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# Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$

$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k)$

$\int\limits_0^{\infty}\int\limits_x^{\infty}\, f(\gamma)\, d\gamma dx=\sum\limits_{k = 2 }^ \infty \frac{1}{k^2} \ln(k)$

$\int\limits_0^{\infty}\int\limits_x^{\infty} f(\gamma)\, d\gamma dx=\sum\limits_{k=2}^ \infty\ln(k^{\frac{1}{k^2}})=\ln(\prod\limits_{k=2}^{\infty}k^{\frac{1}{k^2}})$

$\prod\limits_{k=2}^{ \infty }k^{\frac{1}{k^2}}=\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}=e^{\int\limits_0^{\infty}\int\limits_x^{\infty} f(\gamma) \,d\gamma dx}$

$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k)$

$f(x)=\sum\limits_{k = 1 }^ \infty e^{-(k+1)x} \ln(k+1)$

$f(x)=e^{-x}\sum\limits_{k = 1 }^ \infty e^{-kx} \ln(k+1)$

$f(x)=e^{-x}\sum\limits_{n = 1 }^ \infty \frac{(-1)^{n+1}}{n} \sum\limits_{k = 1 }^ \infty k^n e^{-kx}$

We know that

$\sum\limits_{k = 1 }^ \infty e^{-kx}= \frac{1}{e^{x}-1}$

$\sum\limits_{k = 1 }^ \infty k^n e^{-kx}= (-1)^n\frac{d^n}{dx^n}(\frac{1}{e^{x}-1})$

$f(x)=e^{-x}\sum\limits_{n = 1 }^ \infty \frac{(-1)^{n+1}}{n} \sum\limits_{k = 1 }^ \infty k^n e^{-kx} = e^{-x}\sum\limits_{n = 1 }^ \infty \frac{(-1)^{n+1}}{n} (-1)^n\frac{d^n}{dx^n}(\frac{1}{e^{x}-1})$

$f(x)=-e^{-x}\sum\limits_{n = 1 }^ \infty \frac{1}{n} \frac{d^n}{dx^n}(\frac{1}{e^x-1})$

$\int\limits_0^{\infty}\int\limits_x^{\infty} f(\gamma) \,d\gamma dx= -\int\limits_0^{\infty}\int\limits_x^{\infty} e^{-\gamma}\sum\limits_{n = 1 }^ \infty \frac{1}{n} \frac{d^n}{d\gamma^n}(\frac{1}{e^{\gamma}-1})\, d\gamma dx$

I have lost my way after that.

Is it possible to find a closed form in my way? or I need to follow a different way. I need your mathematical sense. Thanks a lot for answers and advice.

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Try using Euler-Maclaurin formula en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula. –  Jon Jul 25 '12 at 15:06
FWIW: $$-\zeta^\prime(s)=\sum_{k=1}^\infty \frac{\log\,k}{k^s}$$ Now, let $s=2$ and exponentiate your series to turn it into a product... –  Ｊ. Ｍ. Jul 25 '12 at 15:54
It's funny how this question is also on the front page with this question. Now, I see that both questions are yours... –  Ｊ. Ｍ. Jul 25 '12 at 15:57
@unklerhaukus: JFYI instead of \limits one can also just enclosed the expressions in double dollar signs $$...$$ instead of single dollar signs $...$ to force display style instead of inline style. We ask that the maths in titles be written in inline style, but the body text can have displayed formulae. –  Willie Wong Jan 11 '13 at 13:43
@WillieWong that works for the products , but not for the integrals –  Elements in Space Jan 11 '13 at 13:58

Yes $\boxed{\displaystyle e^{-\zeta'(2)}}$ I think.

To prove it start with : $$\zeta(2-x)=\sum_{k=1}^\infty \frac {k^x}{k^2}$$

and compute the derivative!

The trick is that the derivation will create a $\ln(k)$ term at the numerator. At the end take the limit as $x\to 0$.

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@anon: you are right (and it seems you didn't see my last 4 micro-edits...:-)) –  Raymond Manzoni Jul 25 '12 at 14:48
No, I've been kicking back and watching them roll in one after the other :) Still, $s\to2$ makes more sense. –  anon Jul 25 '12 at 14:49
@RaymondManzoni : It is very nice way. I can see now I proved that $\int_0^{\infty}\int_x^{\infty} e^{-\gamma}\sum\limits_{n = 1 }^ \infty \frac{1}{n} \frac{d^n}{d\gamma^n}(\frac{1}{e^{\gamma}-1})d\gamma dx=\zeta'(2)$. Thanks a lot. –  Mathlover Jul 25 '12 at 14:52
@Mathlover: Nice! Note that $\zeta'(2)$ doesn't have a simpler closed form (I think...) and in fact it is used to define the Glaisher-Kinkelin constant so that you found a new expression for this famous constant, felicitations! –  Raymond Manzoni Jul 25 '12 at 14:58
For completeness: $$\zeta'(2)=(\gamma+\log\,2\pi-12\log\,A)\zeta(2)$$ where $A$ is the Glaisher-Kinkelin constant, though I believe $A=\exp\left(\frac1{12}-\zeta^\prime(-1)\right)$ is the more traditional definition. –  Ｊ. Ｍ. Jul 25 '12 at 15:50

Your infinite product equals to : $e^{-{\zeta}'(2)}$.

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G = C2×C5⋊2C16order 160 = 25·5

Direct product of C2 and C5⋊2C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C5⋊2C16
 Chief series C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C2×C5⋊2C16
 Lower central C5 — C2×C5⋊2C16
 Upper central C1 — C2×C8

Generators and relations for C2×C52C16
G = < a,b,c | a2=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×C52C16
Regular action on 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 97)(13 98)(14 99)(15 100)(16 101)(17 124)(18 125)(19 126)(20 127)(21 128)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 123)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(81 144)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)
(1 125 88 80 148)(2 149 65 89 126)(3 127 90 66 150)(4 151 67 91 128)(5 113 92 68 152)(6 153 69 93 114)(7 115 94 70 154)(8 155 71 95 116)(9 117 96 72 156)(10 157 73 81 118)(11 119 82 74 158)(12 159 75 83 120)(13 121 84 76 160)(14 145 77 85 122)(15 123 86 78 146)(16 147 79 87 124)(17 101 58 36 134)(18 135 37 59 102)(19 103 60 38 136)(20 137 39 61 104)(21 105 62 40 138)(22 139 41 63 106)(23 107 64 42 140)(24 141 43 49 108)(25 109 50 44 142)(26 143 45 51 110)(27 111 52 46 144)(28 129 47 53 112)(29 97 54 48 130)(30 131 33 55 98)(31 99 56 34 132)(32 133 35 57 100)
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)

G:=sub<Sym(160)| (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)>;

G:=Group( (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,97)(13,98)(14,99)(15,100)(16,101)(17,124)(18,125)(19,126)(20,127)(21,128)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,123)(33,76)(34,77)(35,78)(36,79)(37,80)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,145)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,125,88,80,148)(2,149,65,89,126)(3,127,90,66,150)(4,151,67,91,128)(5,113,92,68,152)(6,153,69,93,114)(7,115,94,70,154)(8,155,71,95,116)(9,117,96,72,156)(10,157,73,81,118)(11,119,82,74,158)(12,159,75,83,120)(13,121,84,76,160)(14,145,77,85,122)(15,123,86,78,146)(16,147,79,87,124)(17,101,58,36,134)(18,135,37,59,102)(19,103,60,38,136)(20,137,39,61,104)(21,105,62,40,138)(22,139,41,63,106)(23,107,64,42,140)(24,141,43,49,108)(25,109,50,44,142)(26,143,45,51,110)(27,111,52,46,144)(28,129,47,53,112)(29,97,54,48,130)(30,131,33,55,98)(31,99,56,34,132)(32,133,35,57,100), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160) );

G=PermutationGroup([[(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,97),(13,98),(14,99),(15,100),(16,101),(17,124),(18,125),(19,126),(20,127),(21,128),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,121),(31,122),(32,123),(33,76),(34,77),(35,78),(36,79),(37,80),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,145),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(81,144),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143)], [(1,125,88,80,148),(2,149,65,89,126),(3,127,90,66,150),(4,151,67,91,128),(5,113,92,68,152),(6,153,69,93,114),(7,115,94,70,154),(8,155,71,95,116),(9,117,96,72,156),(10,157,73,81,118),(11,119,82,74,158),(12,159,75,83,120),(13,121,84,76,160),(14,145,77,85,122),(15,123,86,78,146),(16,147,79,87,124),(17,101,58,36,134),(18,135,37,59,102),(19,103,60,38,136),(20,137,39,61,104),(21,105,62,40,138),(22,139,41,63,106),(23,107,64,42,140),(24,141,43,49,108),(25,109,50,44,142),(26,143,45,51,110),(27,111,52,46,144),(28,129,47,53,112),(29,97,54,48,130),(30,131,33,55,98),(31,99,56,34,132),(32,133,35,57,100)], [(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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)]])

C2×C52C16 is a maximal subgroup of
C40.10C8  C203C16  C40.2Q8  C10.SD32  C40.5D4  C10.Q32  C40.7Q8  D40.5C4  C16×Dic5  C40.88D4  C8017C4  D101C16  D20.4C8  C40.91D4  C40.92D4  C10.D16  C40.15D4  C20.58D8  C5⋊M6(2)  D5×C2×C16  D20.5C8  C40.70C23  C40.30C23
C2×C52C16 is a maximal quotient of
C203C16  C80.9C4  C40.91D4

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A ··· 8H 10A ··· 10F 16A ··· 16P 20A ··· 20H 40A ··· 40P order 1 2 2 2 4 4 4 4 5 5 8 ··· 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 D5 Dic5 D10 Dic5 C5⋊2C8 C5⋊2C8 C5⋊2C16 kernel C2×C5⋊2C16 C5⋊2C16 C2×C40 C40 C2×C20 C20 C2×C10 C10 C2×C8 C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 2 2 2 2 4 4 16

Matrix representation of C2×C52C16 in GL3(𝔽241) generated by

 1 0 0 0 240 0 0 0 240
,
 1 0 0 0 189 240 0 1 0
,
 197 0 0 0 170 114 0 191 71
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,240],[1,0,0,0,189,1,0,240,0],[197,0,0,0,170,191,0,114,71] >;

C2×C52C16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes_2C_{16}
% in TeX

G:=Group("C2xC5:2C16");
// GroupNames label

G:=SmallGroup(160,18);
// by ID

G=gap.SmallGroup(160,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,50,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^2=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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Strong markov property vs usual markov property.

I was trying to understand the difference between strong Markov property and the usual Markov property for a discrete number of states. I think I understand why the strong Markov property implies the usual one : We have to consider a deterministic stopping time $$T(\omega)=t_0$$, right?
But what would be a simple example (like coin toss, dice toss,...) where we have the Markov property but not the strong Markov property?


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# Two pipes P and Q can fill a tank in 6 hours and 9 hours respectively, while a third pipe R can empty the tank in 12 hours. Initially, P and R are open for 4 hours. Then P is closed and Q is opened. After 6 more hours, R is closed. The total time taken to fill the tank (in hours) is ____.

This question was previously asked in
PY 4: GATE ME 2019 Official Paper: Shift 2
View all GATE ME Papers >
1. 13.50
2. 14.50
3. 15.50
4. 16.50

Option 2 : 14.50
Free
CT 1: Ratio and Proportion
1963
10 Questions 16 Marks 30 Mins

## Detailed Solution

Concept:

Basic Concepts and Pipes & Cisterns formula:

• If a pipe can fill a tank in a hrs, then the part filled in 1 hr =1/a.
• If a pipe can empty a tank in b hrs, then the part of the full tank emptied in 1 hr = 1/b.
• If a pipe can fill a tank in a hrs and the another pipe can empty the full tank in b hrs, then the net part filled in 1 hr, when both the pipes are opened =[1/a - 1/b] ∴ Time taken to fill the tank, when both the pipes are opened = ab/(b - a)
• If a pipe can fill a tank in a hrs and another can fill the same tank in b hrs, then the net part filled in 1 hr, when both pipes are opened = [1/a + 1/b] ∴ Time taken to fill the tank = ab/(a + b)
• If a pipe fills a tank in a hrs and another fills the same tank in b hrs, but a third one empties the full tank in c hrs, and all of them are opened together, the net part filled in 1 hr = [1/a+ 1/b-1/c] ∴ Time taken to fill the tank =abc/(bc + ac – ab) hrs.
• A pipe can fill a tank in a hrs. Due to a leak in the bottom it is filled in b hrs. If the tank is full, the time taken by the leak to empty the tank = ab/(b - a) hrs.

Calculation:

P → 6 hrs (Fill), Q → 9 hrs (Fill), R → 12 hrs (Empty)

In 1 hr, (P + R) can fill: [1/6 - 1/12]

In 4 hrs, (P + R) can fill: $$4\left( {\frac{1}{6} - \frac{1}{{12}}} \right) = \frac{1}{3}$$

After 4 hrs, P is closed and Q is opened for 6 hours:

In 6 hrs, (R + Q) can fill: $$6\left( {-\frac{1}{12} + \frac{1}{{9}}} \right) = \frac{1}{6}$$

Total tank filled in 10 hours (4 + 6 hrs): $$\left( {\frac{1}{3} + \frac{1}{6}} \right) = \frac{1}{2}$$

After 6 more hours, R is closed. So the remaining 1/2 part of the tank will be filled by Q only.

To fill 1/2 part, Q will take: 9/2 = 4.5 hrs

Total time taken to fill the tank = 10 + 4.5 = 14.5 hrs


text:
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Wire-cell 3D pattern recognition techniques for neutrino event reconstruction in large LArTPCs: algorithm description and quantitative evaluation with MicroBooNE simulation
Abstract Wire-Cell is a 3D event reconstruction package for liquid argon time projection chambers. Through geometry, time, and drifted charge from multiple readout wire planes, 3D space points with associated charge are reconstructed prior to the pattern recognition stage. Pattern recognition techniques, including track trajectory and d Q /d x (ionization charge per unit length) fitting, 3D neutrino vertex fitting, track and shower separation, particle-level clustering, and particle identification are then applied on these 3D space points as well as the original 2D projection measurements. A deep neural network is developed to enhance the reconstruction of the neutrino interaction vertex. Compared to traditional algorithms, the deep neural network boosts the vertex efficiency by a relative 30% for charged-current ν e interactions. This pattern recognition achieves 80–90% reconstruction efficiencies for primary leptons, after a 65.8% (72.9%) vertex efficiency for charged-current ν e (ν μ ) interactions. Based on the resulting reconstructed particles and their kinematics, we also achieve 15-20% energy reconstruction resolutions for charged-current neutrino interactions.
Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
Award ID(s):
Publication Date:
NSF-PAR ID:
10336265
Journal Name:
Journal of Instrumentation
Volume:
17
Issue:
01
Page Range or eLocation-ID:
P01037
ISSN:
1748-0221
2. Abstract Atmospheric neutrinos are one of the most relevant natural neutrino sources that can be exploited to infer properties about cosmic rays and neutrino oscillations. The Jiangmen Underground Neutrino Observatory (JUNO) experiment, a 20 kton liquid scintillator detector with excellent energy resolution is currently under construction in China. JUNO will be able to detect several atmospheric neutrinos per day given the large volume. A study on the JUNO detection and reconstruction capabilities of atmospheric $$\nu _e$$ ν e  and $$\nu _\mu$$ ν μ  fluxes is presented in this paper. In this study, a sample of atmospheric neutrino Monte Carlo events has been generated, starting from theoretical models, and then processed by the detector simulation. The excellent timing resolution of the 3” PMT light detection system of JUNO detector and the much higher light yield for scintillation over Cherenkov allow to measure the time structure of the scintillation light with very high precision. Since $$\nu _e$$ ν e  and $$\nu _\mu$$ ν μ  interactions produce a slightly different light pattern, the different time evolution of light allows to discriminate the flavor of primary neutrinos. A probabilistic unfolding method has been used, in order to infer the primary neutrino energy spectrummore »


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1. ## Fourier transformation

Hi

I am trying to calculate the Fourier transformation of the function

f(x)=exp(-x)*(the characteristic function running from 0 to 1, both included)

what I have got so far is this

|f_hat(gamma)|=|int_from -infinty to infinity (f(x)*exp(-2*pi*i*x*gamma)dx)|

= int_o to 1(exp(-x)*exp(-2*pi*i*x*gamma)dx)

and then what? Have I made a mistake somewhere along the way or is it simply a case of having got stuck?

2. Originally Posted by Ase
Hi

I am trying to calculate the Fourier transformation of the function

f(x)=exp(-x)*(the characteristic function running from 0 to 1, both included)

what I have got so far is this

|f_hat(gamma)|=|int_from -infinty to infinity (f(x)*exp(-2*pi*i*x*gamma)dx)|

= int_o to 1(exp(-x)*exp(-2*pi*i*x*gamma)dx)

and then what? Have I made a mistake somewhere along the way or is it simply a case of having got stuck?

Then what ....? Use a well known index law to write the integrand in the form e^{kx} and then integrate.

3. Hi mrfantastic

I am well aware of the index law. The "then what" referred to what happens to the gamma. I can only integrate over x but what happens to gamma?

4. Originally Posted by Ase
Hi mrfantastic

I am well aware of the index law. The "then what" referred to what happens to the gamma. I can only integrate over x but what happens to gamma?
The integral is respect to x so gamma is treated as a constant as far as the integration is concerned. So the result of the integration is a function of gamma, which ought to be no surprise.

5. Ok. Thanks a lot. I was just under the impression that the end result should be a number (not a function).


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Thursday, February 23
1:00pm    SCGP: Math Seminar / Math of Gauge Fields: Katherine Raoux, "τ-invariants for knots in rational homology spheres"
Where:      SCGP 313When:        Thu, Feb 23    1:00pm — 2:00pm
Speaker:   Katherine Raoux

Title:          τ-invariants for knots in rational homology spheres

Abstract:    Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K), which they also showed is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we may obtain a collection of τ-invariants, one for each spin-c structure. We also show that these invariants can be used to obtain a lower bound on the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y

2:30pm    YITP: YITP Seminar, Hiren Patel (UMass, Amherst)
When:        Thu, Feb 23    2:30pm — 3:30pm
"Package-X: A Mathematica package for the analytic calculation of one loop integrals"

"Package-X is a Mathematica package for the fully analytic calculation of one-loop Feynman integrals. In this talk I will highlight the capabilities of Package-X using a selection of familiar examples, and summarize some of the novel algorithms driving the program. I will close the talk with possible directions for future development."

4:00pm    Colloquium: Otis Chodosh - New results on the global geometry of scalar curvature
Where:      Math Tower P-131When:        Thu, Feb 23    4:00pm — 5:00pm
Title:          New results on the global geometry of scalar curvature
Speaker:   Otis Chodosh [Princeton University/IAS]

Abstract:    I'll survey some recent results concerning the global behavior of minimal/constant mean curvature surfaces in three manifolds with non-negative scalar curvature. Co-authors for these works include M. Eichmair, V. Moraru, Y. Shi, and H. Yu.
View Details
Friday, February 24
11:00am    Am.Math.Soc. Chapter Seminar: Thomas Ng - Finite state automata and connections to group theory
Where:      Math Tower 5-127When:        Fri, Feb 24    11:00am — 12:00pm
Title:          Finite state automata and connections to group theory
Speaker:   Thomas Ng [Temple University]

Abstract:    Representing group elements in an efficient manner is impossible due to pathological examples such as those with unsolvable word problem. A finite state automaton(FSA) is a combinatorial tool from computation theory than can encode word theoretic properties of certain "well-behaved" groups for example the fundamental group of a surfaces or a hyperbolic 3-manifolds and braid groups. This structure can be use to algorithmically study algebraic, geometric, and probabilistic properties of these groups. We will start by introducing FSAs and how Thurston's notion of an automatic group provides fast solutions to the word problem and quadratic bounds on the isoperimetric function of the Cayley graph. We will also discuss how spectral properties of the automaton can be used to effectively study the growth of a group as well as random/generic group elements. One theme is that computationally simple groups have nice properties and there are many of them.
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12:00pm    Grad / Postdoc Professional Development Seminar: Robert Lazarsfeld - What does the AMS do, and why should we all belong to it?
Where:      Math Tower P-131When:        Fri, Feb 24    12:00pm — 1:20pm
Title:          What does the AMS do, and why should we all belong to it?
Speaker:   Robert Lazarsfeld [Stony Brook]

Abstract:    This week, Rob Lazarsfeld will give an overview of some of things that the American Mathematical Society does to support mathematics in the United States (in particular various AMS programs aimed at younger mathematicians). As always, pizza will be served.
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2:30pm    Dynamical Systems Seminar: Yuri Bakhtin - Ergodic theory of the Burgers equation with random forcing
Where:      Math Tower P-131When:        Fri, Feb 24    2:30pm — 3:30pm
Title:          Ergodic theory of the Burgers equation with random forcing
Speaker:   Yuri Bakhtin [Courant Institute of Mathematical Sciences]

Abstract:    The Burgers equation is a basic nonlinear evolution PDE of Hamilton--Jacobi type related to fluid dynamics and growth models. I will talk about the ergodic theory of randomly forced Burgers equation in noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the inviscid case) and polymer measures on one-sided infinite trajectories (in the positive viscosity case). This is joint work with Eric Cator, Kostya Khanin, and Liying Li.
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4:00pm    Geometric Analysis Learning Seminar: Shaosai (Arthur) Huang - TBA
Where:      P-131 Math TowerWhen:        Fri, Feb 24    4:00pm — 6:00pm
Title:          TBA
Speaker:   Shaosai (Arthur) Huang [Stony Brook]

Abstract:    TBA
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Monday, February 27
5:30pm    First and Second Year Student Seminar: Aleksander Doan and Ruijie Yang - First and Second Year Student Seminar
Where:      Math Tower P-131When:        Mon, Feb 27    5:30pm — 6:30pm
Title:          First and Second Year Student Seminar
Speaker:   Aleksander Doan and Ruijie Yang [Stony Brook]

Abstract:    As young mathematicians, Aleksander and Ruijie will share some of their experience and tips of doing research.
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Tuesday, February 28
4:00pm    Geometry/Topology Seminar: Christina Sormani - Convergence of Manifolds with Nonnegative Scalar Curvature
Where:      Math Tower P-131When:        Tue, Feb 28    4:00pm — 5:30pm
Title:          Convergence of Manifolds with Nonnegative Scalar Curvature
Speaker:   Christina Sormani [Lehman College and CUNY Graduate Center]

Abstract:    I will survey conjectures, examples and results about sequences of three dimensional manifolds with nonnegative scalar curvature. Some of the conjectures are stated in recent work of Gromov, in recent work of mine, and in joint work of mine with Lee. I will present partial results towards some of these conjectures completed with Huang-Lee, with Stavrov, and with LeFloch, and techniques related to the intrinsic flat convergence of manifolds appearing in joint work with Lakzian, with Portegies, and alone. Finally I will present examples constructed with Basilio. See arxiv1606.08949
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Wednesday, March 1
11:30am    RTG Student Geometry Seminar: John Sheridan - Sections of semi-positive line bundles
Where:      Math Tower 5-127When:        Wed, Mar 1    11:30am — 12:30pm
Title:          Sections of semi-positive line bundles
Speaker:   John Sheridan [Stony Brook]

Abstract:    We will discuss how to construct sections of line bundles on tori using Fourier series. This will allow us to compute the dimension of the space of holomorphic sections of the line bundle in terms of the corresponding data described in the Appel-Humbert Theorem. Finally, we will address the issue of determining ampleness.
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Where:      Math Tower P-131When:        Wed, Mar 1    1:00pm — 2:00pm
Title:          TBA
Speaker:   TBA [Stony Brook]

Abstract:    TBA
View Details

4:00pm    Algebraic geometry seminar: Dawei Chen - Extremal effective divisors on the moduli space of curves
Where:      Math Tower P-131When:        Wed, Mar 1    4:00pm — 5:30pm
Title:          Extremal effective divisors on the moduli space of curves
Speaker:   Dawei Chen [Boston College]

Abstract:    The cone of effective divisors governs central properties of a variety. Knowing the shape of the effective cone amounts to understanding extremal effective divisors that span the boundary rays of the cone. In this talk I will give an introduction to this topic, with a focus on the case when the underlying variety is the moduli space of curves. If time permits I will also talk about extremal effective cycles of higher codimension.
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Thursday, March 2
2:30pm    YITP: YITP seminar, Ciaran Williams
When:        Thu, Mar 2    2:30pm — 3:30pm

4:00pm    Colloquium: Arseniy Akopyan - Waists of balls in different spaces
Where:      Math Tower P-131When:        Thu, Mar 2    4:00pm — 5:00pm
Title:          Waists of balls in different spaces
Speaker:   Arseniy Akopyan [MCCME]

Abstract:    Gromov and Memarian (2003-2011) have established the waist inequality asserting that for any continuous map f from the sphere $S^n \to R^{n-k}$ there exists a fiber $f^{-1}(y)$ such that every its t-neighborhood has measure at least the measure of the t-neighborhood of an equatorial subsphere $S^k$ of $S^n$.

Going to the limit we may say that the (n-k)-volume of the fiber $f^{-1}(y)$ is at least that of the standard sphere $S^k$. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces.

By the volume of preimages for a non-regular map f we mean its lower Minkowski content, some new properties of which will be also presented in the talk.
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Friday, March 3
1:00pm    Grad / Postdoc Professional Development Seminar: Sixth meeting
Where:      Math Tower P-131When:        Fri, Mar 3    1:00pm — 2:30pm
Title:          Sixth meeting
Speaker:   TBA [Stony Brook]

Abstract:
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2:30pm    Dynamical Systems Seminar: Kyounghee Kim - Rational Surface Automorphism: Homology actions of real mappings
Where:      Math Tower P-131When:        Fri, Mar 3    2:30pm — 3:30pm
Title:          Rational Surface Automorphism: Homology actions of real mappings
Speaker:   Kyounghee Kim [Florida State University]

Abstract:    Let f:X -> X be a quadratic rational surface automorphism fixing a cuspidal cubic. Let X_R be the closure of RP^2 inside X. When the multiplier at the invariant cuspidal cubic is real, f induces an automorphism f_R of X_R. In this talk, we discuss an algorithm for calculating the homology action of f_R, and we make several observations on the homology action. This is joint work with J. Diller.
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4:00pm    Geometric Analysis Learning Seminar: Jean-Franois Arbour - TBA
Where:      P-131 Math TowerWhen:        Fri, Mar 3    4:00pm — 6:00pm
Title:          TBA
Speaker:   Jean-Franois Arbour [Stony Brook]

Abstract:    TBA
View Details


text:
# Eight girls A, B, J, K, P, R, L, and T are sitting around a circular table facing the center (not necessarily in the same order). J is sitting opposite P. Either A or B are immediate neighbors of K. T and L are immediate neighbors. J is second to the right of K. Either T or L is to the immediate right of J. If A is to the immediate left of P and L is to the immediate left of A, then which of the following statements about R can be true?

This question was previously asked in
DSSSB PRT General Section Official Paper - 11 Nov 2019 Shift 2
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1. K is second to the right of R
2. R is between K and B.
3. J is to the immediate right of R.
4. R is opposite L

Option 3 : J is to the immediate right of R.

## Detailed Solution

Eight girls A, B, J, K, P, R, L, and T are sitting around a circular table facing the center (not necessarily in the same order).

i) J is sitting opposite P.

ii) J is second to the right of K.

iii) T and L are immediate neighbors.

iv)  If A is to the immediate left of P and L is to the immediate left of A.

v) Either A or B are immediate neighbors of K.

vi) Either T or L is to the immediate right of J.

As A is to the immediate left of P and L is to the immediate left of A thus only case i) is possible and case ii) is discarded.

The final arrangement is:

Hence, the correct answer is "J is to the immediate right of R".


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## FANDOM

17,464 Pages

In the Pokémon games, the player can capture a Pokémon he/she likes to have in his/her team, but mostly the catch doesn't succeed as they hope it to be. Each Pokémon has a catch rate. It depends on the Pokémon, the Pokémon's HP, and the type of Poké Ball the player uses.

The lowest catch rate is 3, which belongs to the Beldum family and almost every legendary Pokémon in the game. A low catch rate means that the Pokémon is very hard to catch, often meaning the player must use over 20 Poké Balls or more. The highest catch rate is 255, which belongs to small Pokémon such as Caterpie or Pidgey, which can be easily caught with a Poké Ball without the need to damage them.

Depending on the amount of HP, current status ailment, level of the Pokémon, and/or the Poké Ball being used, the chances may raise or lower. Because of this, the player may sometimes have to use stronger Poké Balls or more of them to catch the desired Pokémon. If an attempt to catch a Pokémon fails too many times, the Pokémon may flee.

# Formulae

The formulae for catch rate is more complex and different in each generation.

## Generation I

In Generation I, the capture mode differs from the later generations. To determine the catch rate, there are important steps performed below. If, at any point, the Pokémon breaks free, the steps will never be performed.

1. If Master Ball is used, the Pokémon is always caught, no matter what.
2. Generate a random number, N, depending on the type of ball used.

Important formulae (always denoted in Θ (Theta)):

$\Theta_{PokeBall} = rand(255)$

$\Theta_{GreatBall} = rand(200)$

$\Theta_{UltraAndSafariBall} = rand(150)$

3. The Pokémon is caught if...
• If it is asleep or frozen and Θ is less than 25.
• It is paralyzed, burned, or poisoned and Θ is less than 12.
4. Otherwise, if Θ minus the status threshold (above) is greater than the Pokémon's catch rate, the Pokémon breaks free.
5. If not, generate a random value (Denoted by the letter M).

$M = rand(255)$

6. Calculate Λ.

Formula:

$\Lambda = \frac{\left ( \beta_{max} \times 255 \times 4\right )}{(\beta_{current} \times B)}$

beta_max - The maximum HP of a Pokémon
beta_current - The current HP of a Pokémon
B - The Poké Ball's integer

The minimum value of Λ is 1 and its maximum value is 255. The value of B is 8 if a Great Ball is used or 12 otherwise.

7. If Θ is greater than or equal to Λ, the Pokémon is caught. Otherwise, the Pokémon breaks free. In practical terms, lowering the target's HP to 1/3 of its maximum will guarantee capture with a Poké Ball, while lowering it to 1/2 will guarantee capture with a Great Ball.

If the Pokémon broke free, the steps below are performed to determine how many times the ball will shake.

1. Calculate δ:

$\delta = \frac {\Omega \times 100}{B}$

Where the value of Ball is 255 for the Poké Ball, 200 for the Great Ball, or 150 for other balls.

2. If δ is greater than or equal to 256, the ball shakes three times before the Pokémon breaks free.
3. If not, use this formula.

$x = \frac{\delta \times \Lambda}{255 + \rho}$

Where ρ is the status modifier. It is 10 if the Pokémon is asleep or frozen or 5 if it is paralyzed, poisoned, or burned.

4. If...

• x < 10: the Ball misses the Pokémon completely.
• x < 30: the Ball shakes once before the Pokémon breaks free.
• x < 70: the Ball shakes twice before the Pokémon breaks free.
• Otherwise, the Ball shakes three times before the Pokémon breaks free.

### Approximated Probability

Using a simulation of the capture algorithm, a more straightforward and simple formula for the probability of catching a Pokémon was found.

$Probability = \mu_{0} + \mu_{1}$

μ_0:

$\mu_0 = \frac{\rho_{ailment}}{B_{mod} + 1}$

Where:

rho_ailment = 12 if poisoned, burned, or paralyzed, 25 if frozen or asleep, 0 otherwise.
B_mod = 255 if using a Poké Ball, 200 if using a Great Ball, and 150 otherwise.

μ_1:

$\mu_1 = \left ( \frac {(\Omega + 1)}{(B_{mod} + 1)} \right ) \times \left ( \frac {(\Lambda + 1)}{256} \right )$

• Lambda is defined in the above section for the capture method.
• Omega (catch rate) (given as an integer value) is stated on each individual Pokémon's article.

# General Capture Method (Generation II onwards)

The capture algorithms in Generation II and onwards have three crucial parameters: the modified catch rate, the "shake probability", and the "shake checks". Generation II's shake checks handle differently than further generations.

### Modified Catch Rate

The modified catch rate, Ω (Omega), is the catch rate after various factors such as weakening the Pokémon and using stronger Poké Balls are taken into consideration. A modified catch rate may never fall to 0 (that is, render a Pokémon impossible to capture), but it may cause the modified rate to fall below its original unmodified catch rate (such as from high health, Heavy Balls, Baiting in the Safari Zone, or the dark grass penalty in Generation V). In Generation III and Generation IV, the modified catch rate may never fall below 1.[1]

### Shake probability

The shake probability, φ, is a value that determines the probability that a single shake check passes..

### Shake checks

Shake checks are performed to determine whether the Pokémon will be caught or, if the Pokémon breaks free, the number of shakes that will occur before it does so. In Generation II, whether a Pokémon will be caught is determined before any shake checks are performed, and shake checks are only performed if the Pokémon is not caught.

## Capture Method (Generation II)

### Formula

The modified catch rate Ω (Omega) is calculated in Generation II as follows:

$\Omega = \frac {(3 \times \beta_{max} - 2 \times \beta_{current}) \times R}{\left ((3 \times \beta_{max}), 1 \right ) + \rho_{status}}$

with the final value rounded down to the nearest integer, where

beta_max - The maximum HP of the targeted Pokémon
beta_current - The current HP of the targeted Pokémon
rho_status - The modifier for any status condition the Pokémon has (10 for sleep or freeze, 0 otherwise). It was intended to equal 5 for paralyze, poison, or burn, but due to a glitch, the game skips this check.
R - The catch rate of a Pokémon.
• If 3 × Beta_max > 255, then both 3 × Beta_max and 2 × HPcurrent are halved twice (and rounded down after each division) for use in the formula, as the values used are unsigned 8-bit integers. If the latter product is 0, it is set to 1 instead.
• If the Pokémon's HP is 342 or greater, the 3 × Beta_max value will be truncated and the subtraction may underflow, giving bizarre results and even making it possible for the game to freeze; however, no such Pokémon can be legitimately encountered in-game.
• Ω is capped at 255.

### Shake probability

The shake probability φ is determined from the table below, depending on the value of Ω.

 Ω φ 0-1 2 3 4 5 6-7 8-10 11-15 16-20 21-30 31-40 41-50 51-60 61-80 81-100 101-120 121-140 141-160 161-180 181-200 201-220 221-240 241-254 255 63 75 84 90 95 103 113 126 134 149 160 169 177 191 201 211 220 227 234 240 246 251 253 255
[2]

### Shake checks

First, a check is performed to determine whether the Pokémon is caught at all. A random number between 0 and 255 is generated, and if this number is less than or equal to a, the Pokémon is caught.

Shake checks are only performed if the Pokémon is not caught. A single shake check consists of generating a random number between 0 and 255 and comparing it to b. This is done at most three times, but if the number generated in a given shake check is greater than or equal to b, no further shake checks will be performed. The number of times the ball shakes is the same as the number of shake checks that were performed.

# Capture Method (Generation III-IV)

## Modified Catch Rate

The modified catch rate, Ω, is calculated in Generation III and Generation IV as follows:

$\Omega = \frac {(3\times \beta_{max}-2\times \beta_{current}) \times R \times \rho_{ball}}{3 \times \beta_{max}}\times \rho_{status}$

Where:

• beta_max is the number of hit points the Pokémon has at full health,
• beta_current is the number of hit points the Pokémon has at the moment,
• R is the catch rate of the Pokémon (which may be modified due to use of apricorn balls or actions in the Safari Zone),
• rho_ball is the multiplier for the Poké Ball used, and
• rho_status is the multiplier for any status condition the Pokémon has (2 for sleep and freeze, 1.5 for paralyze, poison, or burn, and 1 otherwise).

The formula for R (the catch rates are denoted in η):

$R = (\eta_1 \times n_1)+(\eta_2 \times n_2)...$

equal to the summation of:

$R_{total} = \sum (\eta_{catch} \times n_{pokeballs})$

If a Pokémon could have 0 HP, the maximum value for a would be R × rho_ball × rho_status. For a Pokémon with full health and no status condition, and with a neutral ball used, the minimum value for Omega would be R/3.

The formula is slightly different when applied to the Apricorn balls in HeartGold and SoulSilver. The modifiers for these balls are applied directly to the Pokémon's catch rate, rather than in the formula. In this case, bonusball is always 1, and the catch rate cannot go higher than 255. This means that for Pokémon whose catch rate is already 255, such as Rattata, the Apricorn balls do not make the capture any more likely than a regular Poké Ball.

### Shake Probability

The shake probability φ is calculated as follows:

$\varphi = \frac{1045860}{\sqrt{\sqrt{\frac{16711680}{\Omega}}}}$

The divisions and square roots all round down to the nearest integer.

## Shake checks

To perform a shake check, a random number between 0 and 65535 (inclusive) is generated and compared to b. If the number is greater than or equal to b, the check "fails".

Four shake checks are performed. The Pokémon is caught if all four shake checks succeed. Otherwise, the Poké Ball will shake as many times as there were successful shake checks before the Pokémon breaks free.

If Omega is 255 or greater, the capture will always succeed and no shake checks will be performed.

## Probability of capture

The probability "mu" of catching a Pokémon, given the values Omega and Phi calculated above, approximates φ/255. However, due to rounding errors produced when calculating φ, this approximation can be significantly inaccurate: all a values greater than 200, for instance, yield the same φ value, 65535 (which results in a 99.994% chance of a successful capture).

For a constant probability mu, the probability P that a player can capture the Pokémon with no more than tau tries is:

$P = 1 - e^{\tau \ln(-\mu+1)}$

Note that this is the cumulative probability function for a geometric distribution. The expected value of tau is 1/mu, that is to say, on average, a Pokémon that can be caught with probability mu will be caught with 1/mu tries.

The inverse problem, the number of tries, tau, needed to have a probability P of capturing a Pokémon is:

$\tau=\frac{\ln(1-P)}{\ln(1-\mu)}$

# Capture Method (Generation V)

## Modified Catch Rate

Generation V follows the formula in Generation III-IV, with all divisions above rounded down to the nearest multiple of 1/4096. However, there are three alterations:

• rhostatus is now 2.5 for sleep and freeze (instead of 2).
• Capture Power factor: if a Capture Power is active, the catch rate is multiplied by a factor and rounded down to the nearest multiple of 1/4096.
• Capture Power ↑: ×1.1
• Capture Power ↑↑: ×1.2
• Capture Power ↑↑↑, S, or MAX: ×1.3
• Dark grass factor: This factor is multiplied into the HP factor:

$\Omega_{DarkGrass} = 3\times \beta_{max} - 2 \times \beta_{current}$

above and then rounded to the nearest multiple of 1/4096. This factor is applied if the battle occurs in dark grass, and depends on the number of Pokémon that have been caught in the Pokédex, as shown in the table below, usually making it harder to catch a Pokémon.

Number Caught Multiplier
> 600 1
451-600 3686/4096 (90%)
301-450 3277/4096 (80%)
151-300 2867/4096 (70%)
30-150 2048/4096 (50%)
< 30 1229/4096 (30%)

If the modified catch rate is greater than 255, the Pokémon is guaranteed to be caught (but a critical capture check will be performed anyway).

## Shake Probability

The formula for phi is

$\varphi = \frac {65536}{\sqrt(\sqrt(\frac {255}{\Omega}))}$

where the divisions and square roots are rounded to the nearest 1/4096, and the final value is rounded down to the nearest whole number.

## Shake Checks

Generation V's shake checks work essentially the same as those of Generations III and IV. Before any shake checks are performed, the game checks to see if a critical capture may happen. If not, three shake checks will be performed, otherwise, only one shake check will be performed. In a normal capture, the Pokémon breaks free without shaking if the first check fails or after one shake if the second check fails. If the third check fails, the Pokémon breaks free after three shakes (not two). In a critical capture, the ball will always shake once, and the Pokémon will break free or be caught depending on the result of the shake check.

# Capture Method (Generation VI)

In Generation VI games, on the first route where a player can catch wild Pokémon (Kalos Route 2 in Pokémon X and Y or Hoenn Route 101 in Pokémon Omega Ruby and Alpha Sapphire), wild Pokémon will never break out of a Poké Ball thrown at them.

## Modified Catch Rate

The modified catch rate is essentially identical in all respects to Generation V, except that Capture O-Powers give a 1.5×, 2× or 2.5× boost rather than 1.1×, 1.2× and 1.3×.

## Shake Probability

Unlike Generation V, the shake probability in Generation VI is calculated as follows:

$\varphi = \frac {65536}{(\frac {255}{\Omega})^{0.1875}}$

## Shake checks

The Generation VI games perform four shake checks again, like Generations III-IV. Critical captures function like in Generation V, with only one shake check being performed for a critical capture.

The change to the formula for the shake probability is designed to counteract the change in the number of shake checks: the odds of success for a normal capture are the same in Generation VI as in Generation V, barring small rounding errors. Since each shake check is more likely to succeed, however, critical captures are more effective than before.

## Critical capture

Since Generation V, when a Poké Ball is thrown, a special type of capture that is much more likely to catch a Pokémon, called a critical capture (Japanese: 捕獲クリティカル critical capture), may occur. When a critical capture occurs, the Poké Ball will make a high-pitched whistling sound as it is thrown, then pause in mid-air, and shudder before it drops down to the ground, shaking only once on the ground before the Pokémon escapes or is caught.

The probability of a critical capture occurring depends of the number of Pokémon that have been caught in the Pokédex. First, compute the value of a (the modified catch rate) as shown in the appropriate section above. Then, multiply a by the appropriate multiplier from the table below.

Species Caught Alpha
>600 2.5
451-600 2
301-450 1.5
151-300 1
31-150 0.5
<=30 0

Finally, divide the result by 6 and round down to the nearest integer. Call this final result Gamma. A random number between 0 and 255 is generated; if it is less than Gamma, a critical capture will occur. Overall, this means the critical capture chance ranges from 0 (if 30 or fewer species have been caught) to about 41.67% of the modified capture rate Omega (if over 600 species have been caught).

Critical captures only make one shake check instead of three or four, which means they are considerably more likely to successfully capture the Pokémon. In Generation V, the success chance for a critical capture is the cube root of the success chance for a regular capture; thus, for example, if the regular success chance is 5%, the chance of success with a critical capture will be about 37%, while if the regular success chance is 50%, the chance of success with a critical capture will be just over 79%. In Generation VI, meanwhile, the success chance for a critical capture is instead the fourth root of the success chance for a regular capture: if the regular success chance is 5%, the critical success chance will be around 47%, while if the regular success chance is 50%, the critical success chance will be around 84%. Given the rarity of critical captures, however, the Pokémon is always more likely overall to be caught with a regular capture than a critical one.

Formula:

$\Gamma = \Omega_{CR} \times \alpha$

Community content is available under CC-BY-SA unless otherwise noted.


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# ufo design

This topic is 2249 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Let me know if this is good design for UFO game.
Superclass is UFO.
Subclass is UFO_Up_and_Down. (moves up and down)
Subclass is UFO_Side_to_Side. (moves side to side)
Subclass is Kamikaze_UFO. (goes straight for the player)
Subclass is Slider_UFO. (moves across screen from right to left or vice versa)
Subclass is Sine_Slider_UFO. (moves across screen from right to left but with sine bobbing motion)
Subclass is Hammer_UFO. (slides in from right or left to player's x coordinate… then drops!)
So, there will be a virtual method called move:
virtual void move(void)

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You haven't given me any information to determine if this is "good" or not, remember that something being good is relative.

If I'm correct in assuming this is like Space Invaders, I'd say yes it is good.

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I'm not to sure here if you are looking for programming assistance, or design assistance, but either way you haven't really given enough information to know what your question is really asking.

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This is a programming related question.
Bad design. You can do everything with a single class and a supporter class controlling it. This class could be something like
 class UFOTargetProvider { public: virtual single3 GetTarget() = 0; }; class UFOPatrollerTarget : public UFOTargetProvider { double time; const single3 from, to; public: UFOPatrollerTarget(single3 p0, single3 p1) : from(p0), to(p1) { } void Tick(double inc) { time += inc; } single3 GetTarget() { if(int(time / period) % 1) return to; return from; } }; UFO(UFOPatrollerTarget(SCREEN_LEFT, SCREEN_RIGHT); // equivalent to UFO_Side_to_Side (terrible naming by the way) UFO(UFOPatrollerTarget(SCREEN_UP, SCREEN_DOWN); // equivalent to UFO_Up_and_Down

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You haven't given me any information to determine if this is "good" or not, remember that something being good is relative.

If I'm correct in assuming this is like Space Invaders, I'd say yes it is good.

Yes. Like space invaders.
Thanks.

1. 1
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16
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• Total Topics
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text:
Mathematics Department Colloquium: Quantum minimal surfaces

Colloquium | October 24 | 4:10-5 p.m. | 60 Evans Hall

Maxim Kontsevich, Institut des Hautes Études Scientifiques

Department of Mathematics

Quantum analog of a minimal surface in the Euclidean space $\mathbb R^n$ is a collection of almost-commuting self-adjoint operators $X_1,\dots ,X_n$ satisfying equations $$\forall i\qquad \sum _{j=1}^n [X_j,[X_j,X_i]]=0.\qquad \qquad \qquad$$

Remarkably, the same equations appear in Yang-Mills theory for translationally-invariant connections. This story is well-known in physics, but is barely touched by mathematicians.

I will show some concrete examples, and propose many more or less precise conjectures (based on a joint work with J.Arnlind and J.Hoppe).

holtz@math.berkeley.edu


text:
# 5.2 Recording impulse response

Problem

To measure impulse responses for sound source localization and sound source separation.

Solution

Ingredients

A loudspeaker, a microphone array, an audio device that can simultaneously play and record a sound, and a signal including multiple TSPs are needed in the impulse response measurement. A wave file of a single TSP signal is installed in /usr/bin/harktool3_utils upon installation of harktool with a filename 16384.little_endian.wav. Since this file includes only a single TSP signal, it is necessary to cut and paste to make multiple TSPs (e.g., 8 or 16 times) using waveform editing softwares such as matlab or python. Note that spaces should not be inserted between TSPs.

The number of TSP depends on the environment of the measurement. For example, 8-times TSP is enough in an environment with only a little noise. While 16-times TSP is recommended in an environment with a lot of noise, e.g a running air conditioner.

Similarly, a raw file 16384.little_endian.tsp, which is a 32bit float wave data in a little-endian format is also installed in the same path. The wavesurfer software can be used to read the file. Below is the instruction on the installation and reading files using wavesurfer.

Installation of wavesurfer

 	  \verb+sudo apt-get install wavesurfer+

Opening file using wavesurfer

 	  wavesurfer /usr/bin/harktool3_utils/16384.little_endian.tsp

Then, set the configuration as shown in (Fig. 5.1)

Impulse Response Measurement Procedure

Impulse responses are measured in two steps: (1) playing and recording the TSP signal, and (2) calculating impulse responses from TSP signals.

1. Playing and recording the TSP signal
Playing multiple TSP signals and adding each signal reduces the background noise and vibrations. Files that includes 8-times TSP is named tsp8.wav.

Next is to use the HARK support tool wios  for measurement. For example, to perform a synchronous recording-playback in an ALSA supported 8-channel multi-input device, execute the following command: (see Recording multichannel sound how to use wios  )

	  wios -s -x 0 -i tsp.wav -c 8 -o r100_d000.wav

It is necessary to measure the impulse response of sounds coming from each direction. In about 1-2 meters of concentric circle from the center of the microphone array, move the speakers in order and with an equal angle while recording. For example, when the impulse response is measured in a 1 meter distance from the microphone array with 10 degrees interval, the following files will be generated:

    r100_d000.wav
r100_d010.wav
...
r100_d350.wav

2. Calculating impulse responses from TSP signals.
The impulse response is calculated by convolving the reverse TSP signal with the signal recorded in a way described above, and synchronously adding multiple playbacks to find the average value. harktool can be used in calculating impulse response. See the chapter on harktool in the HARK document for more details.

Discussion

Impulse response is an output of the system following the application of an impulse function to the system. Intuitively, the reverberation you hear upon clapping your hands in a room is a type of impulse response. The problem in actually measuring the “impulse response” is the need for a large amount of energy with a sufficient signal-to-noise ratio. To solve this problem, the impulse response is measured using a Time Stretched Pulse (TSP) signal, whose energy is stretched over time. Impulse response can be determined by recording the TSP and calculating the converse and inverse of the TSP [1,2]. The best interval between measurements is usually around 5 or 10 degrees, although it depends on the microphone array and on the shape and configuration of the room.


text:
# 4.E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Exercise $$\PageIndex{1}$$:

Find the GCD of:

1. 10 and 75
2. 48 and 360

Find the LCM of:

1. 24 and 35
2. 56 and 72

Exercise $$\PageIndex{2}$$:

Using GCD determine how many packages of hotdogs and hotdog buns are needed to have an equal amount if the hotdog package contains 10 hotdogs, and the hotdog bun package contains 12 buns.

Exercise $$\PageIndex{3}$$:

Using Euclidean Algorithm, find the gcd of 1716 and 1260, and the LCM of 1716 and 1260.

Exercise $$\PageIndex{4}$$:

1. Use the Euclidean algorithm to find $$\gcd (270, 504).$$. Find integers $$x$$ and $$y$$ such that $$\gcd (270, 504) =270 x+504 y.$$
2. Use the Euclidean algorithm to find $$\gcd (-270, 504).$$, Find integers $$x$$ and $$y$$ such that $$\gcd (-270, 504) =-270 x+504 y.$$

Exercise $$\PageIndex{5}$$:

Suppose $$a$$ and $$b$$ are relatively prime integers and $$c$$ is an integer such that $$a|c$$ and $$b|c.$$ Prove that $$ab|c.$$

Exercise $$\PageIndex{6}$$:

Suppose $$a$$ and $$b$$ are relatively prime integers and $$c$$ is an integer such that $$a|bc$$. Prove that $$a|c.$$

Exercise $$\PageIndex{7}$$:

1. Let $$a$$ and $$b$$ be non-zero integers with least common multiple $$l$$. Let $$m$$ be any common multiple of $$a$$ and $$b$$. Prove that $$l|m.$$
2. Prove that if $$a, b$$ and $$c$$ are natural numbers, $$\gcd(a, c)=1$$and $$b \mid c$$, then $$\gcd(a, b)=1$$.
3. Suppose $$a$$ and $$c$$ are relatively prime integers and $$b$$ is an integer such that $$b|c.$$ Prove that $$gcd(a, b)=1.$$

Exercise $$\PageIndex{8}$$:

Prove that for any positive integer $$k$$, $$7k+5$$ and $$4k+3$$ are relatively prime.


text:
# There are 6 blocks of jaggery, each of one kilogram. If one family requires one and a half kg jaggery every month, for how many families will these blocks suffice? - Mathematics

Sum

There are 6 blocks of jaggery, each of one kilogram. If one family requires one and a half kg jaggery every month, for how many families will these blocks suffice?

#### Solution

One and a half is 1 + 1/2 = 3/2.

Let us divide to see how many families can share the jaggery.

6 ÷ 3/2 = 6/1 ÷ 3/2 = 6/1 xx 2/3 = 4.

Therefore, 6 blocks will suffice for 4 families.

Is there an error in this question or solution?


text:
# How do you solve the equation 2x^2+10x=-17 by completing the square?

Mar 3, 2018

color(red)(x =  color(green)(-(1/2)(5-3i), (-1/2)(5+3i)

#### Explanation:

$2 {x}^{2} + 10 x = - 17$

Divide by 2 on both sides. ${x}^{2} + 5 x = - \frac{17}{2}$

Rewrite the (xy term as multiple of 2

${x}^{2} + \left(2 \cdot x \cdot \left(\frac{5}{2}\right)\right) = - \frac{17}{2}$

Complete the square on L H S by adding ${\left(\frac{5}{2}\right)}^{2}$ to both sides.

x^2 + (2*(5/2) x+ (5/2)^2 = -17 / 2 + (5/2)^2 = -9/4

(x+5/2)^2 = (sqrt(-9/4)^2 = (i * (3/2)^2

$x + \frac{5}{2} = \pm \left(\frac{3}{2}\right) i$

color(red)(x ) = color(blue)(-5/2 + 3/2i, -5/2 -3/2i or color(green)(-(1/2)(5-3i), (-1/2)(5+3i)#


text:
# [2015] Wondering what is happening at the Adventures of the Mind?

This time it's me and Kishlaya and Trevor nominated to attend Adventures of the Mind from Brilliant.

We'll keep posting our thoughts here.

4 years, 11 months ago

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• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
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Excuse my typing, please. I am a bit tired now and I am just copying this down from my IMs.

We had a burger like thing for lunch plus a slice of pizza plus cookies, and there were a whole lot of stuff whose names I do not really know. I really like the ice cream and the grape juice here.

Our dinner was called the 'ice cream social'. We had some cakes and ice cream and stuff. Then, we had an option between dancing or playing poker or chess. I tried my hand at poker. Trevor was winning the table except that he tried bluffing and the other girlcaught him.

We also had a breakout session on computers, poker or chess. You could choose any. I chose computers [deleted text] But the mentors are great, she was a prof at MIT working with cyber security.

We also had a welcome session and then a talk by Annie and his dad (deleted text) . They are great poker players.

And then here is some important thing she said,

"when you loose a hand, you say bad luck. And when you win one, you say wow I am skilled. But when you see your opponent winning, you say just got lucky and then when he looses you say, he played badly.

Don't do this. That's pretty bad for learning.

Also, this applies to the exam scenario. Everytime, you do bad, you keep saying 'oh that was a tough question' and when you win, you think "you're clever."

- 4 years, 11 months ago

@Agnishom Chattopadhyay , That brought some fond memories, do u guys have a hackaton this time?

- 4 years, 11 months ago

Yep, Tuesday

- 4 years, 11 months ago

- 4 years, 11 months ago

Woke up at 7:00. Had a shower and then the breakfast.

The rosemont campus is beautiful

- 4 years, 11 months ago

@Agnishom Chattopadhyay Would you please tell us about "what is Adventures of the Mind from Brilliant"? And congrats to all of you three(Agnishom, Kishlaya and Trevor) for being nominated.

- 4 years, 11 months ago

Adventures of the Mind is a summer camp where many people across the United States meet some really interesting people. You could google that out.

Brilliant nominates a few persons each year and they sponsor their expenses.

- 4 years, 11 months ago

Congrats to you all!.......

- 4 years, 11 months ago

we had two breakout sessions yesterday. i attended poker and cyber security. plus, there were lectures in the rotwitt theatre. there was the orientation for hackathon by Mayank. I played chess but that didn't turn out to be too good for me. i managed to win three hands at Limit hold'em though

- 4 years, 11 months ago

Hopefully, you will remember all three exactly so we can discuss them.

- 4 years, 11 months ago

I have zero clue of what the hands were but we can play some more.

- 4 years, 11 months ago

Wednesday: After breakfast, we left the Rosemont campus at 9:30 and it was like an hour ride to the University of Pennsylvania college fair. We had lunch in a canteen in their basement. Then, a Penn student called Britney gave us a tour of the Penn campus. She was a really fast talker. Next, we went to the bookstore. Kishlaya bought a number theory book for around \$12. I didn't have any money with me, though. The scary part was that the rest of the adventurers forgot about us while we were still in the bookstore and left us there. But, we were safe, anyway; they came back and got us. Then, we had dinner at Temple at around 5:30. We drove back to Rosemont again, it was a long drive. Then, it was a movie night and they presented a movie called The Joy Luck Club. It is about how chinese-american women are exploited in their relationships. I have no idea on why everyone was laughing so hard.

- 4 years, 11 months ago

@Kishlaya Jaiswal , what's the name of the book you bought?

@Agnishom Chattopadhyay , didn't you have breakout sessions today?

- 4 years, 11 months ago

First to clarify, it isn't any number theory book. It's all about numbers and hidden treasures of maths. The name of the book is "Numbers : A Very Short Introduction" by Peter M. Higgins

Btw that book was more for my lil sister. Actually, her birthday was on 5th so I wanted to present her a gift from Philadelphia when I return back. So, I got her a book and a pen because we never got time (and weren't allowed also) to visit some other places or store. That's more of a reason why I'd kept some extra backup money with me for buying a gift.

- 4 years, 10 months ago

Thursday: This was a day packed with speeches. We began with a speech by Andrew M,, the magician, which was pretty cool. Then, we had also had talks from a Pulitzer winning poet and a Nobel laureate economist who talked a bit about mechanism design and fair division. We had dinner out in the lawn but then it started raining, so we went back to the cardinal hall. There were talks from explorers and neuroscientists too. I attended the breakout session called 'Life in the Lab'. At night, we met a chess grandmaster who calls herself 'The Chess Bitch'. She played in a Chess Simul with around 20 of us: only Anita could win though.

Something sad just happened, I lost my 206 day streak. I managed it to keep it until yesterday, despite how tired I was during examinations, travelling or dates but the packed schedule last day really made me tired.

- 4 years, 11 months ago

Andrew Mayne is there?

Can you ask him if he remembers the boy from Bangladesh from last year who kept following him around and asking him questions?

Losing your streak is sad, specially when you have a long one. You get somewhat attached to it.

- 4 years, 11 months ago

Oh, yeah. I will. What questions did you ask him? Did you know him before you came to AoM?

- 4 years, 11 months ago

He's a really cool and down-to-earth guy. You'll see him around a lot.

- 4 years, 11 months ago

Have you heard of him before?

I walked up to him and asked if he remembered the guy from Bangladesh. He first asked if that was me. I said no and that I am his friend. He said, "Tell him I said hello"

- 4 years, 11 months ago

Not really, no.

Tell him I said hello as well.

- 4 years, 11 months ago

Yeah, I agree that he is really a nice personality

- 4 years, 11 months ago

You will start another streak.

- 4 years, 11 months ago

She didn't call herself The Chess Bitch. It's one of her authored books.

- 4 years, 10 months ago

Friday: The presentations today were excellent. First, we had a panel of artists talking about their career in the arts. Then, a particle physicist with phenomenal work with the neutrino came to give a little talk on his life and research. We all wanted to talk to hm during lunch but he was found nowhere. Then, there was a speech from the chief officer of explorations and science from National Geographic and an egyptologist who was the mastermind behind various famous excavation projects.

Next, we went to see the liberty bell and the independence hall, where the Declaration of Independence was signed. We went to a constitution center where there was a little show called 'We the People' educating us about the American constitution and then there was a large exhibit on American history. We had dinner in the Constitution Center. Personally, I enjoyed playing with the candles there but did not like it so much because the dinner was lacking ice cream.

- 4 years, 11 months ago

LoL. Lacking ice cream...!!!

- 4 years, 11 months ago

Saturday: Today was the last day for all of us. We got a talk from a data scientist, a photographer, and an artist. Then, there were two primatologists talking about Bonobo's and Baboons. I went to the breakout session called the future of Technology where a scientist from the bell labs talked about inventions. Then, we all dressed up and for the dinner but it was just as good as the rest of the dinners for me. I really liked the Talent show by the adventurers where there was awesome musical performances, poetry, and dances. By the way, I messed up one of the poetries thinking that he was telling jokes.

- 4 years, 11 months ago


text:
# mstransform¶

mstransform(vis, outputvis='', createmms=False, separationaxis='auto', numsubms='auto', tileshape=[0], field='', spw='', scan='', antenna='', correlation='', timerange='', intent='', array='', uvrange='', observation='', feed='', datacolumn='corrected', realmodelcol=False, keepflags=True, usewtspectrum=False, combinespws=False, chanaverage=False, chanbin=1, hanning=False, regridms=False, mode='channel', nchan=- 1, start='0', width='1', nspw=1, interpolation='linear', phasecenter='', restfreq='', outframe='', veltype='radio', preaverage=False, timeaverage=False, timebin='0s', timespan='', maxuvwdistance=0.0, docallib=False, callib='', douvcontsub=False, fitspw='', fitorder=0, want_cont=False, denoising_lib=True, nthreads=1, niter=1, disableparallel=False, ddistart=- 1, taql='', monolithic_processing=False, reindex=True)[source]

Split the MS, combine/separate/regrid spws and do channel and time averaging

[Description] [Examples] [Development] [Details]

Parameters
• vis (path) - Name of input Measurement set or Multi-MS.

• outputvis (string=’’) - Name of output Measurement Set or Multi-MS.

• createmms (bool=False) - Create a multi-MS output from an input MS.

createmms = False
• separationaxis (string=’auto’) - Axis to do parallelization across(scan,spw,auto,baseline).

• numsubms ({string, int}=’auto’) - The number of Sub-MSs to create (auto or any number)

• disableparallel (bool=False) - Hidden parameter for internal use only. Do not change it!

• ddistart (int=-1) - Hidden parameter for internal use only. Do not change it!

• taql (string=’’) - Table query for nested selections

• reindex (bool=True) - Hidden parameter for use in the pipeline context only

createmms = True
• separationaxis (string=’auto’) - Axis to do parallelization across(scan,spw,auto,baseline).

• numsubms ({string, int}=’auto’) - The number of Sub-MSs to create (auto or any number)

• disableparallel (bool=False) - Hidden parameter for internal use only. Do not change it!

• ddistart (int=-1) - Hidden parameter for internal use only. Do not change it!

• taql (string=’’) - Table query for nested selections

• reindex (bool=True) - Hidden parameter for use in the pipeline context only

• tileshape (intVec=[0]) - List with 1 or 3 elements giving the tile shape of the disk data columns.

• field ({string, stringVec, int, intVec}=’’) - Select field using ID(s) or name(s).

• spw ({string, stringVec, int, intVec}=’’) - Select spectral window/channels.

• scan ({string, stringVec, int, intVec}=’’) - Select data by scan numbers.

• antenna ({string, stringVec, int, intVec}=’’) - Select data based on antenna/baseline.

• correlation ({string, stringVec}=’’) - Correlation: ‘’ ==> all, correlation=”XX,YY”.

• timerange ({string, stringVec, int, intVec}=’’) - Select data by time range.

• intent ({string, stringVec, int, intVec}=’’) - Select data by scan intent.

• array ({string, stringVec, int, intVec}=’’) - Select (sub)array(s) by array ID number.

• uvrange ({string, stringVec, int, intVec}=’’) - Select data by baseline length.

• observation ({string, stringVec, int, intVec}=’’) - Select by observation ID(s).

• feed ({string, stringVec, int, intVec}=’’) - Multi-feed numbers: Not yet implemented.

• datacolumn (string=’corrected’) - Which data column(s) to process.

datacolumn = model
• realmodelcol (bool=False) - Make real a virtual MODEL column.

datacolumn = all
• realmodelcol (bool=False) - Make real a virtual MODEL column.

datacolumn = data,model,corrected
• realmodelcol (bool=False) - Make real a virtual MODEL column.

• keepflags (bool=True) - Keep completely flagged rows or drop them from the output.

• usewtspectrum (bool=False) - Force creation of the columns WEIGHT_SPECTRUM and SIGMA_SPECTRUM in the output MS, even if not present in the input MS.

• combinespws (bool=False) - Combine the input spws into a new output spw. Only supported when the number of channels is the same for all the spws.

• chanaverage (bool=False) - Average data in channels.

chanaverage = True
• chanbin ({int, intVec}=1) - Width (bin) of input channels to average to form an output channel.

• hanning (bool=False) - Hanning smooth data to remove Gibbs ringing.

• regridms (bool=False) - Transform channel labels and visibilities to a different spectral reference frame. Notice that u,v,w data is not transformed.

regridms = True
• mode (string=’channel’) - Regridding mode (channel/velocity/frequency/channel_b).

• nchan (int=-1) - Number of channels in the output spw (-1=all). Used for regridding, together with 'start' and 'width'.

• start (variant=’0’) - Start of the output visibilities. Used for regridding, together with 'width' and 'nchan'. It can be in different units, depending on the regridding mode: first input channel (mode='channel'), first velocity (mode='velocity'), or first frequency (mode='frequency'). Example values: '5', '0.0km/s', '1.4GHz', for channel, velocity, and frequency modes, respectively.

• width (variant=’1’) - Channel width of the output visibilities. Used for regridding, together with 'start', and 'nchan'. It can be in different units, depending on the regridding mode: number of input channels (mode='channel'), velocity (mode='velocity'), or frequency (mode='frequency'. Example values: '2', '1.0km/s', '1.0kHz', for channel, velocity, and frequency modes, respectively.

• nspw (int=1) - Number of output spws to create in output MS.

• interpolation (string=’linear’) - Spectral interpolation method.

• phasecenter (variant=’’) - Phase center direction to be used for the spectral coordinate transformation: direction measure or field index

• restfreq (string=’’) - Rest frequency to use for output.

• outframe (string=’’) - Output reference frame (‘’=keep input frame).

• veltype (string=’radio’) - Velocity definition.

• preaverage (bool=False) - Pre-average channels before regridding, when the ratio between the output and and input widths is greater than 2.

• timeaverage (bool=False) - Average data in time.

timeaverage = True
• timebin (string=’0s’) - Bin width for time averaging.

• timespan ({string, stringVec}=’’) - Span the timebin across scan, state or both.

• maxuvwdistance (double=0.0) - Maximum separation of start-to-end baselines that can be included in an average. (meters)

• docallib (bool=False) - Enable on-the-fly (OTF) calibration as in task applycal

docallib = True
• callib (string=’’) - Path to calibration library file

• douvcontsub (bool=False) - (Deprecated) Enable continuum subtraction as in task uvcontsub

douvcontsub = True
• fitspw (string=’’) - Spectral window:channel selection for fitting the continuum

• fitorder (int=0) - Polynomial order for the fits

• want_cont (bool=False) - Produce continuum estimate instead of continuum subtracted data

• denoising_lib (bool=True) - Use new denoising library (based on GSL) instead of casacore fitting routines

• nthreads (int=1) - Number of OMP threads to use (currently maximum limited by number of polarizations)

• niter (int=1) - Number of iterations for re-weighted linear fit

Description

Warning

There are Known Issues for mstransform.

mstransform is a multipurpose task that provides all the functionality of split, partition, cvel, hanningsmooth, uvcontsub and applycal with the possibility of applying each of these transformations separately or together in an in-memory pipeline, thus avoiding unnecessary I/O steps. Each transformation can be activated through a boolean parameter in the task interface. Below are the main parameter descriptions. See also how to run mstransform for some use-cases in the Examples section below.

Parameter Descriptions

createmms

When set to True, this parameter will create an output Multi-MS, which is the basic step for running CASA in parallel. See more about this in the Parallelization chapter.

Combination of spws: combinespws

Combine the input spectral windows into a new output spectral window. Whenever the data to be combined has different EXPOSURE values in the spectral windows, mstransform will use the WEIGHT_SPECTRUM for the combination. If WEIGHT_SPECTRUM is not available, it will use the values from the WEIGHT column. Each output channel is calculated using the following equation:

$outputChannel_{j} = \frac{\sum (inputChannel_{i}*contributionFraction_{i}*inputWeightSpectrum_{i})}{\sum(contributionFraction_{i}*inputWeightSpectrum_{i})}$

Warning

ALERT: The combination of spws is only supported when the number of channels is the same for all the spws. Using this option with different numbers of channels for different spws will result in an error.

Channel averaging: chanaverage

Average data across channels. Partially flagged data is not included in the average unless all data contributing to a given output channel is flagged. In this case, mstransform calculates the average of all flagged data, and writes it to the output MS with the corresponding flag set to true. If present, WEIGHT_SPECTRUM/SIGMA_SPECTRUM are used together with the channelized flags (FLAG), to compute a weighted average (using WEIGHT_SPECTRUM for CORRECTED_DATA and SIGMA_SPECTRUM for DATA). The sub-parameter chanbin takes an integer number or a list of integers. If a list is given, each bin applies to one of the selected spws.

Note

NOTE: WEIGHT_SPECTRUM/SIGMA_SPECTRUM will be used (if present) in addition to the flags to compute a weighted average. The calculations is done as follows:

1. When WEIGHT_SPECTRUM/SIGMA_SPECTRUM are not present:

$Average = \frac{\sum(Chan_{i}*Flag_{i})}{\sum(Flag_{i})}$
1. When WEIGHT_SPECTRUM/SIGMA_SPECTRUM are present:

$Average = \frac{\sum(Chan_i*Flag_i*WeightSpectrum_i)}{\sum(Flag_i*WeightSpectrum_i)}$

Hanning smoothing: hanning

This function Hanning smooths the frequency channels with a weighted running average to remove Gibbs ringing. The weights are 0.5 for the central channel and 0.25 for each of the two adjacent channels. The first and last channels are flagged. Inclusion of a flagged value in an average causes that data value to be flagged. By default, all visibility data columns available in the MS will be smoothed, unless a specific column is given in the datacolumn parameter.

Reference frame transformation: regridms

Transform channel labels and visibilities to a different spectral reference frame, which is appropriate for science analysis. For example, transform from TOPO to LSRK to correct for Doppler shifts throughout the time of the observation.

Note

NOTE: U,V,W data are not transformed with this function

The regridding mode can be “channel”, “velocity”, “frequency” or “channel_b”, channel being the default. When set to velocity or frequency, it means that the channels must be specified in the respective units. When set to channel_b it means an alternative channel mode that does not force an equidistant grid, which is faster to process.

The start sub-parameter gives the first channel to use in the output spw (depending on the mode). When mode=’channel’, start means the first channel in the input spw to use when creating the output spw. When mode=’frequency’, start means the lowest frequency of the output spw. If this information is not available, mstransform will calculate it automatically.

The width sub-parameter gives the width of the output channels. The width is expressed in different units, depending on the mode chosen. This way the sub-parameter width can take units of either number of input channels (mode=”channel”), velocity (mode=”velocity”), or frequency (mode=”frequency”). For example, the value of the width parameter can be “4” (mode=”channel”), “-1.0km/s” (mode=”velocity”), or “1.0kHz” (mode=”frequency”).

Warning

NOTE: mstransform will only shift spws with channel widths of the same sign in a single operation. If you are regridding spws with mixed positive and negative channel widths, you should run this task separated for each group of spws. You can verify the channel widths for your MS using listobs for example, and looking at the SPW table, column ChanWidth.

It is possible to separate the spectral windows into a given number of spws. This is achieved with the sub-parameter nspw, which is activate when set to > 1. Internally, the framework will combine the selected spws before separating them so that channel gaps and overlaps are taken into account. This sub-parameter will create a regular grid of spws in the output MS. If nchan is set, it will refer to the number of output channels in each of the separated spws.

Time averaging: timeaverage

Average data across time by setting timeaverage=True and giving the bin for averaging using the sub-parameter timebin. Partially flagged data is not included in the average unless all data contributing to a given output channel is flagged. In this case, mstransform calculates the average of all flagged data, and writes it to the output MS with the corresponding flag set to True. If keepflags=False, the fully flagged data is not written to the output MS. If present, WEIGHT_SPECTRUM/SIGMA_SPECTRUM are used together with the channelized flags (FLAG), to compute a weighted average (using WEIGHT_SPECTRUM for CORRECTED_DATA and SIGMA_SPECTRUM for DATA). Otherwise WEIGHT/SIGMA are used instead to average together data from different integrations.

The timespan sub-parameter will span the timebin across scans, states or both. State is equivalent to sub-scans and one scan may have several state IDs. Another option when doing time averaging is to provide a maximum separation of start-to-end baselines that can be included in an average with the use of the maxuvwdistance sub-parameter.

On-the-fly calibration parameters: docallib

mstransform is able to apply the calibrations on the fly, similar to the applycal task. This is possible by specifying a Cal Library filename that contains the actual specification for the calibrations to be applied. See more about the Cal Library file syntax here. See also the examples section below for how to apply the Cal library in mstransform.

Multi-MS Processing using mstransform

Task mstransform will process an input Multi-MS (MMS) in parallel whenever possible. Each Sub-MS of the MMS will be processed in a separate computer core and the results will be post-processed at the end to create an output MMS. The output MMS will have the same separationaxis of the input MMS, which will be written to the table.info file inside the MMS directory.

Naturally, some transformations available in mstransform require more care when the user first partition the MS. If one wants to do a combination of spws by setting the parameter combinespws=True in mstransform, the input MMS needs to contain all the selected spws in each of the Sub-MSs or the processing will fail. For this, one may set the initial separationaxis to ‘scan’ or use the default ‘auto’ with a proper numsubms set so that each Sub-MS in the MMS is self-contained with all the necessary spws for the combination.

The task will check if the Sub-MSs contain all the selected spws when combinespws=True and if not, it will issue a warning and process the input MMS as a monolithic MS. In this case, the separation axis of the output MMS will be set to ‘scan’, regardless of what the input axis was.

A similar case happens when the separation axis of the input MMS is per ‘scan’ and the user wants to do time averaging with time spanning across scans. If the individual Sub-MSs are not self-contained of the necessary scans and the duration of the scans is shorter than the given timebin, the spanning will not be possible. In this case, the task will process the input MMS as a monolithic MS and will set the axis of the output MMS to spw.

It is important that the user sets the separation axis correctly when first partitioning the MS. See the table below for when it is possible to process the input MMS in parallel or not, using mstransform.

 input MMS axis combinespws = True nspw > 1 timeaverage = True timespan = ‘scan’ scan YES YES NO spw NO NO YES auto maybe maybe maybe

Note

NOTE: If mstransform decides it’s not possible to process the MMS in parallel, it will still create an output but the processing will run serially without any parallelization involved.

Examples

Split out a single channel:

mstransform(vis='ctb80-vsm.ms', outputvis='mychn.ms',
datacolumn='data', spw='0:25')

Combine the selected spws into a single output spw:

mstransform(vis='Four_ants.ms', outputvis='myspw.ms',
combinespws=True, spw='0~3')

Combine two spws and regrid one field, using two input channels to make one output:

mstransform(vis='jupiter6cm.demo.ms',outputvis='test1.ms',datacolumn='DATA',field='11',
spw='0,1', combinespws=True, regridms=True, nchan=1, width=2)

Combine 24 spws and regrid in frequency mode to create 21 output channels, change the phase center:

mstransform(vis='g19_d2usb_targets_line.ms',
outputvis='test2.ms', datacolumn='DATA', combinespws=True,
regridms=True, mode='frequency', nchan=21, start='229587.0MHz',
width='1600kHz', phasecenter="J2000 18h25m56.09 -12d04m28.20")

Apply Hanning smoothing to an MS:

mstransform(vis='g19_d2usb_targets_line.ms',
outputvis='test3.ms', datacolumn='DATA', hanning=True)

Change the reference frame and apply Hanning smoothing after combining all spws:

mstransform(vis='g19_d2usb_targets_line.ms',
outputvis='test4.ms', datacolumn='DATA', combinespws=True,
regridms=True, mode="channel", outframe="BARY",
phasecenter="J2000 18h25m56.09 -12d04m28.20", hanning = True)

Apply time averaging using a bin of 30 seconds on the default CORRECTED column:

mstransform(vis='g19_d2usb_targets_line.ms',
outputvis='test5.ms', timeaverage=True, timebin='30s')

Apply OTF calibration to ngc5921 using a calibration library:

mstransform(vis='ngc5921.ms',
outputvis='ngc5921_calibrated.ms',docallib=True,
callib='./ngc5921_callib.txt')

The calibration file (ngc5921_callib.txt) used in the above example contains the following information:

caltable='ngc5921_regression/ngc5921.bcal' calwt=True tinterp='nearest'
caltable='ngc5921_regression/ngc5921.fluxscale' calwt=True tinterp='nearest' fldmap='nearest'
caltable='ngc5921_regression/ngc5921.gcal' calwt=True field='0' tinterp='nearest' fldmap=[0]
caltable='ngc5921_regression/ngc5921.gcal' calwt=True field='1,2' tinterp='linear' fldmap='1'

Development

Parameter Details

Detailed descriptions of each function parameter

vis (path) - Name of input Measurement set or Multi-MS.
outputvis (string='') - Name of output Measurement Set or Multi-MS.
createmms (bool=False) - Create a multi-MS output from an input MS.
separationaxis (string='auto') - Axis to do parallelization across(scan,spw,auto,baseline).
numsubms ({string, int}='auto') - The number of Sub-MSs to create (auto or any number)
tileshape (intVec=[0]) - List with 1 or 3 elements giving the tile shape of the disk data columns.
field ({string, stringVec, int, intVec}='') - Select field using ID(s) or name(s).
spw ({string, stringVec, int, intVec}='') - Select spectral window/channels.
scan ({string, stringVec, int, intVec}='') - Select data by scan numbers.
antenna ({string, stringVec, int, intVec}='') - Select data based on antenna/baseline.
correlation ({string, stringVec}='') - Correlation: ‘’ ==> all, correlation=”XX,YY”.
timerange ({string, stringVec, int, intVec}='') - Select data by time range.
intent ({string, stringVec, int, intVec}='') - Select data by scan intent.
array ({string, stringVec, int, intVec}='') - Select (sub)array(s) by array ID number.
uvrange ({string, stringVec, int, intVec}='') - Select data by baseline length.
observation ({string, stringVec, int, intVec}='') - Select by observation ID(s).
feed ({string, stringVec, int, intVec}='') - Multi-feed numbers: Not yet implemented.
datacolumn (string='corrected') - Which data column(s) to process.
realmodelcol (bool=False) - Make real a virtual MODEL column.
keepflags (bool=True) - Keep completely flagged rows or drop them from the output.
usewtspectrum (bool=False) - Force creation of the columns WEIGHT_SPECTRUM and SIGMA_SPECTRUM in the output MS, even if not present in the input MS.
combinespws (bool=False) - Combine the input spws into a new output spw. Only supported when the number of channels is the same for all the spws.
chanaverage (bool=False) - Average data in channels.
chanbin ({int, intVec}=1) - Width (bin) of input channels to average to form an output channel.
hanning (bool=False) - Hanning smooth data to remove Gibbs ringing.
regridms (bool=False) - Transform channel labels and visibilities to a different spectral reference frame. Notice that u,v,w data is not transformed.
mode (string='channel') - Regridding mode (channel/velocity/frequency/channel_b).
nchan (int=-1) - Number of channels in the output spw (-1=all). Used for regridding, together with 'start' and 'width'.
start (variant='0') - Start of the output visibilities. Used for regridding, together with 'width' and 'nchan'. It can be in different units, depending on the regridding mode: first input channel (mode='channel'), first velocity (mode='velocity'), or first frequency (mode='frequency'). Example values: '5', '0.0km/s', '1.4GHz', for channel, velocity, and frequency modes, respectively.
width (variant='1') - Channel width of the output visibilities. Used for regridding, together with 'start', and 'nchan'. It can be in different units, depending on the regridding mode: number of input channels (mode='channel'), velocity (mode='velocity'), or frequency (mode='frequency'. Example values: '2', '1.0km/s', '1.0kHz', for channel, velocity, and frequency modes, respectively.
nspw (int=1) - Number of output spws to create in output MS.
interpolation (string='linear') - Spectral interpolation method.
phasecenter (variant='') - Phase center direction to be used for the spectral coordinate transformation: direction measure or field index
restfreq (string='') - Rest frequency to use for output.
outframe (string='') - Output reference frame (‘’=keep input frame).
veltype (string='radio') - Velocity definition.
preaverage (bool=False) - Pre-average channels before regridding, when the ratio between the output and and input widths is greater than 2.
timeaverage (bool=False) - Average data in time.
timebin (string='0s') - Bin width for time averaging.
timespan ({string, stringVec}='') - Span the timebin across scan, state or both.
maxuvwdistance (double=0.0) - Maximum separation of start-to-end baselines that can be included in an average. (meters)
docallib (bool=False) - Enable on-the-fly (OTF) calibration as in task applycal
callib (string='') - Path to calibration library file
douvcontsub (bool=False) - Enable continuum subtraction as in task uvcontsub. Note: this parameter is deprecated and will be removed in an upcoming release. The task uvcontsub can be used as an alternative.
fitspw (string='') - Spectral window:channel selection for fitting the continuum
fitorder (int=0) - Polynomial order for the fits
want_cont (bool=False) - Produce continuum estimate instead of continuum subtracted data
denoising_lib (bool=True) - Use new denoising library (based on GSL) instead of casacore fitting routines
nthreads (int=1) - Number of OMP threads to use (currently maximum limited by number of polarizations)
niter (int=1) - Number of iterations for re-weighted linear fit
disableparallel (bool=False) - Hidden parameter for internal use only. Do not change it!
ddistart (int=-1) - Hidden parameter for internal use only. Do not change it!
taql (string='') - Table query for nested selections
monolithic_processing (bool=False) - Hidden parameter for internal use only. Do not change it!
reindex (bool=True) - Hidden parameter for use in the pipeline context only


text:
# Propositional calculus and intuitionist logic

Having never had much exposure to formal mathematical logic, I have decided to embark on a quest to rectify this; unfortunately having been exposed to concepts from Intuitionistic Logic through my dabbles in functional programming, I find myself running into issues understanding "how it all fits together".

I am familiar with the notion of rejecting the law of the excluded middle (LEM); from the little I know of intuitionistic logic, I understand that this is done in the sense that LEM is not treated as an axiom within whatever deductive system is being used. However, looking at the treatments of propositional calculus within pretty much any text that I can lay my hands on, I find that after a discussion of the syntax of a zeroth-order logic, the author starts to talk about "semantics", in many cases illustrating this with truth tables.

At this point I run into an issue, because clearly the semantics presented for example in Goldrei (Propositional and Predicate Calculus: A Model of Argument) mean that LEM is a tautology.

When presenting "the semantics of propositional calculus", is the author in fact presenting a specific "logic" (i.e. classical logic), or are these semantics a fundamental part of of any propositional calculus? It seems to me that the latter cannot be true (as intuitionistic propositional logic definitely exists), but I cannot find an introductory text in which the author addresses this issue.

I am likely falling into the trap of a little knowledge being a dangerous thing and it's quite possible that everything I think I know about this is wrong, but I find myself currently unable to pass beyond this point without serious doubts as to what I'm reading actually refers to.

• "Obviously", introductory textbooks deal with classical logic, where LEM is a tautology according to the "standard" semantics for classical logic. – Mauro ALLEGRANZA Jan 14 at 20:24
• You can see van Dalen for an introductory textbook with a chapter dedicated to Intuitionsitic Logic and the appropriate semantics. – Mauro ALLEGRANZA Jan 14 at 20:25

Truth table semantics is particular to classical propositional logic. So if the author were being specific, they would have added this qualifier. (However it’s probably a good rule of thumb that if you just see “propositional calculus,” in most contexts they are referring to classical.)

While “propositional” is a descriptor that mainly refers to the language (in particular, the absence of non logical symbols other than sentence letters), as soon as we start talking about proof systems or semantics, we have a particular logic in mind: classical, intuistionistic, minimal, modal, etc.

In classical propositional logic, a semantic interpretation can be more generally provided by a Boolean algebra, which is a set $$X$$ with constants $$\top, \bot$$, unary operation $$\lnot$$, and binary operations $$\wedge, \vee$$ satisfying certain axioms. The prototypical Boolean algebra is $$\{ T, F \}$$ with the obvious interpretations.

In intuitionistic propositional logic, the corresponding construction is a Heyting algebra, which is a set $$X$$ with constants $$\top, \bot$$, and binary operations $$\wedge, \vee, \rightarrow$$ satisfying certain axioms. Given a Heyting algebra, we can then often define $$\lnot x := (x \rightarrow \bot)$$; however, in a general Heyting algebra, some of the Boolean algebra axioms or derived identities such as $$x \vee \lnot x = \top$$, $$\lnot(\lnot x) = x$$, and $$\lnot(x \wedge y) = (\lnot x) \vee (\lnot y)$$ no longer hold.

An interesting example of a Heyting algebra which is not a Boolean algebra is if $$X$$ is the set of open subsets of $$\mathbb{R}$$, with $$\top := \mathbb{R}$$, $$\bot := \emptyset$$, $$U \wedge V := U \cap V$$, $$U \vee V := U \cup V$$, and $$U \rightarrow V := \operatorname{int}((\mathbb{R} \setminus U) \cup V)$$. In this Heyting algebra, for example if we set $$U := \mathbb{R} \setminus \{ 0 \}$$, then $$\lnot U = (U \rightarrow \emptyset) = \operatorname{int}(\{ 0 \}) = \emptyset$$, so $$U \vee \lnot U = \mathbb{R} \setminus \{ 0 \} \neq \top$$, and $$\lnot (\lnot U) = \mathbb{R} \ne U$$. (Of course, there's nothing particular special about $$\mathbb{R}$$ in this example: the same construction will work for the open subsets of any topological space.)

In case you're not comfortable with topology or real analysis, there's also a finite example that's often used: $$X := \{ 0, \frac{1}{2}, 1 \}$$; $$\top := 1$$; $$\bot := 0$$; $$x \vee y := \max(x,y)$$; $$x \wedge y := \min(x,y)$$; and $$x \rightarrow y$$ is given by a table. In this Heyting algebra, $$\lnot(\frac{1}{2}) = (\frac{1}{2} \rightarrow 0) = 0$$, so $$\frac{1}{2} \vee \lnot(\frac{1}{2}) = \frac{1}{2} \ne \top$$.

• What is ${\rm int}$? – DanielV Jan 15 at 3:43
• @DanielV Interior. – Daniel Schepler Jan 15 at 5:55
• This probably looks like gibberish to those who don't know or don't quite remember the conditions (a postulate set) that a Boolean and a Heyting Algebra must satisfy. Also, that the axioms are 'certain axioms' I'm not so sure qualifies as correct. The term 'certain' might imply a finite set of axioms, but Boolean Algebra has an infinite set of possible axioms. – Doug Spoonwood Jan 17 at 18:30

What author are you reading? The author may be referring to semantics in the context of category theory and type theory, where various notions of equality in type theory have the same structural property in category theory which are various notions of semantics.