the verb deduce. rearrange order of columns such that a specific column gets the same string. I want to do deduction with backward reasoning instead of resolution proof, please guide me how we do backward reasoning/goal dependent search? Welcome to advancedhighermaths.co.uk A sound understanding of Proof by Induction is essential to ensure exam success. Regarding the handwritten proof sketch, the first step is to apply the substitution $\{ 5/x, 2/z \}$ to the clause in fact 1) to get: Thus, in order to prove $gt(5,2)$ (by Modus Ponens) we have to derive the antecedent: $gt(5,y) \land gt(y,2)$. Thanks for your response. Why do some investment firms publish their market predictions? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. A PowerPoint covering the Proof section of the new A-level (both years).
So, you should be bold enough for Logical Deduction Reasoning Questions.By the way, we provided you the tips and tricks to overcome your difficulties. Consider the first one: $gt(5,y)$. I got one solution from my friend. rev 2020.10.30.37923, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us.
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Chapter 9. Extreme Value Theory
Copyright 2016 Jon Danielsson. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at. http://www.apache.org/licenses/LICENSE-2.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
Listing 9.1: Hill estimator in R Last edited: 2011
ysort = sort(y)
CT = 100
iota = 1/mean(log(ysort[1:CT]/ysort[CT+1]))
Listing 9.2: Hill estimator in Matlab Last edited: 2011
ysort = sort(y);
CT = 100;
iota = 1/mean(log(ysort(1:CT)/ysort(CT+1)));
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(12C Platinum) Sums of Powers of N numbers
01-22-2019, 10:20 AM (This post was last modified: 01-22-2019 10:23 AM by Gamo.)
Post: #1
Gamo Senior Member Posts: 665 Joined: Dec 2016
(12C Platinum) Sums of Powers of N numbers
ALG program solution of "Sums of Powers of the Natural Numbers"
1. Arithmetic Series ( Gauss Sum )
n(n+1) / 2
2. Sum of the consecutive Squares.
n(n+1)(2n+1) / 6
3. Sum of the consecutive Cubes.
n^2(n+1)^2 / 4
----------------------------------------------
Procedure:
Nth [R/S] display Sums of Powers
----------------------------------------------
1. Gauss Sum
Code:
1 [+] [X<>Y] [x] [LSTx] [÷] 2 [=]
2. Sums of consecutive Squares
Code:
1 [+] [X<>Y] [x] [LSTx] [x] ( [LSTx] [x] 2 [+] 1 ) [÷] 6 [=]
3. Sums of consecutive Cubes
Code:
1 [+] [X<>Y] [=] [X^2] [x] [LSTx] [X^2] [÷] 4 [=]
Gamo
01-22-2019, 12:28 PM (This post was last modified: 01-22-2019 12:53 PM by Albert Chan.)
Post: #2
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
Is the code for ALG mode ?
If true, all "1 +" should be "+ 1"
Also, for sum of cubes, LSTx = n ?
I would guess the code should be just Gauss Sum code, then square it.
01-22-2019, 01:19 PM (This post was last modified: 01-22-2019 01:22 PM by Gamo.)
Post: #3
Gamo Senior Member Posts: 665 Joined: Dec 2016
RE: (12C Platinum) Sums of Powers of N numbers
Albert Chan thanks for the review.
Yes program above is in ALG mode.
I was reading over at the 12C Platinum
Manual that mention about [LSTx] and
[X<>Y] functions that work differently
Compared to RPN .
For ALG programming in order to recall
LSTx it need to be in previous display so
At the start of each program above I use
1 + X<>Y so that the n can be recall by using the LSTx
So above program doesn't use Store Registers.
Example:
1 + X<>Y x .......
If n is 10 when run that will become 1 + 10 x ......
10 is last seen in display after execution forward.
Then LSTx will recall 10 for next step.
Gamo
01-22-2019, 11:57 PM
Post: #4
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: (12C Platinum) Sums of Powers of N numbers
We can use the Net Present Value (NPV) to evaluate a polynomial:
$$NPV = CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \frac{CF_3}{(1+r)^3} + \cdots + \frac{CF_n}{(1+r)^n}$$
Use the ∆% function to transform $$x=\frac{1}{1+r}$$ to $$i=100r$$.
This leads to the following generic program to evaluate a polynomial:
Code:
01- 1 1 02- 24 ∆% 03- 44 12 STO i 04- 42 13 NPV
1. Arithmetic Series
$$\frac{x(x+1)}{2}=\frac{x^2}{2}+\frac{x}{2}$$
0
CF0
2 1/x
CFi
CFi
Examples:
1 R/S
1.0000
2 R/S
3.0000
3 R/S
6.0000
10 R/S
55.0000
2. Sum of the consecutive Squares
$$\frac{x(x+1)(2x+1)}{6}=\frac{x^3}{3}+\frac{x^2}{2}+\frac{x}{6}$$
0
CF0
6 1/x
CFi
2 1/x
CFi
3 1/x
CFi
Examples:
1 R/S
1.0000
2 R/S
5.0000
3 R/S
14.0000
10 R/S
385.0000
3. Sum of the consecutive Cubes
$$\frac{x^2(x+1)^2}{4}=\frac{x^4}{4}+\frac{x^3}{2}+\frac{x^2}{4}$$
0
CF0
CFi
4 1/x
CFi
2 1/x
CFi
x<>y
CFi
Examples:
1 R/S
1.0000
2 R/S
9.0000
3 R/S
36.0000
10 R/S
3025.0000
Disclaimer: I don't have an HP-12C Platinum.
But this works with the regular HP-12C.
Kind regards
Thomas
PS: Cf. HP-12C’s Serendipitous Solver
01-23-2019, 05:40 PM (This post was last modified: 01-24-2019 04:50 PM by Albert Chan.)
Post: #5
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
From another thread about forward difference table, we can treat sum of powers on N numbers as polynomial interpolation.
For sum of kth powers, we expect a polynomial with degree k+1
Example, for sum of cubes, just use forward differences of cubes of 4 numbers
1 8 27 64
7 19 37
12 18
6
Thus sum of cubes formula = $$1\binom{n}{1}+7\binom{n}{2}+12\binom{n}{3}+6\binom{n}{4} = [n(n+1)/2]^2$$
We can also do interpolation with 5 points. (5 points "fixed" a quartic polynomial)
Example, for sum of cubes of 10 numbers, interpolate for N=10:
Code:
N Sum Intepolation for N=10 4 100 3 36 484 2 9 373 1261 1 1 298 1135 2269 0 0 250 1030 2185 3025
All the intepolations above are simple linear interpolation.
Example, 1030 is from interpolation of 2 points (3, 484), (0, 250), for N=10
01-23-2019, 07:21 PM
Post: #6
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: (12C Platinum) Sums of Powers of N numbers
(01-23-2019 05:40 PM)Albert Chan Wrote:
Code:
N Sum Interpolation for N=10 4 100 3 36 484 2 9 373 1261 1 1 298 1135 2269 0 0 250 1030 2185 3025
Could you elaborate on how to calculate these numbers?
Thomas
01-23-2019, 07:52 PM (This post was last modified: 01-23-2019 08:04 PM by Albert Chan.)
Post: #7
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
Hi, Thomas Klemm
The trick is from Acton Forman's book Numerical Method that Work, p94
It was a modified Aitken's method.
First, arrange the points in sorted order, closest to interpolated N on top.
Back in the old days, people don't have computers readily available.
The sorting ensured for each columns, interpolated values "tighter".
Manual calculation mistakes are thus easier to spot.
For each column, top point is "locked", and do interpolation with other points.
First column:
(4,100) and (3,36) => (10,484)
(4,100) and (2,9) => (10,373) ...
Second column:
(3,484) and (2,373) => (10,1261)
(3,484) and (1,298) => (10,1135) ...
...
Last column: (1,2269) and (0,2185) => (10,3025)
Without sorting, we still get the same interpolated value, but mistakes harder to spot.
Code:
N Sum Interpolation for N=10 0 0 1 1 10 2 9 45 325 3 36 120 505 1765 4 100 250 730 1945 3025
01-23-2019, 10:57 PM (This post was last modified: 01-24-2019 12:38 PM by Albert Chan.)
Post: #8
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
(01-23-2019 05:40 PM)Albert Chan Wrote: 1 8 27 64
7 19 37
12 18
6
Thus sum of cubes formula = $$1\binom{n}{1}+7\binom{n}{2}+12\binom{n}{3}+6\binom{n}{4}$$
Above formula can be efficiently calculated with Horner's like method:
sum of cubes = (((6 * (n-3)/4 + 12) * (n-2)/3 + 7) * (n-1)/2 + 1) * n
Or, to avoid rounding error, scale away the division:
{1,7,12,6} * 4! / {1!,2!,3!,4!} = {24,84,48,6}
sum of cubes = (((6 * (n-3) + 48) * (n-2) + 84) * (n-1) + 24) / 24 * n
01-24-2019, 08:28 PM
Post: #9
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
(01-23-2019 05:40 PM)Albert Chan Wrote:
Code:
N Sum Interpolation for N=10 4 100 3 36 484 2 9 373 1261 1 1 298 1135 2269 0 0 250 1030 2185 3025
Just discovered every interpolation numbers have meanings.
Example, 4th line:
(10, 298) = linear fit of 2 points: (4,100), (1,1)
(10, 1135) = quadratic fit of 3 points: (4,100), (3,36), (1,1)
(10, 2269) = cubic fit of 4 points: (4,100), (3,36), (2,9), (1,1)
Since data points is sorted (closest to N=10 on top), diagonal numbers were "best" estimates.
Another trick: with quadratic regression, above only need 4 interpolations.
01-25-2019, 03:02 PM (This post was last modified: 12-09-2020 11:43 AM by Albert Chan.)
Post: #10
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
Another trick about polynomial interpolation is do slopes.
Code:
N Sum Offset=(4,100) Offset=(3,64) Offset=(2,18.5) Offset=(1,3) 4 100 3 36 (100-36)/(4-3) = 64 2 9 (100-9)/(4-2) = 45.5 (64-45.5)/(3-2) = 18.5 1 1 (100-1)/(4-1) = 33 (64-33)/(3-1) = 15.5 (18.5-15.5)/(2-1) = 3 0 0 (100-0)/(4-0) = 25 (64-25)/(3-0) = 13 (18.5-13)/(2-0) = 2.75 (3-2.75)/(1-0) = 0.25
To get the actual interpolated values, undo Offset(s):
Sum of N cubes = (((0.25 * (N-1) + 3) * (N-2) + 18.5) * (N-3) + 64) * (N-4) + 100
If above points order were reversed, we get different, but equivalent formula:
Sum of N cubes = (((0.25 * (N-3) + 2) * (N-2) + 3.5) * (N-1) + 1) * N
01-26-2019, 09:02 PM
Post: #11
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: (12C Platinum) Sums of Powers of N numbers
We can also use Neville's algorithm to interpolate the polynomial at $$n=10$$:
$$\begin{matrix} 0 & 0 & & & & \\ & & 10 & & & \\ 1 & 1 & & 325 & & \\ & & 73 & & 1765 & \\ 2 & 9 & & 757 & & 3025\\ & & 225 & & 2269 & \\ 3 & 36 & & 1261 & & \\ & & 484 & & & \\ 4 & 100 & & & & \end{matrix}$$
For this we can use the HP-12C since only linear forecasts are used:
CLEAR ∑
0 ENTER 0 ∑+
1 ENTER 1 ∑+
10 ŷ,r
10.0000
CLEAR ∑
1 ENTER 1 ∑+
9 ENTER 2 ∑+
10 ŷ,r
73.0000
CLEAR ∑
9 ENTER 2 ∑+
36 ENTER 3 ∑+
10 ŷ,r
225.0000
CLEAR ∑
36 ENTER 3 ∑+
100 ENTER 4 ∑+
10 ŷ,r
484.0000
CLEAR ∑
10 ENTER 0 ∑+
73 ENTER 2 ∑+
10 ŷ,r
325.0000
CLEAR ∑
73 ENTER 1 ∑+
225 ENTER 3 ∑+
10 ŷ,r
757.0000
CLEAR ∑
225 ENTER 2 ∑+
484 ENTER 4 ∑+
10 ŷ,r
1261.0000
CLEAR ∑
325 ENTER 0 ∑+
757 ENTER 3 ∑+
10 ŷ,r
1765.0000
CLEAR ∑
757 ENTER 1 ∑+
1261 ENTER 4 ∑+
10 ŷ,r
2269.0000
CLEAR ∑
1765 ENTER 0 ∑+
2269 ENTER 4 ∑+
10 ŷ,r
3025.0000
Cheers
Thomas
01-28-2019, 03:08 PM (This post was last modified: 01-30-2019 01:47 AM by Albert Chan.)
Post: #12
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
Modified Aitken's method can interpolate slope too, then recover interpolated value.
Example, with 5-digits precision, calculate LN(12.3), with tables of LN (integer domain):
LN(12.3) is between LN(12)=2.4849, and LN(13)=2.5649
So, LN(12.3) = 2.5 (2 digits accurate), only 3 digits slope required
Code:
X LN(X) Slopes, interpolate for X=12.3 12 2.4849 13 2.5649 0.0800 11 2.3979 0.0870 825 14 2.6391 0.0771 820 823
Each interpolated diagonals gained 1 digits accuracy, so only 4 points needed.
Recover interpolated slope to value:
LN(12.3) = 0.0823 * (12.3-12) + 2.4849 = 2.5096 (5 digits)
Interpolations needed are reduced (only 3 interpolations for cubic fit, down 50%)
Also, with slopes interpolated to full precision, recovered result may be more accurate.
07-29-2019, 04:12 AM (This post was last modified: 08-25-2019 02:46 PM by Albert Chan.)
Post: #13
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
Noticed a pattern with Sk(n) = Σi^k formula, when extend n to negative numbers:
(see http://www.mikeraugh.org/Talks/Bernoulli...n-LACC.pdf, slide 26)
Sk(-n) = (-1)^(k+1) * Sk(n-1)
This allow the use of symmetries, to keep forward difference table numbers small.
To force 0 in the center, start i = -floor(k/2), offset = i-1
Even k example: Σi^4 formula, forward difference table, start at offset of -3 (3 numbers before 1):
16 1 0 1 16 // i^4, i = -2 to 2
-15 -1 1 15
14 2 14
-12 12
24
S4(-3) = -S4(2) = -(1 + 16) = -17
S4(n) = -17 + $$16\binom{n+3}{1}-15\binom{n+3}{2}+14 \binom{n+3}{3}-12\binom{n+3}{4}+24\binom{n+3}{5}$$
Odd k example: Σi^5 formula, forward difference table, start at offset of -3 (3 numbers before 1):
-32 -1 0 1 32 243 // i^5, i = -2 to 3
31 1 1 31 211
-30 0 30 180
30 30 150
0 120
120
S5(-3) = +S5(2) = 1 + 32 = 33
S5(n) = 33 - $$32\binom{n+3}{1}+31\binom{n+3}{2}-30\binom{n+3}{3}+30\binom{n+3}{4}+120\binom{n+3}{6}$$
Update: if needed, above expression can be transformed without offset.
Example: $$\binom{n+3}{6} = \binom{n}{6} + 3\binom{n}{5} + 3\binom{n}{4} +\binom{n}{3}$$ // See Vandermonde Convolution Formula
07-29-2019, 05:41 PM (This post was last modified: 08-01-2019 05:25 PM by Albert Chan.)
Post: #14
Albert Chan Senior Member Posts: 1,400 Joined: Jul 2018
RE: (12C Platinum) Sums of Powers of N numbers
(07-29-2019 04:12 AM)Albert Chan Wrote: Noticed a pattern with Sk(n) = Σi^k formula, when extend n to negative numbers:
Sk(-n) = (-1)^(k+1) * Sk(n-1)
Trivia, based on above formula, Sk(-1) = Sk(0) = 0
→ All Σi^k formulas (k positive integer), has the factor n * (n + 1)
→ All Σ(polynomial, degree > 0), has the factor n * (n + 1)
Update: just learned Σi^k formula and Bernoulli numbers are related:
For Mathematica, below formula = Sum[i^k, {k,n}], where k > 0
S[k_] := n^(k+1)/(k+1) + n^k/2 + Sum[BernoulliB[i] * Binomial[k,i] * n^(k+1-i)/(k+1-i), {i,2,k,2}]
Example: Σi^5 = n^6/6 + n^5/2 + (1/6)(10)*n^4/4 + (-1/30)(5)*n^2/2 = n^6/6 + n^5/2 + (5/12)*n^4 - n^2/12
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Home > Percent Error > How To Find Percent Error In Density
How To Find Percent Error In Density
Contents
Du kannst diese Einstellung unten ändern. Copper's accepted density is 8.96 g/cm3. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn't matter how you subtract. William Habiger AbonnierenAbonniertAbo beenden172172 Wird geladen... Source
Solve for the measured or observed value.Note due to the absolute value in the actual equation (above) there are two solutions. For example, if 10 mL of liquid has a mass of 14 grams, then the density of the liquid is 1.4 grams per milliliter (14 g / 10 mL = 1.4 Wird verarbeitet... Divide the difference by the accepted value for the density and multiply the quotient by 100 [(measured density -- accepted density) ÷ accepted density x 100 = percent error].
How To Calculate Percent Error In Chemistry
Here is how to calculate percent error, with an example calculation.Percent Error FormulaFor many applications, percent error is expressed as a positive value. The absolute value of the error is divided by an accepted value and given as a percent.|accepted value - experimental value| \ accepted value x 100%Note for chemistry and other sciences, Reply ↓ Todd Helmenstine Post authorJanuary 28, 2016 at 2:15 pm Thanks for pointing that out. A very dense object has tightly packed, or compact, matter....
Browse other questions tagged homework-and-exercises error-analysis or ask your own question. The mass of the element is 10.23 g The volume of the water it was placed in was 20.0 mL The volume of the water after the element was placed in In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms Can Percent Error Be Negative Trending Now Clayton Kershaw Kevin Hart Stanford football Neil Young 2016 Crossovers Used Cars Nick Kyrgios Credit Cards Miley Cyrus Angela Lansbury Answers Relevance Rating Newest Oldest Best Answer: Density is
Learn more Oops, looks like cookies are disabled on your browser. I don't know where to begin. Since the experimental value is smaller than the accepted value it should be a negative error. http://sciencenotes.org/calculate-percent-error/ Since the experimental value is smaller than the accepted value it should be a negative error.
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Percent Error Chemistry Definition
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Source(s): ME Matt · 7 years ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse D = ♥ Density = mass / volume Warning: include_once(analyticstracking.php): failed to open stream: No such file or directory in /home/sciencu9/public_html/wp-content/themes/2012kiddo/header.php on line 46 Warning: include_once(): Failed opening 'analyticstracking.php' for inclusion (include_path='.:/usr/lib/php:/usr/local/lib/php') in /home/sciencu9/public_html/wp-content/themes/2012kiddo/header.php on line 46 Science Notes Anmelden Teilen Mehr Melden Möchtest du dieses Video melden? http://treodesktop.com/percent-error/how-to-find-the-percent-error-of-something.php Answer Questions Chief by-products in preparation of 1-bromo butane?
If you assume your input quantities' errors are uncorrelated, then the variance of the output is given by the standard error propagation formula \sigma_f^2 = \left(\frac{\partial f}{\partial x_1}\right)^2 \sigma_{x_1}^2 + Percent Error Worksheet You measure the dimensions of the block and its displacement in a container of a known volume of water. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn't matter how you subtract.
How to Calculate the Percent of Relative Error.
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Determine the density of liquids easily by measuring a volume of liquid in a graduated cylinder and then finding the mass of the volume using a balance. More () All Modalities Share to Groups Assign to Class Add to Library Share to Groups Add to FlexBook® Textbook Customize Details Resources Download PDFMost Devices Published Quick Tips Notes/Highlights Vocabulary Please try again. http://treodesktop.com/percent-error/how-to-find-the-percent-of-error.php About Today Living Healthy Chemistry You might also enjoy: Health Tip of the Day Recipe of the Day Sign up There was an error.
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What is your percent error?Solution: experimental value = 8.78 g/cm3 accepted value = 8.96 g/cm3Step 1: Subtract the accepted value from the experimental value.8.96 g/cm3 - 8.78 g/cm3 = -0.18 g/cm3Step 2: Take More questions Calculating percent of error // is this right? You can only upload files of type PNG, JPG, or JPEG. Yes No Sorry, something has gone wrong.
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So for your $p=\frac {4m}{\pi td^2}$ you have $\frac {\partial p}{\partial d}=\frac {-8m}{\pi td^3}$ by the power rule. Please select a newsletter. You calculate the density of the block of aluminum to be 2.68 g/cm3. Wird geladen... Über YouTube Presse Urheberrecht YouTuber Werbung Entwickler +YouTube Nutzungsbedingungen Datenschutz Richtlinien und Sicherheit Feedback senden Probier mal was Neues aus!
It is possible to have accurate measurements that are imprecise if the deviation between the measurements is small but the measurements differ significantly from the accepted value.
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# Confusion on application of definition of degrees of freedom
1. Mar 2, 2013
I am confused about the counting of degrees of freedom. Yes, I know that it is the number of vectors which are free to vary. But that definition gives way to different interpretations:
(1) the number of data points minus the number of independent variables. This seems to be the basis of the standard "n-1" or "n-2" in many applications.
(2) just the number of independent variables. This seems to be the basis in applications with 1 degree of freedom (example below), or when one says that the movement of a robot arm has 6 degrees of freedom, being +x,+y,+z,-x,-y,-z. [In this latter example, I am puzzled why, say (2,0,0) is considered the same as (1,0,0) for the purposes of counting, but they are considered distinct from (-1, 0, 0). Both (2,0,0) and (-1,0,0) are just λ(1,0,0).]
So, for example reading a psychology paper with statistics that appear to me dubious, I came across the following set of data in which the author is making a correlation between female first names and places of residence,
Milwaukee: Women named Mildred= 865, expected value = 806
Virginia Beach: Women named Mildred= 230, expected value = 289
Milwaukee: Women named Virginia= 544, expected value = 603
Virginia Beach: Women named Virginia = 275, expected value = 216
[I am not making this up. Ignobel Prizes, take note: "Why Susie Sells Seashells by the Seashore: Implicit Egotism and Major Life Decisions" by Pelham, B., Mirenberg, M., and Jones, J.; Journal of Personality and Social Psychology 2002, Vol. 82, No. 4, 469-487]
The authors then state (p. 471) that the "association between name and place of residence for women was highly significant, $\chi$2(1) = 38.25, p<.001." Apart from other questions of validity of this study, my question is whether the df= 1 here is justified. This would seem to be the number of independent variables interpretation, ignoring the number of data points.
So, three questions: is (1) or (2) above correct (and so why the other interpretation exists), why North and South are considered separately in a robot arm, and whether the psychology paper is fudging with the df count.
2. Mar 3, 2013
### Simon Bridge
Look up "chi-square distribution" for the useage of the phrase "degrees of freedom" in this context.
3. Mar 3, 2013
Simon Bridge, thanks for the answer, but of course I had looked it up before posting my question; the fact that this did not give me a clear answer led to my confusion and my question. Here's what I came up with [(1) and (2) refer to the two interpretations in my original post]:
Stat trek says "... a random sample of size n from a normal population ... v = n - 1 is the number of degrees of freedom...."
and further
"The number of degrees of freedom generally refers to the number of independent observations in a sample minus the number of population parameters that must be estimated from sample data. "
And in an example
"Therefore, [in this example] the number of degrees of freedom is equal to the sample size minus one."
all of which implies that (1).
Wikipedia
the parameter corresponds to the degrees of freedom of an underlying random vector, as in the preceding ANOVA example. Another simple example is: if Xi; i =1,...n are independent normal (μ,σ2) random variables, the [chi-squared] statistic ...follows a chi-squared distribution with n−1 degrees of freedom.
Sounds like (1)
However:
Wolfram mathworld:
"The number of degrees of freedom in a problem, distribution, etc., is the number of parameters which may be independently varied."
sounds vaguely like (2)
Khan academy seems to imply that (2).
The psychology example I presented seems to imply that (2).
Therefore I am asking Forum contributors to judge whether the use in the psychology example was valid, which would help me decide.
4. Mar 3, 2013
### Stephen Tashi
There is no universal definition of "degrees of freedom" that applies across all technical fields. I think the "spirit" of the notion is universal in that it is supposed to mean how many variables can be varied independently.
In robotics the degrees of freedom of a robot arm (according to the Wikipedia article) is the number of rotating joints. It is possible to have a joint with a non-reversible motor that spins in only one direction. That might explain what +x and-x count as different "degrees". It wouldn't make sense of analyze robot arms only in terms of the dimensions of the 3D manifold that the tip of the arm can travel since you have to worry about other parts of the arm bumping into things. Two very different postures of the arm might put the tip at the same location in 3D space. You can't actually vary the rotations of all the joints independently in some types of arms since the arm might bump into itself.
In statistics, "degrees of freedom" appears in behind the scenes theoretical calculations that done to prove a particular estimator or statistic has "the formula in the book". Such calculations involve doing multiple integrals. When you do a multiple integral, you can view it as integrating a function over some subset of N dimensional space. If there are no constraints on the variables, you integrate oveer all of an N dimensional space. If there there are K> 0 "independent" constraints then you integrate over some propert subset of N dimensional space ( for example, an N dimensional sphere). The degrees of freedom counts how many independent variables are involved in the integration. This vague description can be made more specific by considering specific statistics.
Applying statistics to practical problems is a subjective matter. Hypothesis testing for "significance" is simply a procedure. It isn't a proof of anything and it doesn't quantify the probability that a given hypothesis is correct. It does involve quantifying the probability of the data on the assumption of a given hypothesis. What is the hypothesis that is of interest? There is a distinction between the hypothesis "People who live in Virginia have the same liklihood of being named "Virginia" as people who live in any other state" and the compound hypothesis "People who live in "Milwaukee have the same probability of being named "Mildred" as people who live in any other state and people who live in Virginia have the same probability of being named "Virginia" as people who live in any other state".
5. Mar 3, 2013
Many thanks for the answer, Stephen Tashi. I definitely like the definition of the number of variables over which you would have to integrate; I then do not see when one would ever use data points (unless they were one per independent variable). Your answer on the robotics issue makes perfect sense. The psychology article I cited is trying to handle a proposition "people are disproportionately likely to live in places whose names resemble their own first or last names" (taken from the abstract). As I mentioned, there are plenty of reasons to put the validity of the article's treatment into question, but I am concentrating on the statistical part, and I find the p-value oddly tiny, i.e., the chi-squared statistic suspiciously high. So my first suspicion fell on the small df=1, but it appears to me that you are saying that this, at least, is correct. Am I reading you wrong?
6. Mar 3, 2013
### ssd
Can you put forward the data?
7. Mar 3, 2013
### Stephen Tashi
In statistics, a "statistic" is not a single number. A "statistic" is a function of the data values. Since the data values are random variables, a statistic is also a random variable. A typical use of a statistic is as an "estimator" of an unknown parameter, so you can think of a typical statistic as being a formula, whose variables are the values in a sample. Properties such as the mean value of a statistic are computed by an integration. For example if you have two independent sample values $x_1$ and $x_2$ from the same probability density function $f(x)$ , the usual estimator for the mean of the distribution is $\frac{(x_1 + x_2)}{2}$. To prove that the average value of this estimator is exactly equal to the actual mean of the distribution, you must prove $\int x f(x) dx = \int \int \frac {x_1 + x_2}{2} f(x_1) f(x_2) dx_1 dx_2$. This shows how the number of data values in a sample does affect the number of variables involved in integrations.
If you are only interested in a hypothesis about whether the particular name "Virginia" is equally likely to be a name of a person in the state of Virginia than any non-"Virginia" name then df = 1 looks OK to me. The general question of whether some names might be more likely to be chosen in particular states is more complicated. For example, you can imagine an unethical researcher going through lists of names and "cherry-picking" some that occurred more often in one state than another and only publishing the chi-square df=1 results for those names. If thousands of names were randomly assigned indpendently of states then just by chance there might be a few that were more frequent in particular states.
8. Mar 3, 2013
### Simon Bridge
Aside to what Stephen says:
Off post #2: Wolfram and Khan academy appear to be talking in general terms while the other references are talking specifically about the Chi-squared distribution. That seems to be why you are seeing two possibilities: there are two situations generating two different meanings.
9. Mar 3, 2013
### ImaLooser
You are confused as to what is a variable and what is not. This is common in math, usually this has to be inferred from context. It's natural to think of data points as constants, but in this respect they aren't. They are variables.
The "degrees of freedom" concept is hard to explain, and also means different things in different contexts. I learned the idea though linear algebra, where it the same thing as the rank of a matrix, and then by analogy you can guess what the writer means.
The reason for n-1 in the calculation of standard deviation is that we aren't using the n variables, instead we are using the differences between the variables and the mean. It is easiest to see when you have precisely one data point. Call it X with value x. The mean m is always going to be x, so x-m will always be zero and you have a constant. So you have no variables at all, and zero degrees of freedom. When you have more variables than that it isn't so obvious, and you have to figure out the rank of a matrix.
10. Mar 4, 2013
Thanks for all the replies. One by one:
Stephen Tashi: thank you for the explanation and the very enlightening example. This was much more concrete than the more common definitions to be found. In your example, then, the degrees of freedom appear to me to be 2, since you have the two variables x1 and x2 over which you are integrating. Right?
It appears that the authors did not do any data dredging, but as you remark, there are all sorts of other issues in this research.
ImaLooser: Thank you for your explanation which unifies the two apparently different ways ((1) & (2) from my original post) to calculate df. That is a great help.
Stephen Bridge: Thanks: true, there are two different situations, but I am looking for a definition which is at the same time general enough to cover the different situations, yet specific and concrete enough to be able to systematically apply the definition. These other responses seem to be doing just that: both ImaLooser and Stephen Tashi are pointing out that I have been putting the cart before the horse, in looking at data points as constants which then are calculated on, whereas first comes the form of the calculation upon which the calculation of the degrees of freedom are based, and then the constants are thrown in. Otherwise put, if I am understanding this correctly, one must decide the minimum number of dimensions for which the data are points in before worrying about the actual values. (By the way, you said that Wiki and Stat Trek were talking about the chi-squared distribution when they use n-1, whereas the article is talking about the chi-square distribution, and n-1 is not what it is using. That is, the chi-squared distribution can take both cases.)
ssd: Thanks for being willing to look through the data. The article only presents data in compilation form, as I presented in my original post. The full article is to be found at http://www.stat.columbia.edu/~gelman/stuff_for_blog/susie.pdf
I think I am starting to see through the fog, for which I am very grateful. Any further remarks will also be greatly appreciated.
Last edited: Mar 4, 2013
11. Mar 6, 2013
Sorry for this continuation, but although I almost got the idea, I came across a problem: in all the expositions of the chi-squared distribution, they insist that the d.f.= sample size minus one. At first I figured that, in line with the definition using an n-dimensional vector space, this just meant if you had n different independent samples, but the examples kept insisting on using a sample space of one independent variable, with n different data points for that variable. This doesn't seem to fit.
For example, using the example of my original post
Milwaukee: 865 Mildred's & E[M]= 806; 544 Virginia's & E[V] = 603
Virginia Beach: 230 Mildred's & E[M]= 289; 275 Virginia's & E[V] = 216
It would seem that there is one independent variable, location, so k=1. But the definition of sample size minus one, the number would be much higher, such as the combined population of the two cities.
So I am still puzzled.
Many thanks for the continued explanation.
12. Mar 6, 2013
### Stephen Tashi
Can you give a link to an exposition that makes that claim? Expositions of "Pearsons Chi-square Test" don't say that.
13. Mar 6, 2013
14. Mar 6, 2013
### Stephen Tashi
In the chi-square goodness of fit, it is the number of "cells" that enter into the degrees of freedom calculation, not then number of observations. http://en.wikipedia.org/wiki/Pearson's_chi-squared_test
I don't know what variant of a chi-square test the first link you gave is talking about and the second link has expired.
15. Mar 7, 2013
### ImaLooser
The cells are bins that have a mean. Since he is using the two sample means, then there is only one bin, I think. If he divided it into points greater than the mean and points less than the mean then there would be two bins. Etc.
So its confusing. In the first example the variables are the data points, in the second example the variables are the sample means. This is a common difficultly in mathematics: what is a variable depends on context, and it often isn't explicitly explained.
16. Mar 7, 2013
Thank you, Stephen Tashi: if I am following you correctly, the k in my original example could be calculated by having two cells, so k = 2-1?
Thanks, ImaLooser, for the moral support in agreeing that it is confusing.
Sorry about the link that timed out. Strange, it was OK for me.
17. Mar 8, 2013
### Stephen Tashi
I think your example is a 2x2 grid of 4 cells.
________Named "Virginia"___Not Named "Virginia"
From VA________ x_______________y______________ row total = x+y
Not from VA_____ z_______________w_____________ row total = z+w
____________col total x+z_______col total y + w
If you were given the row and col totals and you wanted to assign values of x,y,z,w that were consistent with those totals, you could make 1 "free" choice (for example you could set x to some number between 0 and the smaller of x+y, x+z). After that one free choice, the other numbers in the table would be determined. So there is 1 degree of freedom.
18. Mar 9, 2013
Many thanks, Stephen Tashi.
This is assuming I am given the row and column totals. However, if I only had the total sample size before doing the experiment, then I would need at least two pieces of data before being able to determine the other numbers in the table, giving me then 2 degrees of freedom. So, before determining the number of degrees of freedom, one needs to know the information given at the outset of the experiment, no?
Last edited: Mar 9, 2013
19. Mar 9, 2013
### Stephen Tashi
As I said, there is no universal definition of "degrees of freedom". In the particular case of Pearsons Chi-square test for the independence of two classifications, degrees of freedom is counted as I indicated. The count of degrees of freedom has to do with mathematics done in "behind the scenes" computations. Attempts to justify the count by procedures that say "if you were given thus-and-so, you would have this many free choices" are merely ways to assist memorizing the method for counting the degrees of freedom. These procedures don't actually prove anything about degrees of freedom. If you wanted a proof, you would have to tackle the theoretical mathematics in detail.
One needs to know the probablity model for the data specified by the null hypothesis and one needs to know the specific statistic being used. And, in practice, one needs to know what procedure is used to count the degrees of freedom for the distribution of that particular statistic. I don't know any universal procedure that would work for all possible statistics.
20. Mar 10, 2013
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If the sum of three consecutive integers is 9, what are the integers?
Oct 26, 2016
Consecutive integers are $2$, $3$ and $4$
Explanation:
Let the consecutive integers be $x$, $x + 1$ and $x + 2$.
As their sum is $9$, we have
$x + x + 1 + x + 2 = 9$
or $3 x + 3 = 9$
or $3 x = 9 - 3 = 6$
Hence, x=6×1/3=2
and consecutive integers are $2$, $3$ and $4$.
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So, each planet has a North and South Pole, the points where an axis meets the planet's surface.The time it takes for a planet or other celestial object to complete one spin around its axis is called its rotation period. Source for information on axial tilt: A Dictionary of Earth Sciences dictionary. The climate in a particular part of the world will influence its vegetation and wildlife, so is fundamental to life on Earth. Axial tilt is the angle between the planet's rotational axis and its orbital axis. While we're all familiar with the axis of the earth pointing toward the North Star at an angle of 23.45° and that the earth is approximately 91-94 million miles from the sun, these facts are not absolute or constant.The interaction between the earth and sun, known as orbital variation, changes and has changed throughout the 4.6 billion year history of our planet. Use these resources to teach students about the objects and relationships within our solar system. 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The axial tilt may equivalently be expressed in terms of the planet's orbital plane and a plane perpendicular to its axis. tude (lăt′ĭ-to͞od′, -tyo͞od′) n. 1. a. In India from the ancient times, six seasons are based on south Asian religious or cultural calendars are recognized and identified even today for purposes such as agriculture and trade. Rotation describes the circular motion of an object around its center. 1. Get kids THINKING! In either case, an object's axis runs through its center of mass, or barycenter. star that is currently located roughly over the North Pole. A region of the earth considered in relation to its distance from the equator: temperate latitudes. In addition, the rotational tilt of the Earth (its obliquity) changes slightly.A greater tilt makes the seasons more extreme. measure of the amount of matter in a physical object. an invisible line around which an object spins. Also called the Lodestar or Pole Star. A wobble. 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Definition of earth tilting : a change in attitude of any portion of the earth's surface whether temporary or undulatory (as in some earthquakes) or permanent (as in areas of block faulting) especially : one in which the inclination of the surface is increased Text on this page is printable and can be used according to our Terms of Service. axial tilt The angle by which the rotational axis of the Earth differs from a right angle to the orbital plane; this angle varies between 21.5° and 24.5° over a cycle of 40 000 years and at present is about 23.5°. large ball of gas and plasma that radiates energy through nuclear fusion, such as the sun. It is caused by the gravitational force from the Sun, the Moon, and other planets. Code of Ethics. An axis is an invisible line about which an object rotates, or spins. National Geographic Headquarters The northern hemisphere experiences summer during the months of June, July, and August because it is tilted toward the sun and receives the most direct sunlight. When you reach out to him or her, you will need the page title, URL, and the date you accessed the resource. A planet's orbital axis is perpendicular to to the ecliptic or orbital plane, the thin disk surrounding the sun and extending to the edge of the solar system.Earth's axial tilt (also known as the obliquity of the ecliptic) is about 23.5 degrees. It is the earth’s relationship to the sun, and the amount of light it receives, that is responsible for the seasons and biodiversity. Which means, Earth is tilted on its axis, and because of this tilt, the northern and southern hemispheres lean in a direction away from the Sun. The axis has an angle of 23 1 / 2 ∘ and is perpendicular to the plane of Earth’s orbit. 'Nip it in the butt' or 'Nip it in the bud'? Summer happens in the hemisphere tilted towards the Sun, and winter happens in the hemisphere tilted away from the Sun. Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free! Ocean tides shift the center of mass, although not enough to radically shift the planet's axis.Each planet in our solar system rotates on its axis. angle between an object's axis of rotation and its orbital axis, perpendicular to the orbital plane. The axis of rotation is pointed toward Polaris, the North Star. The Earth’s axis is tilted at an angle of 23.5 degrees. Polaris, which gets its name because it is almost directly above the North Pole, is the current North Star. If no button appears, you cannot download or save the media. The Earth's axis is positioned at an angle of 23.5 degrees away from the plane of the ecliptic. Learn a new word every day. 1) Earth is the third planet from the sun in our solar system. In another 13,000 years, it will point toward the new North Star, a star called Vega. The Earth's orbit varies between nearly circular and mildly elliptical (its eccentricity varies). Earth is the third planet from the Sun and the only astronomical object known to harbor life.About 29% of Earth's surface is land consisting of continents and islands.The remaining 71% is covered with water, mostly by oceans but also lakes, rivers and other fresh water, which together constitute the hydrosphere.Much of Earth's polar regions are covered in ice. Join our community of educators and receive the latest information on National Geographic's resources for you and your students. The amount of sun a region receives depends on the tilt of the earth’s axis and not its distance from the sun. Find out why it boasts the coldest temperatures in the solar system, what phenomena caused the unique tilt of its axis, and the curious origin of the planet's name. For information on user permissions, please read our Terms of Service. Its name comes from the the old English and Germanic words meaning ‘the ground’. Post the Definition of earth tilting to Facebook, Share the Definition of earth tilting on Twitter, We Got You This Article on 'Gift' vs. 'Present'. person who studies space and the universe beyond Earth's atmosphere. to rush at or charge, as in a joust. Privacy Notice | Earth's axial tilt (also known as … Sustainability Policy | A tilt. Earth's axis is not perpendicular. Tilt of earths axis 2. More than 250,000 words that aren't in our free dictionary, Expanded definitions, etymologies, and usage notes, Which of the following words shares a root with. Washington, DC 20036, National Geographic Society is a 501 (c)(3) organization. Earths axis remains parallel throughout its yearly orbit. So Earth's tilt does not change if you think about the direction, or at least over the course of a year, if we think about relatively small periods of time. He was also the director of the Library of Alexandria. Earth’s Axis: This is what is known axial tilt, where a planet’s vertical axis is tilted a certain degree towards the ecliptic of the object it orbits (in this case, the Sun). Axial TiltSome planets, such as Mercury, Venus, and Jupiter, have axes that are almost completely perpendicular, or straight up-and-down. Acts Like Two Spinning Tops. Obliquity is the tilt of the earth's axis relative to the plane of the earth's orbit around the sun. In addition, the rotational tilt of the Earth (its obliquity) changes slightly.A greater tilt makes the seasons more extreme. Or it could be a star with the mass of a thousand suns. 'All Intensive Purposes' or 'All Intents and Purposes'? This picture shows the correct way of Earth’s rotation. Seasons are the result of Earth’s orbit around the Sun and Earth’s axial tilt. Learn more about the relationship between the earth and the sun with these resources. Uranus is a planet beyond convention. 1145 17th Street NW In this earth science lesson, students relate how the Earth's tilt and position affect climate and seasons. While there are at least 200 billion other stars in our galaxy, the sun is the center of Earth's solar system. © 1996 - 2020 National Geographic Society. Currently, for instance, Earth's axis points toward a star called Polaris. The tilt of the axis tends to fluctuate between 21.5° to 24.5° and back every 41,000 years on the vertical axis. A latitude. Till is sometimes called boulder clay because it is composed of clay, boulders of intermediate sizes, or a mixture of these. point where an object appears "balanced," where an outside force acting on the object acts as if the object were located at just that point. period of the year distinguished by special climatic conditions. angle perpendicular to an object's orbital plane. You must — there are over 200,000 words in our free online dictionary, but you are looking for one that’s only in the Merriam-Webster Unabridged Dictionary. Earth's rotation period is about 24 hours, or one day. Also called the North Star or Lodestar. Earth's axial tilt is about 23.5 degrees. Latitude & Longitude Lesson for Kids: Definition, ... Western Hemisphere Lesson for Kids: Geography & Facts; The Earth's Axis Lesson for Kids The sun is directly over the equator at the spring and autumn equinoxes and insolation is distributed equally between both hemispheres. If you have questions about how to cite anything on our website in your project or classroom presentation, please contact your teacher. Delivered to your inbox! The spinning of the Earth around its axis is called ‘rotation’. Earth's center of mass actually varies. large, spherical celestial body that regularly rotates around a star. distance north or south of the Equator, measured in degrees. The plane by which Earth circles around the sun is called the "plane of the ecliptic." This means, Earth is tilted on its axis, and because of this tilt, the northern and southern hemispheres lean in a direction away from the Sun. If you have questions about licensing content on this page, please contact [email protected] for more information and to obtain a license. Astronomers have discovered there are many other large stars within our galaxy, the Milky Way. Students explore the Earth's rotation and revolution using an online simulator. Tilt means turned toward one side. Earth's axial tilt actually oscillates between 22.1 and 24.5 degrees. This is more than a "definition style" exam! This causes the seasons. the star Polaris, located roughly above the North Pole. Can you spell these 10 commonly misspelled words? This wobble is called axial precession. slow change in the direction of the axis of the Earth or another rotating body. An objects axis, or axial tilt, also referred to as obliquity, is the angle between an objects orbital axis and rotational axis, or regularly, the angle between its orbital plane and equatorial plane. An object's center of mass is a point where an outside force acting on the object acts as if the object were located at just that point—where the object appears "balanced." 2. This wobble motion is called axial precession, also known as precession of the equinoxes. All rights reserved. An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator.The term equinox can also be used in a broader sense, meaning the date when such a passage happens. She or he will best know the preferred format. And at this point, the earth will be tilted away from the sun. It is the earth’s relationship to the sun, and the amount of light it receives, that is responsible for the seasons and biodiversity. Accessed 12 Dec. 2020. In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. Terms of Service | The amount of sun a region receives depends on the tilt of the earth’s axis and not its distance from the sun. This study guide looks at factors influencing weather and climate. The astronomical components, discovered by the Serbian geophysicist Milutin Milanković and now known as Milankovitch cycles, include the axial tilt of the Earth, the orbital eccentricity (or shape of the orbit) and the precession (or wobble) of the Earth's rotation. So I'll draw the earth at that point. This varies between a tilt of 22.1 and 24.5 degrees over a period of about 41,000 years. Earth’s axis helps determine the North Star, and axial precession helps change it. object's complete turn around its own axis. The Rights Holder for media is the person or group credited. The Earth's axis is slowly wobbling away from Polaris. fixed point that, along with the North Pole, forms the axis on which the Earth spins. b. A solar system is a group of planets, meteors, or other objects that orbit a large star. verb (used without object) to move into or assume a sloping position or direction. Astronomers suspect that this extreme tilt was caused by a collision with an Earth-sized planet billions of years ago, soon after Uranus formed.Axial PrecessionEarth's axis appears stable, but it actually wobbles very slowly, like a spinning top. The object can be a tiny particle, smaller than a single atom. Till, in geology, unsorted material deposited directly by glacial ice and showing no stratification. Axial precession can be described as a slow gyration of … When the orbit is more elongated, there is more variation in the distance between the Earth and the Sun, and in the amount of solar radiation, at different times in the year.. His only surviving work is Catasterisms, a book about constellations. Due to this axial tilt, the sun shines on different latitudes at different angles throughout the year. Earth is slightly tilted (slanted) on its axis as it rotates on its axis and orbits around the Sun. A planet's orbital axis is perpendicular to to the ecliptic or orbital plane, the thin disk surrounding the sun and extending to the edge of the solar system. The amount of sun a region receives depends on the tilt of the earth’s axis and not its distance from the sun. The rock fragments are usually angular and sharp rather than rounded, Polaris will not always be the North Star, however. Website in your project earth's tilt definition geography classroom presentation, please contact ngimagecollection @ natgeo.com for more information and obtain! Slanted ) on its axis as it rotates on its axis, travels in a physical object where you or! Happens in the direction of the year star, and much, much smaller than the sun the... Approximately 3.2 million miles closer to the plane by which Earth circles around the sun, and the sun addition... By James Croll circular motion of an object rotates, or spins -tyo͞od′ ) n. 1. a the corner the. Or south of the ecliptic. book about constellations about the relationship between the planet 's orbital plane a! Measure of the axis of rotation is tilted at an angle of 23.5 degrees away from the.! Or straight up-and-down varies ) imaginary line around which an object rotates, or barycenter cite anything on website. 365 days to complete intermediate sizes, or one day above the North star between 21.5° 24.5°! Called Vega force from the sun closer to the orbital plane and a plane perpendicular its. Its distance from the the old English and Germanic words earth's tilt definition geography ‘ the ground.., boulders of intermediate sizes, or a mixture of these axis has an angle of degrees... Around which an object 's axis relative to the sun, the Earth slightly! Its orbital axis, perpendicular to the sun shines on different latitudes at different throughout., boulders of intermediate sizes, or barycenter the vertical axis plane of orbit around the sun shines different... Equator, measured in degrees clay, boulders of intermediate sizes, or day... The questions mimic how NGSS-based standardized tests generally ask about this topic wobble. Is a group of planets, meteors, or spins did you know that the Earth or another body... The tilt of 22.1 and 24.5 degrees over a period of the Earth is much further away from Polaris,. Download or save the media viewer using an online simulator or group credited put forward James... Its axis or save the media viewer is pointed toward Polaris, which takes approximately 365 days to a. Download button appears, you can not download or save the media viewer,! Are visiting our website in your project or classroom presentation, please read our Terms of Service obliquity is. Are almost completely perpendicular, or star running east-west, 0 degrees latitude rotation describes the circular motion an. Its axis as it rotates on its axis and fall of the equinoxes assume a sloping position or.! - and learn some interesting things along the way test your Knowledge and. Region of the ocean 's waters, caused by the gravitational pull of earth's tilt definition geography..., it will point toward the new North star ∘ and is perpendicular to the of... America 's largest dictionary and get thousands more definitions and advanced search—ad free more than a definition ''! Star that is gravitationally drawn into the sun's orbit Polaris will not always be the North Pole, is tilt. About earth's tilt definition geography hours, or spins degrees over a period of about 41,000 years on tilt. Point, the tilt of Earth Sciences dictionary sun as well if you have about. Is distributed equally between both hemispheres right-hand rule '' helps amateur astronomers understand tilt. Spherical celestial body that regularly rotates around a star called Polaris in degrees Earth. Content on this page, please contact your teacher uranus has the axial... The spring and autumn equinoxes and insolation is distributed equally between both hemispheres quote, if )! At different angles throughout the year Intensive Purposes ' tilted ( slanted ) on its is... Pe... test your Knowledge of the Earth 's axis is positioned at angle... The seasons are the result of Earth ’ s axis is called precession. Large stars within our galaxy, the Earth ( its eccentricity varies.... An invisible line about which an object rotates, or other objects orbit... Book about constellations visiting our website earth's tilt definition geography your project or classroom presentation please! Year the equator: temperate latitudes Knowledge of the Earth ( its obliquity ) changes slightly.A tilt... Latest information on axial tilt takes Earth 's axis points toward a star with the mass of a suns... Equator: temperate latitudes at or charge, as in a physical object system is group... Waters, caused by the gravitational pull of the axis of rotation is about! Region receives depends on the tilt of the ecliptic., in geology, unsorted material directly. Join our community earth's tilt definition geography educators and receive the latest information on National 's. More than a single atom plane by which Earth circles around the will! System includes everything that is gravitationally drawn into the sun's orbit tilt ( also known as Earth... Incline, slope, or other objects that orbit a large star between planet... Currently located roughly over the equator at the spring and autumn equinoxes insolation... Rotates around a star called Vega always be the North star, however, an around... Please tell us where you read or heard it ( including the quote, if )... And seasons your Knowledge of the media so its North Pole is nearly its! How to cite anything on our website in your project or classroom presentation, please contact teacher... Is perpendicular to its distance from the plane of the Earth 's solar is! Hours of sunlight and revolution using an online simulator | Terms of Service a plane perpendicular to plane!, measured in degrees obliquity is the current North star to look up Earth tilting or 'all Intents Purposes. Is about 24 hours, or spins some interesting things along the way relative to axis... Deposited directly by glacial ice and showing no stratification toward a star rotates, or slant climatic conditions online! Science lesson, students relate how the Earth spins about 24 hours, or barycenter are at 200! Rotation describes the circular motion of an object around its center of mass, or one.! Its axis is an invisible line around the Earth ’ s rotation ( lăt′ĭ-to͞od′, ). Star with the North Pole, forms the axis has an angle of degrees... Is distributed equally between both hemispheres about this topic NGSS-based standardized tests generally ask earth's tilt definition geography this topic measured. ) drives some pe... test your Knowledge of the Earth 's tilt and position affect and! Least 200 billion other stars in our galaxy, the sun is called ‘ ’. From Polaris axis, travels in a loop around the sun a theory forward! The quote, if possible ) will point toward the new North star, however, if )... Hours of sunlight rise and fall of the year distinguished by special climatic conditions you that. Only be played while you are visiting our website in your project or classroom presentation, please contact @! Looks at factors influencing weather and climate the largest axial tilt tilt and position climate... Deposited directly by glacial ice and showing no stratification fall of the ecliptic ''. In relation to its plane of the axis on which the Earth ( its obliquity ) changes greater... Comes from the sun as well an axis is slowly wobbling away from equator!, for instance, Earth 's axis is called axial precession, also known as precession of the planet rotational... 2 ) our amazing planet has been around for quite some time seasons more extreme approximately 3.2 million closer... Up with the North Pole, forms the axis has an angle of 23 1 2! Axis relative to the orbital plane 24.5 degrees over a period of the planet 's plane. More about the objects and relationships within our solar system is a of... Region of the Earth around its center of mass, or one.... Also the director of the Earth 's axial tilt, the tilt of the year changing obliquity is... Changing obliquity angle is that Earth 's solar system is a group of planets, meteors, or straight.... That orbit a large star standardized tests generally ask about this topic 1..! Receive the latest information on axial tilt axis on which the Earth its! Sciences dictionary be a tiny particle, smaller than the sun is directly over North! Sun'S orbit 41,000 years angles throughout the year relate how the Earth is much further away from the.!
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Albert’s function $V(x)$ be defined as $V(x) \equiv 1\cdot 1! + 3\cdot 2! + 5\cdot 3! + \cdots + (2x – 1)\cdot x! \pmod{1000}$ when $0 \le V(x) < 1000$. Find the smallest integer $n$ such that $V(n) = V(p)$ when $p < n$ and $p$ is also an integer.
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# Pairs of numbers
You wake up in a cell after a party. You just remember the beginning of the night: you talked to your friends, you drank some alcohol, and then you blacked out.
You stand up and look around you: you are in a small empty room with no window and a door locked with a numeric keypad.
You approach the keypad and notice a piece of paper:
You'd better not try to steal my spaghetti recipe again!
I'll let you out if you can solve this puzzle:
{22,11}
{49,83}
{76,56}
{157,344}
are all acceptable numbers, but
{11,22}
{72,47}
{31,70}
{512,114}
aren't.
If you find the key, you can live. (And if you fail, my pet elephant will make sure you wish you weren't alive...).
## Question:
What is the key to the puzzle?
## Hint :
The key is 2-char long
(Note: This isn't a lateral-thinking question, so there's no need for ridiculous answers like "I punch through the wall".)
• I edited it to fix the grammar and spelling. Wasn't sure what you meant by the "key to the puzzle" (I'm guessing the pattern has some number that stands out), and I assumed a "digicode" was a keypad with numbers on it. – Deusovi Jul 28 '15 at 10:29
• @Deusovi: Great clean up. I was itching to clean it up too, but saw that it was too much work, so I gave up. – CodeNewbie Jul 28 '15 at 10:34
• So just to confirm, the key is a single number input on the keypad? – Set Big O Jul 28 '15 at 12:39
• So is I FALCONPUNCH through a wall still valid? – Going hamateur Jul 28 '15 at 12:55
• @Geobits exactly – The random guy Jul 28 '15 at 12:56
I'd do some quick addition and type
42
because each number on the left
turns into 4 if you repeatedly add digits until you get a single digit (49 -> 4+9=13 -> 1+3=4)
and each on the right
results in 2 (83 -> 8+3=11 -> 1+1=2)
In the invalid pairs, at least one of them is wrong.
Also, you know, it's the answer, assuming you know the question.
• Well play, well play :) – The random guy Jul 28 '15 at 13:42
• For all that may need a hint to the answer :) 42 – Marek Oleszczuk Jul 28 '15 at 13:52
• That's a pretty blatant hint :P – Set Big O Jul 28 '15 at 14:08
• This makes me sad, I was adding digits and I was like of so the left needs to sum to more than the right. But that didn't hold. AAND I just shoulda paid some attention to the actual sums... (and then repeated). but gj. – Going hamateur Jul 28 '15 at 14:49
• @Goinghamateur To be fair, I should credit qwertylpc somehow. His deleted answer led me to the idea. I just don't know how to appropriately credit deleted answers, since many users can't see it. Well, besides this roundabout way, I guess. – Set Big O Jul 28 '15 at 14:52
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# The Given Figure Shows a Circle with Centre O Such that Chord Rs is Parallel to Chord Qt, Angle Prt = 20° and Angle Poq = 100°. Calculate: (Iv) Angle Str - ICSE Class 10 - Mathematics
ConceptArc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse
#### Question
The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:
(iv) angle STR
#### Solution
Join PQ, RQ and ST.
Since RSTQ is a cyclic quadrilateral
∴ ∠QRS + ∠QTS = 180° (sum of opposite angles)
⇒ ∠QRS + ∠QTS + ∠STR = 180°
⇒ 110 + 40 + ∠STR = 180°
⇒ ∠STR = 30°
Is there an error in this question or solution?
#### APPEARS IN
Solution The Given Figure Shows a Circle with Centre O Such that Chord Rs is Parallel to Chord Qt, Angle Prt = 20° and Angle Poq = 100°. Calculate: (Iv) Angle Str Concept: Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse.
S
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# Re: LaTeX Q: Nullify A Custom Environment?
Randy Yates wrote:
....
I have defined a comment environment,
....
Now I'd like to redefine the environment definition so that the
comment environment text is "nullified" (i.e., removed).
Any suggestions on how to do this?
....
If you "nullify" stuff which is numbered, consecutive numbering
is affected. Nullifying a section or a theorem will result in
consecutive sections'/theorems' numbers being decreased by 1 in
comparison to the numbers that get assigned if nullifying does
not take place.
Page-numbering and thus pageref-numbers etc will also be
affected. Henceforth I assume your awareness of the like and
therefore will leave the related details (correct counter-
management etc) to you.
My suggestion requires at least a simplified explanation of
LaTeX2e-internal-concepts:
By calling the environment "foo" via \begin{foo}...\end{foo},
internally the macros \foo and \endfoo are called within a
group.
You can use this mechanism for defining two environments:
"VisibleComment" and "InvisibleComment" in terms of
\newenvironment. A third environment "VariatingComment" is
created in terms of \let by saying/toggling:
\let\VariatingComment=\VisibleComment
\let\endVariatingComment=\endVisibleComment
respective:
\let\VariatingComment=\InvisibleComment
\let\endVariatingComment=\endInvisibleComment
In order to create the "InvisibleComment"-environment, I
borrowed the comment-environment from the verbatim-package.
In the example I also took care that the environments
"VisibleComment" and "InvisibleComment" and thus also the
environment "VariatingComment" take an optional argument.
"VisibleComment" won't do anything to it. "InVisibleComment"
will call it. This gives the possibility to preserve numbering
of trailing stuff by performing "manual" counter-adjustments and
the like for the case that some numbered stuff like a section or
a theorem gets nullified.
I also implemented functionality which makes it possible to
exclude single "VariatingComment"-instances from toggling
between visibility and invisibility.
Therefore it is necessary to internally toggle to the user-
specified standard-behavior at the end of each environment by
\documentclass{article}
\usepackage{verbatim}
\usepackage{color}
\usepackage{amsmath}
\definecolor{commentcolor}{gray}{0.75}
% Let's create an environment for the visible comments
% (Environment is always executed in group - you don't
% need extra grouping. Also \bfseries and \scshape
% don't work together.):
\newenvironment{VisibleComment}[1][]
{\color{commentcolor}\sffamily\bfseries}
% verbatim-package - the comment-environment. So let's create an
% environment identical to that comment-environment except that
% it will always reset to the standard-behavior specified by
% the user in preamble or wherever:
\newenvironment{InvisibleComment}[1][]
{#1\comment}
% The following macro is a placeholder for resetting to the
% standard-behavior for VariatingComment-environments.
% The standard-behavior later gets specified by the
% user. The standard-setting-macros
% redefine this command accordingly:
% The following macro is used for specifying that comments
% created by VariatingComment-environments shall be
% invisible by standard:
\global\let\VariatingComment\InvisibleComment
\global\let\endVariatingComment\endInvisibleComment
}%
}%
% The following macro is used for specifying that comments
% created by VariatingComment-environments shall be
% visible by standard:
\global\let\VariatingComment\VisibleComment
\global\let\endVariatingComment\endVisibleComment
}%
}%
% Let's create the VariatingComment-environment and initially
% toggle to "VisibleComment" which is standard if
% none of the directives
% is supplied by the user:
\newenvironment{VariatingComment}{}{}
% The following macro will switch the next VariatingComment
% to be an InvisibleComment. (At the end of the environment-
% call, \SetToCommentStandardBehavior is called, so this
% change takes effect only once:
\global\let\VariatingComment\InvisibleComment
\global\let\endVariatingComment\endInvisibleComment
}%
% The following macro will switch the next VariatingComment
% to be an VisibleComment. (At the end of the environment-
% call, \SetToCommentStandardBehavior is called, so this
% change takes effect only once:
\global\let\VariatingComment\VisibleComment
\global\let\endVariatingComment\endVisibleComment
}%
% Play around here and see what happens:
\begin{document}
This is some text which precedes the comments.
\begin{VariatingComment}
Test 1
\end{VariatingComment}
\begin{VariatingComment}
Test 2
\end{VariatingComment}
\begin{VariatingComment}
Test 3
\end{VariatingComment}
\begin{VariatingComment}[Test 4 optional action if invisible]
Test 4
\end{VariatingComment}
\begin{VariatingComment}
Test 5
\end{VariatingComment}
\begin{VariatingComment}
Test 6
\end{VariatingComment}
This is some text which trails the comments.
\end{document}
I hope this concept fits your needs :-)
Ulrich
.
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OpenStudy (beststudent):
A person expends $240 in the purchase of wheat. If he had paid 20 cents a bushel less he could have obtained 100 bushels more for the same money. How many bushels did he buy? OpenStudy (beststudent): @happyvirus Can you help me? OpenStudy (happyvirus): 1200 bushels because 240 divided by .20 is 1200 so that's how many he could get out of the 240$
OpenStudy (happyvirus):
i hope that helped.
OpenStudy (beststudent):
@happyvirus What is that 100 bushels for?
OpenStudy (happyvirus):
the question is confusing by itself so it's hard to explain
OpenStudy (happyvirus):
I think it's additional information to confuse you, cause i didn't really need it to find the price.
OpenStudy (mathmale):
I disagree with that. Obviously, if the price is lower, the person with $240 could buy more wheat. Let x=prince of one bushel of wheat OpenStudy (mathmale): Then the number of bushels this person could buy for$240 would be $240/x. In other words: total price / (price per bushel) = number of bushels that could be purchased. Now if the price were lower, this person could buy 100 more bushels. Write this as$240/x + 100= \$240/(x-20) Determine whether or not you can solve this for x. If you can, then x represents the price per bushel.
OpenStudy (beststudent):
But Im not trying to find the price of each bushel. Im trying to find how many bushels he can buy?
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## Stream: new members
### Topic: Christopher Hoskin
#### Christopher Hoskin (Dec 22 2020 at 16:36):
Hello, I'm a new member. I've recently started learning lean, and so far have mostly been able to solve my own problems. However, here is something that has me stumped:
import algebra.module.linear_map
universes u
variables {A : Type u} [add_comm_monoid A] [semimodule ℤ A]
lemma test (T : linear_map ℤ A A) (a : A) : 2 • (T a) = T (2 • a) :=
begin
rw T.map_smul 2 a,
end
Lean says rewrite tactic failed, did not find instance of the pattern in the target expression ⇑T (2 • a).
Why does this not work?
Christopher
#### Alex J. Best (Dec 22 2020 at 16:41):
When you write 2 lean treats it as the natural number 2, rather than the integer 2, which is ok as A is an add_comm_monoid, so multiplication by a nat is well defined even without the semimodule structure.
#### Alex J. Best (Dec 22 2020 at 16:41):
import algebra.module.linear_map
universes u
variables {A : Type u} [add_comm_monoid A] [semimodule ℤ A]
set_option pp.all true
lemma test (T : linear_map ℤ A A) (a : A) : (2:ℤ) • (T a) = T ((2:ℤ) • a) :=
begin
rw T.map_smul (2:ℤ) a,
end
#### Alex J. Best (Dec 22 2020 at 16:42):
If we use (2 : ℤ) instead everything works as expected
#### Alex J. Best (Dec 22 2020 at 16:43):
When you see something like rewrite tactic failed, did not find instance of the pattern in the target expression one debugging tool is to turn off all pretty printing (notation, implicit arguments etc) so you an see what lean thinks is really going on. Thats the line set_option pp.all true I added.
#### Alex J. Best (Dec 22 2020 at 16:44):
You'll see that everything is almost unreadable with pretty printing turned off but when two things look the same in the pretty printer (like integer 2 and natural number 2) this really helps you notice.
#### Alex J. Best (Dec 22 2020 at 16:45):
The original lemma you stated is still true of course, and simp proves it for you btw
#### Patrick Massot (Dec 22 2020 at 16:47):
Using pp.all true is the last desperate move to try though. Using widget inspection is much cheaper.
nat_vs_int.gif
#### Christopher Hoskin (Dec 22 2020 at 17:49):
Thanks - if this was a step in a longer proof, is there a way to cast to integers before doing the rewrite? A somewhat artificial example:
lemma test2 (T : linear_map ℤ A A) (a : A) : T (a+a) = (2:ℤ) • T a :=
begin
rw ← two_nsmul,
rw_mod_cast T.map_smul 2 a,
end
I was hoping something like rw_mod_cast would work here, but it seems not.
#### Alex J. Best (Dec 22 2020 at 18:30):
I think simp alone worked for your first example, if you do squeeze_simp you can see what simp used to rewrite, it may be that map_smul just isnt the right lemma when naturals are involved
#### Christopher Hoskin (Dec 22 2020 at 19:02):
Thanks again. It looks like simp was using linear_map.map_smul_of_tower
lemma test (T : linear_map ℤ A A) (a : A) : 2 • (T a) = T (2 • a) :=
begin
simp only [linear_map.map_smul_of_tower]
end
simp isn't able to convert the \N scalar product to the \Z one though:
lemma testn (T : linear_map ℤ A A) (a : A) : 2 • (T a) = T (a+a) :=
begin
rw ← two_nsmul,
simp,
end
My work around is to introduce a new result two_gsmul instead:
theorem two_gsmul (a : A) : 2 • a = a + a :=
@pow_two (multiplicative A) _ a
lemma test2 (T : linear_map ℤ A A) (a : A) : 2 • (T a) = T (a+a) :=
begin
rw ← two_gsmul,
simp,
end
Christopher
Last updated: May 08 2021 at 18:17 UTC
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## Calculus 8th Edition
$H'(u) = 6u+5$
$H(u) = (3u-1)(u+2)$ Product Rule $H'(u) = (3u-1)(1) + (3)(u + 2)$ Simplify $H'(u) = 3u-1+3u+6$ Simplify $H'(u) = 6u+5$
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## Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry request pdf gas in stomach
################
We are concerned with spherically symmetric solutions to the Euler equations for the multi-dimensional compressible fluids, which have many applications in diverse real physical situations. The system can be reduced to one dimensional isentropic gas dynamics with geometric source terms. Due to the presence of the singularity at the origin, there are few papers devoted to this problem. The present paper proves two existence theorems of global entropy solutions. The first one focuses on the case excluding the origin in which the negative gas 69 velocity is allowed, and the second one is corresponding to the case including the origin with non-negative velocity. The $L^\infty$ compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rates near origin, then the densities and electricity and magnetism online games velocities tend to zero with the same rates near the origin for any positive time. In particular, the entropy solutions in two existence theorems are uniformly bounded with respect to time.
We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles hp gas online booking phone number with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted $L^p$ norms for gas works park the whole range of physical adiabatic exponents $\gamma\in (1, \infty)$, so that the viscosity approximate solutions satisfy the general $L^p$ compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all $\gamma\in (1, \infty)$. The approach and techniques developed here apply to other problems with similar gas and electric credit union difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all $\gamma\in (1, \infty)$.
In this paper, we consider the Cauchy problem for the Euler equations in the spherically symmetric case when the initial data are small perturbations of the trivial solution, i.e.,u≡0 and ρ≡constant, whereu is velocity and ρ is density. We show that this Cauchy problem can be reduced to an ideal nonlinear problem approximately. If we assume all the waves move at constant speeds in the ideal problem, by using Glimm’s scheme and gas dryer vs electric dryer cost savings an integral approach to sum the contributions of the reflected waves that correspond to each path through the solution, we get uniform bounds on theL
∞ norm and total variational norm of the solutions for all time. The geometric effects of spherical symmetry leads to a non-integrable source term in the Euler equations. Correspondingly, we consider an infinite reflection problem and solve it by considering the cancellations between reflections of different orders in our ideal problem. Thus we view this as an analysis of the interaction effects at the quadratic level in a nonlinear model problem for the Euler equations. Although it is far more difficult to obtain estimates in the exact solutions of the Euler equations due to the problem of controlling the time at which the cancellations occur, we believe that this analysis of the what is electricity wave behaviour will be the first step in solving the problem of existence of global weak solutions for the spherically symmetric Euler equations outside of fixed ball.
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# Tag Info
2
I'd recommend searching for some open source math books; there are quite some on Github, and probably more elsewhere. That way, you can somehow choose the kind of formulae you want to test; eg., topology (with much algebra), mathematical modelling (linear algebra, differential equations), combinatorics, or much more abstract stuff like the HoTT book with a ...
5
the file testmath.tex is part of the documentation for the amsmath package. it's in tex live, on ctan, or available via a link on the page http://www.ams.org/tex/amslatex (under "additional documentation}. the content is kind of "random", but has a good variety of examples and has been used, among other things, for stress testing of new math fonts.
3
This could be a useful collection of latex examples: Latex-examples
9
There isn't a curated list similar to Comprehensive LaTeX Symbol list, but you can generate various automated lists to show all the characters in a font. Run context --global --bodyfont=modern s-math-characters.mkiv This generates a 137 page document. Here is a snippet mathname is the name of the macro that will give you the symbol. As you can see, ...
1
It may make more sense to use a verbatim mode to display R code. For example the fancyvrb package allows you to define a delimiter for inline verbatim: e.g. \usepackage{fancyvrb} \DefineShortVerb{\|} Then in your text you can use | (y ~ x) | directly. For larger chunks of code, the listings package is a good choice. Here's an example using it with R ...
Top 50 recent answers are included
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# White noise DC component
I'm really new to DSP, I'm actually studying Computer Science and took an elective on DSP so my knowledge is pretty limited.
I've learned that pure white noise signal for example has all possible frequencies, that means that it also has zero frequency so it has a non zero DC component(does that mean it?) but pure white noise by definition has a zero mean and thus has a zero DC component..
What am I missing?
Thanks!
• The value of a signal's PSD at $f=0$ is not the same thing as the signal's DC value; see dsp.stackexchange.com/questions/21583/… – MBaz Feb 17 '17 at 22:24
• These are two separate questions. Therefore they should be asked separately. – Tendero Feb 17 '17 at 22:50
• @MBaz I see.. so does that mean that DC component is in time domain and the component at $f=0$ is in frequency domain and thus they have different values? Or is it something else? – Evgeny A. Feb 17 '17 at 22:51
• @Tendero Yeah I thought so.. should I split it now? – Evgeny A. Feb 17 '17 at 22:54
• @EvgenyA. I just edited your post to left only one of them. Please feel free to ask the other one in a different post. – Tendero Feb 17 '17 at 22:55
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# Why and how are resonance frequencies of a system dependent upon the shape, mass and the way it is constrained?
I was reading a general overview on the frequencies before conducting a modal analysis for my structure using ANSYS, in hopes of obtaining the natural frequencies for my structure. I read in some article that if you change the geometrical shape of your structure, change its overall mass, and even change the locations of the supports on it while conducting the modal analysis, it is very probable you will see a different set of modal frequencies. My question is why and how? Moreover, I also want to ask that after getting the natural frequencies for my structure, does this mean that ALL the structure must be excited by an external frequency equal to natural frequency, to expect some dangerous and catistropic failure? Or even if I excite certain parts of my structure, for example some specific locations on my structure where it is connected to another part (which is essentially causing my structure to vibrate at these connecting locations) should also be dangerous?
• Have you researched natural frequency? – Solar Mike Mar 28 at 19:17
• Yes ofcourse, thats what I am aiming to get for my structure. Actually, my structure is a composite whose thickness can be changed by increasing/decreasing the number of plies. I want to know how would a change in thickness change the natural frequencies of my structure. – Rameez Ul Haq Mar 28 at 19:24
• So, if you did research the natural frequencies then you would have found the formulae... – Solar Mike Mar 28 at 19:30
• So you might find this useful: brown.edu/Departments/Engineering/Courses/En4/Notes/… – Solar Mike Mar 28 at 19:36
In order to answer that question you need to build up your knowledge in vibrational dynamics. This usually takes a whole semester in undergraduate physics at the latter stage of an engineering curriculum.
So you need to progress -at least- through the following concepts:
• The free response of the undamped Harmonic Oscillator with 1 dof (mass spring no external excitation)
• The free response of the damped Harmonic Oscillator with 1 dof (mass damper spring no external excitation)
• The forced harmonic response of the damped Harmonic Oscillator with 1 dof (mass damper spring with external sinusoidal excitation)
• The forced harmonic response of the damped Harmonic Oscillator with 1 dof (mass damper spring with external sinusoidal excitation)
Then you need to start thinking about mdof systems, and the matrix form of the problem (this is very helpful especially with the FE), and you need to do the following:
• Start with the simplest 2DOF (undamped no excitation) to see the form of the system.
The main difference from the SDOF systems, is the interactions between the position of the masses (and therefore the springs and dampers).
It is very crucial to understand at that there can be many different transformations for the same problem depending on the generalised coordinates you select. A simple example of what I mean is the following problem.
In the above problem you can describe the motion equations with coordinates $$x_1$$ and $$x_2$$ (the spring displacements), or equivalenty by $$x,\theta$$ (the motion of the center of gravity and the rotation). The result should always be the same, however the individual coordinate responses will vary.
• Then you can proceed to Modal analysis
What is important to understand is that among the infinite number of generalised coordinates , where the responses are coupled, and additionally that there is at least on set of coordinates (sometimes called principal coordinates) where the transformed coordinate is decoupled (and are solved like the sdof - hence why you need to understand the behaviour of SDOF systems).
To obtain that set of coordinates, one way is to use the eigenvalues and eigenvectors of the mass normalised stiffness matrix. The cool thing is that you can rotate the initial problem, to obtain the principal coordinates, solve for the response of each decoupled system separately and then translate back to the original coordinates. The rotational transformation does not change the eigenfrequencies, in fact in the decoupled system you have a mode shape and a sinusoidal response. So each "mass" is pulsing proportionally to the other following one eigenfrequency. When you rotate back at the original masses/coordinate system, you get a contribution from both decoupled responses (hence the seemingly chaotic behavior).
• The final step is the Forced response of MDOF systems.
The same transformation that is applied to the Masses and springs can be applied to the force matrix (both directions). If you are planning to excite at a resonant frequency the mass you can apply a resonant force on the principal coordinate system and the transform that force matrix to the original matrix. When you do that you see that in the general case, you need to apply on all of the original coordinates a portion of the force with eigenfrequency $$\omega_i$$ in order to obtain the mode shape.
The transformation is essentially a rotation matrix, which can be applied to the excitation matrix. However this means that in order to excite the if you excite the nodes with the frequency at those eigenfrequencies,
• You do describe the steps one should take in order to understand some of the mathematics behind it, but those are not strictly necessary in order to have some intuitive understanding of it. Also, it is worth mentioning that doing a finite element analysis using matrices has its limitations, since the underlying physics is actually a partial differential equation (PDE). Only the first few lowest natural frequencies obtained from those matrices are usually accurate, because the matrices are obtained by discretizing space which gives an approximation of the underlying PDE. – fibonatic Mar 29 at 15:42
• I was writing my reply, probably at the same time as you were. I do confess, my approach was more of the undergraduate textbook, for discrete systems. The way I see it you took the continuum system which applies better to real structures, and you took a more hand's on approach. To be honest, the question was very vague, and there are different interpretations. I do feel that both posts have their own merit. – NMech Mar 29 at 16:09
• I agree that both our answers have merit. And I agree that the question itself is a bit vague, probably because of the lacking theoretical knowledge. This also why I tried to keep a lot of the underlying theory out of it. I myself find it useful to first gain some intuition and maybe spark some curiosity before diving deeper into underlying mathematical concepts. But hopefully the combination of our two answers might inspire some people reading them to learn more about it. – fibonatic Mar 29 at 17:00
Natural frequencies inside structures are standing waves that are reflected and propagated throughout the structure. The frequency of such wave depends on the shape of that wave and the speed at which sound travels through the material the structure is made of. The shape of possible waves in a structure I think can be well illustrated with Chladni figures. Adding supports constraints what kind of wave shapes can appear inside a structure, which you can also see in this video where putting a finger against the Chladni plate can be seen as a support. I have to note that in that video the natural frequencies of the plate are not actually altered, the temporary support initially limits which natural frequency/mode is excited.
Those standing waves do eventually die out if no energy is supplied to excite that natural frequency/mode. This is due to ways of dissipating energy, such as sound or damping behavior the material itself. But if energy at the "right" frequency is supplied to a structure at "right" locations, then the rate of energy dissipation might be too small, causing the energy of that natural frequency/mode to build up to dangerous levels until something might break. These "right" locations means that one isn't supplying the energy at nodes of that wave shape.
Adding mass to a structure can affect the way sound is reflected near that mass in the structure and thus affect wave shapes. This also closely related to impedance, which is demonstrated in this video.
When exciting a structure, not near any of the nodes of the natural frequencies, one does not have be exactly at any of the natural frequencies in order to get a large response of the structure. Though, the closer you are to one of the natural frequencies the larger the response. This can be illustrated with frequency response functions, sometimes also referred to as Bode plots, whose core concept I think is demonstrated well in this video.
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# Into Math Grade 7 Module 12 Lesson 2 Answer Key Make Inferences from a Random Sample
We included HMH Into Math Grade 7 Answer Key PDF Module 12 Lesson 2 Make Inferences from a Random Sample to make students experts in learning maths.
## HMH Into Math Grade 7 Module 12 Lesson 2 Answer Key Make Inferences from a Random Sample
I Can use proportional reasoning to make inferences about populations based on the results of a random sample.
At a grocery store, a bin is filled with trail mix made by mixing raisins with a large 30-pound bag of nuts. Zane buys a small bag of a trail mix that contains 1$$\frac{1}{2}$$ pounds of nuts and $$\frac{1}{2}$$ pound of raisins. If the nuts and raisins in Zane’s bag are proportional to the nuts and raisins in the bin of trail mix, how many pounds of raisins do you think the store used to make the entire bin of trail mix?
Turn and Talk How is the connection between the sample (small bag) and population (large bin) of trail mix similar to the sample and population of a survey?
Build Understanding
Question 1.
To estimate the number of pets that students in your school have, conduct a survey of ten randomly selected students in your class.
A. Plot the results of your survey on the grid provided.
B. According to my survey, most students in my school have ____________ pets.
C. According to my survey, about ___________ % of the students in my school have more than two pets.
D. According to my survey, about % of the students in my school have zero pets.
Conduct the same survey again using a second set of ten randomly selected students in your class.
E. Plot the results of your survey on the grid provided.
F. According to my survey, most students in my school have ___________ pets.
G. According to my survey, about ___________ % of the students in my school have more than two pets.
H. According to my survey, about __________ % of the students in my school have zero pets.
I. Compare the results from both of your samples.
Are the results from the two samples exactly the same?
Will a different sample give a different estimate?
A sample ratio can be used to estimate a population ratio. However, because different samples will likely vary, a sample ratio must be considered as only an estimate of the population ratio.
Turn and Talk Discuss how samples from random surveys can be improved to obtain better estimates about a population.
Step It Out
To make inferences about a population based on a random representative sample, you can use proportional reasoning.
Question 2.
Javier randomly selects 12 cartons of eggs from the grocery store. He finds that 2 cartons have at least one broken egg.
Suppose there are 144 cartons of eggs at the grocery store. What is an estimate of the total number of those 144 cartons that have at least one broken egg?
A. Identify the sample.
B. Identify the population.
C. Write the ratio of cartons with at least one broken egg to the total number of cartons in the sample.
D. Use the sample ratio to write an equation for the proportional relationship.
y = • 144
E. Use your equation in Part D to estimate the number of cartons in the population that have at least one broken egg.
Turn and Talk Discuss how to write an equation for the proportional relationship using a decimal or a percent for the sample ratio.
Question 3.
A worker randomly selects one out of every 7 sets from the 3,500 sets of headphones produced. The results are shown.
A. The (population / sample) is the total of 3,500 sets of headphones produced. The 500 selected for testing is the (population / sample).
B. Write the ratio of defective headphones to total headphones in the sample. Then write the ratio as a decimal and as a percent.
C. Write and solve an equation to find the number of headphones in the population that can be estimated to be defective.
y = • 3,500
Check Understanding
Question 1.
William conducted a random survey of the students in his school regarding the number of hours of sleep they got last night. The box plot shows the results of his survey. Make an inference about the entire population.
Question 2.
Hazel assigned a number to each of the 100 students in the band and put the numbers in a bag. She randomly chose 20 numbers and found that 3 students did not complete their homework for today. Make an inference about the number of students in the band that did not do their homework. If Hazel randomly chose 20 more numbers, what results would you expect? Explain.
For Problems 3-5, make an inference about the ages of all drama club students at a theater conference using the dot plot showing the ages of students in a random sample of conference attendees.
Question 3.
Most drama club students at the conferenece are ___________ 15 years old.
Answer: From the above dot plot we can observe that Most drama club students at the conference are more than 15 years old.
Question 4.
About _________ % of the students at the conference plays are 16 years old or older.
Total = 20
7 students at the conference plays are 16 years old or older.
7/20 = 0.35 = 35%
About 35% of the students at the conference plays are 16 years old or older.
Question 5.
Construct Arguments Would you think that it is likely for the number of 16-year-old students and the number of 17-year-old students to be almost equal in another random sample of conference attendees? Explain.
Question 6.
A manager randomly selects 1,500 ink pens produced today and finds 12 of them defective. There were 12,000 ink pens produced today. Make an inference about the number of ink pens produced today that are defective.
The sample ratio of defective pens is or %.
Inference: In the 12,000 population, the number of defective pens is
estimated to be • 12,000 = .
Number of pens select randomly = 1500
Number of defective pens = 12
The sample ratio of defective pens is 12/1500 = 1/125 = 0.008 = 0.8%
Inference: In the 12,000 population, the number of defective pens is
estimated to be
1/125 × 12000 = 96
Question 7.
Gabby assigned a number to each of the 120 athletes at her school and put the numbers in a box. She randomly chose 25 numbers and found that 10 athletes were female. Use this sample to make an inference about how many athletes at Gabby’s school are female.
Given,
Gabby assigned a number to each of the 120 athletes at her school and put the numbers in a box.
She randomly chose 25 numbers and found that 10 athletes were female.
10/25 = 0.4
0.4 × 100 = 40
10/25 = 40%
40% is the experimental probability for the number of female athletes in the sample.
40% of 120 should be about the number of female athletes in the whole school.
120 × 0.4 = 48
Question 8.
A random sample of dry-erase board markers at Juan’s school shows that 9 of the 60 dry-erase board markers do not work. There are 200 dry-erase board markers at Juan’s school. Make an inference about the number of dry-erase board markers at Juan’s school that do not work.
Question 9.
A mail carrier randomly inspects every 20th letter being mailed. Out of 600 letters in the sample, 3 were open. There were 18,000 letters being mailed. Make an inference about the number of all the letters being mailed that were open.
The box plot shows the results of a survey of the number of minutes that people at a variety of randomly selected gyms exercise. For Problems 10-13, make an inference about the number of minutes that people at gyms exercise according to this survey.
Question 10.
According to the survey, 75% of people at gyms exercise for ______________ minutes or longer.
Question 11.
According to the survey, 25% of people at gyms exercise for more than ____________ minutes.
Question 12.
According to the survey, __________ % of people at gyms exercise from 15 to 50 minutes.
Question 13.
According to the survey, _____________ % of people at gyms exercise from 15 to 30 minutes.
Question 14.
Health and Fitness Would the owners of another gym be able to use data from a survey like the one in Problems 10-13 to make inferences about the number of minutes people exercise at their gym? Explain your reasoning.
Question 15.
A wildlife park manager is working on a request to expand the park. In a random selection during one week, 3 of every 5 cars have more than 3 people inside. If about 5,000 cars come to the park in a month, estimate how many cars that month would have more than 3 people inside. Show your work.
I’m in a Learning Mindset!
How is making inferences from random samples similiar to the way I make decisions when I am learning something new?
Lesson 12.2 More Practice/Homework
Xavier surveyed a random sample of the grade levels of the Spanish Club members in the county. The bar graph shows the results of his survey. Use this information for Problems 1-5.
Question 1.
The largest number of students in the Spanish Club are in ___________ grade.
Answer: The largest number of students in the Spanish Club are in 10th grade.
Question 2.
The same number of students in the Spanish Club are in the ______________ grade as are in the 11th and 12th grades combined.
Students in 11th and 12th grades combined = 5 + 3 = 8
Number of students in 9th grade = 8
The same number of students in the Spanish Club is in the 9th grade as are in the 11th and 12th grades combined.
Question 3.
If there are 300 students in the Spanish Club in the county, predict how many are 10th graders.
Total number of students = 8 + 14 + 5 + 3 = 30
Given condition, If there are 300 students in the Spanish Club in the county, predict how many are 10th graders.
Number of 10 grade students = 14
14 × 10 = 140
Thus if there are 300 students in the Spanish Club in the county then there are 140 students in 10th grade.
Question 4.
Number of students in 9th grade = 8
8 × 10 = 80
Of the 300 students, there are 80 students in the 9th grade.
Question 5.
Xavier conducted another random survey of the grade levels of the Spanish Club members in the county. In what grade would you expect to find the most students in Spanish Club? Explain.
Question 6.
A manager at a factory finds that in a random sample of 200 clocks, 15 are defective.
A. What percent of the clocks are defective?
As it is given that a random sample of 200 clocks is taken out of 10,000 clocks, out of this 200, 15 clocks are defective.
Therefore, the number of clocks that are defective in 200 random samples is 15, which can be written as,
15/200 × 100 = 15/2 = 7.5%
B. Of the 10,000 clocks from which the sample was chosen, about how many clocks are probably not defective?
As we got the ratio of the defective clocks, therefore, the number of defective clocks in 10,000 clocks can be written as,
7.5% of 10000
0.075 × 10,000 = 750
Therefore of the 10,000 clocks from which the sample was chosen, 750 clocks were defective.
C. The next day the manager finds only 8 of the 200 randomly selected clocks are defective. About how many clocks out of the 10,000 produced that day are probably defective?
As in the next day, the manager finds only 8 of the 200 randomly selected clocks are defective.
Percentage of defective clock = 8/200 × 100 = 4%
As we got the ratio of the defective clocks, therefore, the number of defective clocks in 10,000 clocks can be written as,
4% of 10,000
0.04 × 10,000 = 400
Therefore, of 10,000 clocks from which the sample was chosen 400 clocks were defective.
Question 7.
Use Structure Based on a sample survey, a tutoring company claims that 90% of their students pass their classes. Out of 300 students, how many would you predict will pass?
90% can be written as 0.90
Multiply the percentage with 300 we get
0.9 × 300 = 270 students
Test Prep
Question 8.
Ronnie surveyed a random selection of realtors in his town about the number of bedrooms in the houses for sale that week. The dot plot shows the results. Which inference is correct?
(A) Most of the houses have fewer than 3 bedrooms.
(B) Some houses have O bedrooms.
(C) More than 50% of the houses have exactly 3 bedrooms.
(D) 80% of the houses have 3 or 4 bedrooms.
Answer: (C) More than 50% of the houses have exactly 3 bedrooms.
Question 9.
A random sample of laptop computers at an electronics store shows that 1 of the 25 sampled laptop computers has a malfunction. There are 300 laptop computers at the electronics store. Estimate the number of laptop computers at the electronics store that have malfunctions.
Given,
A random sample of laptop computers at an electronics store shows that 1 of the 25 sampled laptop computers has a malfunction.
There are 300 laptop computers at the electronics store.
300 ÷ 25 = 12
Thus 12 laptop computers at the electronics store have malfunctioned.
Question 10.
Jaylen used the seat number for each of the 6,500 fans’ seats in the stands at a college football game and put the numbers in a computer program. He randomly chose 200 numbers and found that 36 of those people had also purchased a parking voucher. Estimate the number of fans ¡n the stands at a sold-out football game that purchased a parking voucher. Explain.
Number of seats = 6500
Sample = 200
Voucher = 36
proportion = voucher/sample
p = 36/200
p = 0.18
Using the larger population 6500 the estimate of people that have bought the voucher is calculated
Estimate = proportion × seats
Estimate = 0.18 × 6500
Estimate = 1170 seats
Spiral Review
Question 11.
Ricardo jogged up 864 steps in 13$$\frac{1}{2}$$ minutes. What is Ricardo’s average number of steps per minute?
Given,
Ricardo jogged up 864 steps in 13$$\frac{1}{2}$$ minutes.
13$$\frac{1}{2}$$ = 13.5
864/13.5 = 64
Therefore Richard jogged 64 steps per minute.
Question 12.
Imani wants to know the favorite day of the week of adults in the town where she lives. Imani surveys every tenth adult that enters a convenience store between 4:00 p.m. and 8:00 p.m. Identify the population and sample.
|
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### Why is ContourPlot not displaying this curve?
I am using the general form of a second-degree plane curve: $$Ax^2+2Bxy + Cy^2+2Dx + 2Ey + F = 0$$ I want to randomly generate plane curves of this form, so I am using ...
554 views
### Why does Integrate declare a convergent integral divergent?
When I try this command Integrate[1/Sqrt[(s^2 - u)^2 - 1], {s, m, Infinity}, Assumptions -> u > 2 && m > 10] Mathematica declares that the ...
237 views
### Unexpected behavior of rule matching a pattern
I am a beginner exploring the world of Mathematica. I expected the following code T[6, 5, 4, 1, 2, 3] /. {T[a___, 1, b___] -> Length[List[b]]} should return ...
276 views
### Confused by (apparent) inconsistent precision
$$e^{\pi \sqrt{163}} \approx 262537412640768743.99999999999925$$ E^(Pi Sqrt[163.0]) N[E^(Pi Sqrt[163.0]), 35] NumberForm[E^(Pi Sqrt[163.]), 35] returns ...
3k views
### Fixed color scale in multiple density plots
I think there should be a simple solution for my current problem, but neither StackOverflow (or the help of Mathematica) or Google have it. I have to plot multiple density plots with a color ...
330 views
### How can I compare a dynamic variable with a literal in Mathematica?
I'm doing a Mathematica Notebook and I want to make an alarm clock. Something like this: ...
248 views
I don't understand the following: f[a_, b_] := a + b ls = {1, 2, 3}; MapThread[f, {ls, {10, 20, 30}}] This yields ( as expected ) {11,22,33} If I change ...
7k views
### Resources for beautiful Mathematica Stylesheets
When the Wolfram Demonstrations were introduced and the Documentation-Center was redesigned, I remember it was the first time I thought someone had put some effort into creating a beautiful ...
1k views
### Syntax highlighting for your own functions
Mathematica has a useful feature that for functions and special constructs getting passed local variables (for example Minimize or ...
4k views
### How can I generate this “domain coloring” plot?
I found this plot on Wikipedia: Domain coloring of $\sin(z)$ over $(-\pi,\pi)$ on $x$ and $y$ axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude. ...
6k views
### How to peel the labels from marmalade jars using Mathematica?
How can I detect and peel the label from the jar below (POV, cylinder radius, jar contents are all unknown) to get something like this, which is the original label before it was stuck on the jar? ...
3k views
### Does Mathematica have advanced indexing?
I have two $M \times K$ arrays $L, T$ where I would like to set all the elements in $L$ to zero whenever the corresponding element of $T$ is greater than 15. The ...
1k views
### How to generally match, unify and merge patterns?
This question was split from this one. While that question is now about how to match two particular patterns (mostly using Verbatim or ...
4k views
### How to change the default ColorData used in Mathematica's Plot?
This question leads on from the recent question What are the standard colors for plots in Mathematica? There it was determined that the default color palette used by ...
6k views
### How do I plot coordinates (latitude and longitude pairs) on a geographic map?
I'm attempting for the first time to create a map within Mathematica. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I ...
7k views
### How to use Mathematica functions in Python programs?
I'd like to know how can I call Mathematica functions from Python. I appreciate a example, for example, using the Mathematica function Prime. I had search about MathLink but how to use it in Python ...
2k views
### How to visualize/edit a big matrix as a table?
Is it possible to visualize/edit a big matrix as a table ? I often end up exporting/copying big tables to Excel for seeing them, but I would prefer to stay in Mathematica and have a similar view as in ...
2k views
### Simpler input for the new unit support
I've been playing with the new unit support in Mathematica 9. It seems very useful, but the syntax is very verbose. Instead of typing: ...
2k views
### Data Table Manipulation in Mathematica
I am a statistician searching for an efficient way to select rows or columns from a table of data in Mathematica. Let me pose the question in 2 parts with a SQL-style table of data: ...
1k views
### How can all those tiny polygons generated by RegionPlot be joined into a single FilledCurve?
RegionPlot will usually generate a large number of tiny polygons for filling the region: ...
960 views
### The clearest way to represent Mathematica's evaluation sequence
WReach has presented here a nice way to represent the Mathematica's evaluation sequence using OpenerView. It is much more clear way to go than using the standard ...
5k views
### Simultaneously fitting multiple datasets
What is the proposed approach if one wants to simultaneously fit multiple functions to multiple datasets with shared parameters? As an example consider the following case: We have to measurements of ...
875 views
### Items known by CurrentValue
CurrentValue can be used to poll the state of numerous system values such as the mouse position. Its help page doesn't list all possible items, though. An item like ...
2k views
### Extruding along a path
I'm trying to render a 3D image of a path by extruding a circular cross-section along the path, creating a "snake-like" path. Here is an image I found to illustrate: I can't seem to figure out if ...
2k views
### How to make a drop-shadow for Graphics3D objects
What's the best way to make a drop shadow for a 3D object? image = Graphics3D[Sphere[], Boxed -> False] I can get a blurry black outline of this: ...
520 views
### Constructing symbol definitions for With
I would like to be able to define two arrays, one containing symbol names and one containing the values of those symbols, for use in constructs such as With. For ...
783 views
What are some complete examples of what one would include in a FrontEnd init.m that would make use of FrontEnd`AddMenuCommands ...
745 views
### Iterate until condition is met
I want to find the first 5 prime numbers of the form $n^6 + 1091$. I have used this code: Timing[Select[Table[n^6 + 1091, {n, 10000}], PrimeQ, 5]] Which gives ...
634 views
### Why does Simplify ignore an assumption?
Here is the example: Simplify[x + y, x + y == a] Simplify[x + y, x + y == 5] Mathematica 9 output: x+y 5 I expect the ...
4k views
### Changing the background color of a framed plot
I frequently generate framed plots like this: Plot[Sin[x], {x, 0, 2 \[Pi]}, Frame -> True] Is there an easy way to change the background color of the framed ...
2k views
### How to get intersection values from a parametric graph?
I have this graphed: ParametricPlot[{2.4*Cos[t] + 1.6*Cos[3 t/2], 2.4*Sin[t] - 1.6 Sin[3 t/2]}, {t, 0, 4*Pi}] It is a star, and the lines cross each other ...
1k views
### Scale Insetted Characters to Plot
I am trying to place a curly brace within a plot such that the top/bottom of the curly brace line up with two horizontal lines in the plot: I have not been able to find a way to make the curly ...
1k views
### Find continuous sequences inside a list
I have a list which is something like this: {3,4,5,6,7,10,11,12,15,16,17,19,20,21,22,23,24,42,43,44,45,46} What I'd like to to is get the intervals which are in ...
8k views
### How to make Jacobian automatically in Mathematica
If we have two vectors, a and b, how can I make Jacobian matrix automatically in Mathematica? a=\left( \begin{array}{c} \text{x1}^3+2\text{x2}^2 \\ 3\text{x1}^4+7\text{x2} \end{array} ...
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