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# Math Help - solve the equation

1. ## solve the equation

Solve for z.

$i^{z} = z$ where $z \in \mathbb{C}$

2. I'm not sure if a hint is needed, or if this problem is not challenging at all. I'm leaning towards the latter.

3. Originally Posted by Random Variable
I'm not sure if a hint is needed, or if this problem is not challenging at all. I'm leaning towards the latter.
The complex power of a complex number is multi-valued. So there could be an infinite number of solutions. Today is too late, we can discuss this sometime later. ;-)

4. The complex power of a complex number is multi-valued. So there could be an infinite number of solutions.
True. It depends which branch you pick.

5. Originally Posted by Random Variable
Solve for z.

$i^{z} = z$ where $z \in \mathbb{C}$
Has closed form solution in terms of Lambert's W

CB

6. Originally Posted by CaptainBlack
Has closed form solution in terms of Lambert's W

CB
I know.

Spoiler:

$i^{z} = z$

$i^{-z} = \frac{1}{z}$

$z i^{-z} = 1$

$-z i^{-z} = -1$

$-ze^{-z \ln i} = -1$

$-z \ln i \ e^{-z \ln i} = - \ln i$

$W(-z \ln i e^{-z \ln i}) = W (- \ln i)$

$-z \ln i = W(-\ln i)$

$z = \frac{W(-\ln i)}{-\ln i}$

7. Originally Posted by Random Variable
I know.

Spoiler:

$i^{z} = z$

$i^{-z} = \frac{1}{z}$

$z i^{-z} = 1$

$-z i^{-z} = -1$

$-ze^{-z \ln i} = -1$

$-z \ln i \ e^{-z \ln i} = - \ln i$

$W(-z \ln i e^{-z \ln i}) = W (- \ln i)$

$-z \ln i = W(-\ln i)$

$z = \frac{W(-\ln i)}{-\ln i}$

Horrible, try putting $i=e^{\pi i/2}$

(the term horrible here is not intended to be insulting, but is a term often used to describe an aesthetically unpleasing piece of mathematics)

CB

8. OK. Do this instead. Let a be any constant (real or complex).

$a^{z} = z$

$a^{-z} = \frac{1}{z}$

$z a^{-z} = 1$

$-z a^{-z} = -1$

$-ze^{-z \ \text{Log a}} = -1$

$-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}= - \text{Log a}$

$W(-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}) = W (- \text{Log a})$

$-z \text{Log a} = W(-\text{Log a})$

$z = \frac{W(-\text{Log a})}{-\text{Log a}}$

$\text{Log i} = \ln|i| + i \frac{\pi}{2} = i\frac{\pi}{2}$

$\frac {W(-i \frac{\pi}{2})}{-i\frac{\pi}{2}} \approx 0.4382829366+i0.3605924718$

Is that satisfactory?

9. Originally Posted by Random Variable
OK. Do this instead. Let a be any constant (real or complex).

$a^{z} = z$

$a^{-z} = \frac{1}{z}$

$z a^{-z} = 1$

$-z a^{-z} = -1$

$-ze^{-z \ \text{Log a}} = -1$

$-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}= - \text{Log a}$

$W(-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}) = W (- \text{Log a})$

$-z \text{Log a} = W(-\text{Log a})$

$z = \frac{W(-\text{Log a})}{-\text{Log a}}$

$\text{Log i} = \ln|i| + i \frac{\pi}{2} = i\frac{\pi}{2}$

$\frac {W(-i \frac{\pi}{2})}{-i\frac{\pi}{2}} \approx 0.4382829366+i0.3605924718$

Is that satisfactory?
This looks good to me!

10. But what is $W(z)$ ?

11. Originally Posted by simplependulum
But what is $W(z)$ ?
Lambert W-Function

It's the function W(z) that solves $z = W(z)e^{W(z)}$ for a given value of z.

12. Originally Posted by Random Variable
OK. Do this instead. Let a be any constant (real or complex).

$a^{z} = z$

$a^{-z} = \frac{1}{z}$

$z a^{-z} = 1$

$-z a^{-z} = -1$

$-ze^{-z \ \text{Log a}} = -1$

$-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}= - \text{Log a}$

$W(-z \ (\text{Log a}) \ e^{-z \ \text{Log a}}) = W (- \text{Log a})$

$-z \text{Log a} = W(-\text{Log a})$

$z = \frac{W(-\text{Log a})}{-\text{Log a}}$

$\text{Log i} = \ln|i| + i \frac{\pi}{2} = i\frac{\pi}{2}$

$\frac {W(-i \frac{\pi}{2})}{-i\frac{\pi}{2}} \approx 0.4382829366+i0.3605924718$

Is that satisfactory?
It is better in that you eliminate the $\log$ 's in the final answer but I would still prefer a solution that did not have to worry about which branch of the $\log$ was in use where. That can be done by the the first step that I proposed (which, while that is not the way I think of it, absorbs the taking of logarithms while effectively fixing the branch).

Also as $W$ is multi-valued the right hand side of final line is just one of the solutions that correspond to the left hand side (as you know, and I know you know).

CB

13. Originally Posted by CaptainBlack
It is better in that you eliminate the $\log$ 's in the final answer but I would still prefer a solution that did not have to worry about which branch of the $\log$ was in use where. That can be done by the the first step that I proposed (which, while that is not the way I think of it, absorbs the taking of logarithms while effectively fixing the branch).

Also as $W$ is multi-valued the right hand side of final line is just one of the solutions that correspond to the left hand side (as you know, and I know you know).

CB
The multi-valued-ness needs to be discussed and dealt with, because that is the key here.

14. Originally Posted by Random Variable
Solve for z.

$i^{z} = z$ where $z \in \mathbb{C}$
I like this method:

As $i=e^{(2n+1/2)\pi i}=e^{-\mu_n}$ where $\mu_n=-(2n+1/2)\pi i$ for $n\in\mathbb{Z}$ we have:

$e^{-\mu_n z}=z$

or:

$\mu_n z\; e^{\mu_n z}=\mu_n$

so:

$\mu_n z=W(\mu_n)$.

Hence:

$z=\frac{1}{\mu_n}W(\mu_n); \ \ \mu_n= -(2n+1/2)\pi i,\ n \in \mathbb{Z}$

(notes:
1. this is at least the third refinement of the original idea scribbled in my note book
2. the multi-valuedness of the solution is not only dependent on $n$, but the Lambert-W is itself multi-valued)

CB


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## anonymous 4 years ago The drawing shows two displacements, 3.0 km due north and 4.0 km due east, and the corresponding resultant vector R. What is the angle θ that R makes with respect to due north?

1. anonymous

Do you do cos 3/5?

2. asnaseer

|dw:1327177003361:dw| In this diagram we can find $$\theta$$ from:$\tan(\theta)=\frac{4}{3}$

format: plain


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# Question #cb1e2

Feb 25, 2018

Given $\sin \theta = \frac{12}{13}$

We know, $\sin \theta = \left(\text{opposite side")/("hypotenuse}\right)$

Therefore, $\text{opposite side} = 12$ and $\text{hypotenuse} = 13$

so the third side, $x$ can be found by applying Pythagoras theorem.

i.e. ${\left(\text{hypotenuse")^2=("opposite side}\right)}^{2} + {x}^{2}$
${x}^{2} = {13}^{2} - {12}^{2} = 169 - 144 = 25$
$x = 5$ units.

Now, $\cos \theta = \left(\text{adjacent side")/("hypotenuse}\right)$
$\cos \theta = \frac{5}{13}$


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# Cancellation laws for function composition

Okay I was asked to make a conjecture about cancellation laws for function composition. I figured it would go something like "For all sets $A$ and functions $g: A \rightarrow B$ and $h: A \rightarrow B$, $f \circ g = f \circ h$ implies that $g=h$."

I'm pretty sure $g=h$ isn't always true, but is there a property of $f$ that makes this true?

-
HINT: Consider the properties of injectivity and surjectivity. –  Ittay Weiss Nov 14 '12 at 6:10

Notice that if there are distinct $b_1,b_2\in B$ such that $f(b_1)=f(b_2)$, you won’t necessarily be able to cancel $f$: there might be some $a\in A$ such that $g(a)=b_1$ and $h(a)=b_2$, but you’d still have $(f\circ g)(a)=(f\circ h)(a)$. Thus, you want $f$ to be injective (one-to-one). Can you prove that that’s sufficient?
Okay so if $f$ is one-to-one, then for all sets A and functions g:A→B and h:A→B, f∘g=f∘h implies that g=h. I'm having trouble coming up with a proof for this. I guess that for every $c \in C$ there is at most one $b \in B$ such that $f$$b$$=f$$a$$$. I'm not sure where to go after that though. –  GAtechWizard Nov 14 '12 at 14:14
@GAtechWizard: Do the usual thing: suppose that $g\ne h$. Then there is some $a\in A$ such that $g(a)\ne h(a)$. Let $b_1=g(a)$ and $b_2=h(a)$. Then $$f(b_1)=f\big(g(a)\big)=(f\circ g)(a)=(f\circ h)(a)=f\big(h(a)\big)=f(b_2)\;,$$ contradicting the assumption that $f$ is one-to-one. –  Brian M. Scott Nov 14 '12 at 14:21


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 Original Article Korean Journal of Epidemiology 2004;26(2): 71-80.
 Incidence of complication and societal cost of measles in Korea, 2000. Weon Young Lee, Moran Ki, Jin Kyoung Oh, Jong Koo Lee, Myung Bin Kim, Bo Youl Choi 1Department of Preventive Medicine, Chungang UniversityCollege of Medicine, Korea.2Department of Preventive Medicine, Eulji University Schoolof Medicine, Korea. kimoran@eulji.ac.kr3National Cancer Center, Korea.4Ministry of Health and Welfare, Korea.5Kim Myung Bin Clinic, Korea.6Department of Preventive Medicine, Hanyang UniversityCollege of Medicine, Korea. Abstract OBJECTIVES: To estimate the complication rate and the societal cost of measles, a survey was conducted in Seoul, Jeonju city, and Kyonggi province in 2001. METHODS: A telephone survey was conducted in Jeonju city (175/307) and four local areas of Kyonggi province (793/1,238) to gather information on the complications of measles. To estimate societal cost, the telephone survey was conducted for the sampled subject by complication type and the age group. The response rate was 78% (180/230). Paid bills were examined for direct cost estimation, and the time cost and the transportation expense were analyzed for indirect cost estimation. RESULTS: The incidence of a complication of measles was 3.1% which found to be higher in younger age group. The incidence of pneumonia, otitis media, and encephalitis were 2.1%, 0.8%, and 0.2% respectively. Direct and indirect costs of in-patients without a complication were $417.00 (US$1.00 = 1,000 won) and $256.00 per case, respectively, and the out-patients who have no complication were$54.00 and $65.00, respectively. The average cost for a patient with measles without complication was$119.00 as the result. The societal cost of encephalitis was high as $6,660. Estimated total societal cost of measles ranges from$14 million to \$69 million in the year 2000. CONCLUSION: Complication rate of measles was fairly low compared to foreign countries. The lower rate could result from the difference in vaccination rate and the age distribution of the measles patients. The cost of measles without complication was not high. However, the cost for the complication and the total disease burden caused by measles shown to be high in the year 2000. Keywords: Measles; Complications; Cost of illness


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## G = C4×Dic11order 176 = 24·11

### Direct product of C4 and Dic11

Aliases: C4×Dic11, C11⋊C42, C442C4, C22.3D22, C22.3(C2×C4), (C2×C44).7C2, (C2×C4).6D11, C2.2(C4×D11), (C2×C22).3C22, C2.2(C2×Dic11), (C2×Dic11).4C2, SmallGroup(176,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C4×Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C4×Dic11
 Lower central C11 — C4×Dic11
 Upper central C1 — C2×C4

Generators and relations for C4×Dic11
G = < a,b,c | a4=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×Dic11
Regular action on 176 points
Generators in S176
(1 105 145 171)(2 106 146 172)(3 107 147 173)(4 108 148 174)(5 109 149 175)(6 110 150 176)(7 89 151 155)(8 90 152 156)(9 91 153 157)(10 92 154 158)(11 93 133 159)(12 94 134 160)(13 95 135 161)(14 96 136 162)(15 97 137 163)(16 98 138 164)(17 99 139 165)(18 100 140 166)(19 101 141 167)(20 102 142 168)(21 103 143 169)(22 104 144 170)(23 120 67 65)(24 121 68 66)(25 122 69 45)(26 123 70 46)(27 124 71 47)(28 125 72 48)(29 126 73 49)(30 127 74 50)(31 128 75 51)(32 129 76 52)(33 130 77 53)(34 131 78 54)(35 132 79 55)(36 111 80 56)(37 112 81 57)(38 113 82 58)(39 114 83 59)(40 115 84 60)(41 116 85 61)(42 117 86 62)(43 118 87 63)(44 119 88 64)
(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)(155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 127 12 116)(2 126 13 115)(3 125 14 114)(4 124 15 113)(5 123 16 112)(6 122 17 111)(7 121 18 132)(8 120 19 131)(9 119 20 130)(10 118 21 129)(11 117 22 128)(23 167 34 156)(24 166 35 155)(25 165 36 176)(26 164 37 175)(27 163 38 174)(28 162 39 173)(29 161 40 172)(30 160 41 171)(31 159 42 170)(32 158 43 169)(33 157 44 168)(45 139 56 150)(46 138 57 149)(47 137 58 148)(48 136 59 147)(49 135 60 146)(50 134 61 145)(51 133 62 144)(52 154 63 143)(53 153 64 142)(54 152 65 141)(55 151 66 140)(67 101 78 90)(68 100 79 89)(69 99 80 110)(70 98 81 109)(71 97 82 108)(72 96 83 107)(73 95 84 106)(74 94 85 105)(75 93 86 104)(76 92 87 103)(77 91 88 102)

G:=sub<Sym(176)| (1,105,145,171)(2,106,146,172)(3,107,147,173)(4,108,148,174)(5,109,149,175)(6,110,150,176)(7,89,151,155)(8,90,152,156)(9,91,153,157)(10,92,154,158)(11,93,133,159)(12,94,134,160)(13,95,135,161)(14,96,136,162)(15,97,137,163)(16,98,138,164)(17,99,139,165)(18,100,140,166)(19,101,141,167)(20,102,142,168)(21,103,143,169)(22,104,144,170)(23,120,67,65)(24,121,68,66)(25,122,69,45)(26,123,70,46)(27,124,71,47)(28,125,72,48)(29,126,73,49)(30,127,74,50)(31,128,75,51)(32,129,76,52)(33,130,77,53)(34,131,78,54)(35,132,79,55)(36,111,80,56)(37,112,81,57)(38,113,82,58)(39,114,83,59)(40,115,84,60)(41,116,85,61)(42,117,86,62)(43,118,87,63)(44,119,88,64), (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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,127,12,116)(2,126,13,115)(3,125,14,114)(4,124,15,113)(5,123,16,112)(6,122,17,111)(7,121,18,132)(8,120,19,131)(9,119,20,130)(10,118,21,129)(11,117,22,128)(23,167,34,156)(24,166,35,155)(25,165,36,176)(26,164,37,175)(27,163,38,174)(28,162,39,173)(29,161,40,172)(30,160,41,171)(31,159,42,170)(32,158,43,169)(33,157,44,168)(45,139,56,150)(46,138,57,149)(47,137,58,148)(48,136,59,147)(49,135,60,146)(50,134,61,145)(51,133,62,144)(52,154,63,143)(53,153,64,142)(54,152,65,141)(55,151,66,140)(67,101,78,90)(68,100,79,89)(69,99,80,110)(70,98,81,109)(71,97,82,108)(72,96,83,107)(73,95,84,106)(74,94,85,105)(75,93,86,104)(76,92,87,103)(77,91,88,102)>;

G:=Group( (1,105,145,171)(2,106,146,172)(3,107,147,173)(4,108,148,174)(5,109,149,175)(6,110,150,176)(7,89,151,155)(8,90,152,156)(9,91,153,157)(10,92,154,158)(11,93,133,159)(12,94,134,160)(13,95,135,161)(14,96,136,162)(15,97,137,163)(16,98,138,164)(17,99,139,165)(18,100,140,166)(19,101,141,167)(20,102,142,168)(21,103,143,169)(22,104,144,170)(23,120,67,65)(24,121,68,66)(25,122,69,45)(26,123,70,46)(27,124,71,47)(28,125,72,48)(29,126,73,49)(30,127,74,50)(31,128,75,51)(32,129,76,52)(33,130,77,53)(34,131,78,54)(35,132,79,55)(36,111,80,56)(37,112,81,57)(38,113,82,58)(39,114,83,59)(40,115,84,60)(41,116,85,61)(42,117,86,62)(43,118,87,63)(44,119,88,64), (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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,127,12,116)(2,126,13,115)(3,125,14,114)(4,124,15,113)(5,123,16,112)(6,122,17,111)(7,121,18,132)(8,120,19,131)(9,119,20,130)(10,118,21,129)(11,117,22,128)(23,167,34,156)(24,166,35,155)(25,165,36,176)(26,164,37,175)(27,163,38,174)(28,162,39,173)(29,161,40,172)(30,160,41,171)(31,159,42,170)(32,158,43,169)(33,157,44,168)(45,139,56,150)(46,138,57,149)(47,137,58,148)(48,136,59,147)(49,135,60,146)(50,134,61,145)(51,133,62,144)(52,154,63,143)(53,153,64,142)(54,152,65,141)(55,151,66,140)(67,101,78,90)(68,100,79,89)(69,99,80,110)(70,98,81,109)(71,97,82,108)(72,96,83,107)(73,95,84,106)(74,94,85,105)(75,93,86,104)(76,92,87,103)(77,91,88,102) );

G=PermutationGroup([[(1,105,145,171),(2,106,146,172),(3,107,147,173),(4,108,148,174),(5,109,149,175),(6,110,150,176),(7,89,151,155),(8,90,152,156),(9,91,153,157),(10,92,154,158),(11,93,133,159),(12,94,134,160),(13,95,135,161),(14,96,136,162),(15,97,137,163),(16,98,138,164),(17,99,139,165),(18,100,140,166),(19,101,141,167),(20,102,142,168),(21,103,143,169),(22,104,144,170),(23,120,67,65),(24,121,68,66),(25,122,69,45),(26,123,70,46),(27,124,71,47),(28,125,72,48),(29,126,73,49),(30,127,74,50),(31,128,75,51),(32,129,76,52),(33,130,77,53),(34,131,78,54),(35,132,79,55),(36,111,80,56),(37,112,81,57),(38,113,82,58),(39,114,83,59),(40,115,84,60),(41,116,85,61),(42,117,86,62),(43,118,87,63),(44,119,88,64)], [(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),(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,127,12,116),(2,126,13,115),(3,125,14,114),(4,124,15,113),(5,123,16,112),(6,122,17,111),(7,121,18,132),(8,120,19,131),(9,119,20,130),(10,118,21,129),(11,117,22,128),(23,167,34,156),(24,166,35,155),(25,165,36,176),(26,164,37,175),(27,163,38,174),(28,162,39,173),(29,161,40,172),(30,160,41,171),(31,159,42,170),(32,158,43,169),(33,157,44,168),(45,139,56,150),(46,138,57,149),(47,137,58,148),(48,136,59,147),(49,135,60,146),(50,134,61,145),(51,133,62,144),(52,154,63,143),(53,153,64,142),(54,152,65,141),(55,151,66,140),(67,101,78,90),(68,100,79,89),(69,99,80,110),(70,98,81,109),(71,97,82,108),(72,96,83,107),(73,95,84,106),(74,94,85,105),(75,93,86,104),(76,92,87,103),(77,91,88,102)]])

C4×Dic11 is a maximal subgroup of
Dic11⋊C8  C88⋊C4  D444C4  C44.56D4  C42×D11  C42⋊D11  C23.11D22  C23.D22  Dic114D4  Dic11.D4  Dic22⋊C4  C44⋊Q8  Dic11.Q8  C44.3Q8  C4⋊C47D11  D44⋊C4  C4⋊C4⋊D11  C23.21D22  C44.17D4  C44⋊D4  Dic11⋊Q8  C44.23D4
C4×Dic11 is a maximal quotient of
C42.D11  C88⋊C4  C22.C42

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 11A ··· 11E 22A ··· 22O 44A ··· 44T order 1 2 2 2 4 4 4 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 1 1 1 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D11 Dic11 D22 C4×D11 kernel C4×Dic11 C2×Dic11 C2×C44 Dic11 C44 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 5 10 5 20

Matrix representation of C4×Dic11 in GL4(𝔽89) generated by

 34 0 0 0 0 55 0 0 0 0 88 0 0 0 0 88
,
 88 0 0 0 0 1 0 0 0 0 88 1 0 0 80 8
,
 34 0 0 0 0 88 0 0 0 0 65 13 0 0 79 24
G:=sub<GL(4,GF(89))| [34,0,0,0,0,55,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,1,0,0,0,0,88,80,0,0,1,8],[34,0,0,0,0,88,0,0,0,0,65,79,0,0,13,24] >;

C4×Dic11 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{11}
% in TeX

G:=Group("C4xDic11");
// GroupNames label

G:=SmallGroup(176,10);
// by ID

G=gap.SmallGroup(176,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,20,46,4004]);
// Polycyclic

G:=Group<a,b,c|a^4=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

׿
×
𝔽


text:
# Postdoc position in Ljubljana

A postdoc position in the Effmath research project is available at the University of Ljubljana, Faculty of Mathematics and Physics. The precise topic is flexible, but should generally be aligned with the project (see project description). Possible topics include:

•  reasoning about computational effects
•  implementation of computational effects
• proof assistants and formalization of mathematics

The candidate should have a PhD degree in mathematics or computer science, with background knowledge relevant to the project area. The position is available for a period of one year with possibility of extension, preferably starting in early 2016. No knowledge of the Slovene language is required.

The candidates should contact Andrej Bauer by email as soon as possible, but no later than January 8th 2016. Please include a short CV and a statement of interest.

# The HoTT book

The HoTT book is finished!

Since spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. You can get the book freely at http://homotopytypetheory.org/book/. Mike Shulman has written about the contents of the book, so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research.

# A hott thesis

Egbert Rijke successfully defended his master thesis in Utrecht a couple of weeks ago. He published it on the Homotopy type theory blog (here is a direct link to the PDF file (revised)). The thesis is well written and it contains several new results, but most importantly, it is a gentle yet non-trivial introduction to homotopy type theory. If you are interested in the topic but do not know where to start, Egbert’s thesis might be perfect for you. As far as I know it is the first substantial piece of text written in (informal) homotopy type theory.

What I find most amazing about the work is that Egbert does not have to pretend to be a homotopy type theorist, like us old folks. His first contact with type theory was homotopy type theory, which impressed on his mind a new kind of geometric intuition about $\Pi$’s, $\Sigma$’s and $\mathrm{Id}$’s. If we perform enough such experiments on young bright students, strange things will happen.

# Eff 3.0

Matija and I are pleased to announce a new major release of the eff programming language.

In the last year or so eff has matured considerably:

• It now looks and feels like OCaml, so you won’t have to learn yet another syntax.
• It has static typing with parametric polymorphism and type inference.
• Eff now clearly separates three basic concepts: effect types, effect instances, and handlers.
• How eff works is explained in our paper on Programming with Algebraic Effects and Handlers.
• We moved the source code to GitHub, so go ahead and fork it!

# Bob Harper has a blog

Bob Harper of CMU, has recently started a blog, called Existential Type, about programming languages. He is a leading expert in Programming Languages. I remember being deeply inspired the first time I heard him talk. I was an incoming graduate student at CMU and he presented what the programming languages people at CMU did. His posts are fun to read, unreserved and very educational. Highly recommended!


text:
IWARA2022 - 10th International Workshop on Astronomy and Relativistic Astrophysics

Sep 5 – 9, 2022
America/Guatemala timezone

New tools for better characterizing astroparticle physics sites and detectors for the Latin American Giant Observatory

Sep 6, 2022, 9:00 AM
20m
Talk (in-person)

Hernán Asorey

Description

The Latin American Giant Observatory (LAGO) is an extended astroparticle observatory, consisting of a synchronised network of water Cherenkov detectors operating in Latin America and covering a wide range of altitudes above sea level and geomagnetic rigidity cut-offs[1]. LAGO is operated by the LAGO Collaboration, a highly collaborative organization with 100 members from more than 30 Iberoamerican institutions. LAGO research objectives are focused on studying of high-energy astrophysical and space weather and climate phenomena, by indirectly measuring the temporal evolution of the flux of galactic cosmic rays from ground level[2].
The interaction of such cosmic rays with the atmosphere produces a large number of secondary particles via radiative and decay processes. These true cascades of particles, collectively known as Extensive Air Showers, could reach up to $10^{11}$ particles at the instant of their maximum development.
To accomplish these tasks we have developed several computational tools[3]; that take advantage of the increasing computational capabilities available at high-performance computing facilities and in cloud-based computing environments, such as the European Open Scientific Cloud (EOSC)[4]. With these tools, we can calculate the expected particle flux at any place in the World, including real-time atmospheric and geomagnetic effects, reproducing the expected signals in different types of detectors. We can also calculate the impact in the expected flux due to the occurrence of transient astrophysical phenomena, such as those related to Solar Activity[5] or the high energy component of Gamma-Ray Bursts[6].
In this contribution, we will show how this very complex sequence of simulations is helping us to characterize new sites for our Observatory and to design new and improved astroparticle detectors for the LAGO detection network.

Primary authors

Antonio Juan Rubio Montero (CIEMAT (Research Centre for Energy, Environment and Technology)) Raúl Pagán-Muñoz (CIEMAT) Alfonso Pardo-Díaz (CETA-CIEMAT) Ivan Sidelnik (CONICET (Consejo Nacional de Investigaciones Cientifícas y Técnicas) - Centro Atómico Bariloche - Argentina) Rafael Mayo-García (CIEMAT)


text:
## Atomic orbitals [ENDORSED]

$sp, sp^{2}, sp^{3}, dsp^{3}, d^{2}sp^{3}$

JELCI_BARRAZA_1C
Posts: 17
Joined: Wed Sep 21, 2016 3:00 pm

### Atomic orbitals

why is magnetic quantum number important?

Victoria Draper 1G
Posts: 52
Joined: Fri Sep 29, 2017 7:05 am
Been upvoted: 1 time

### Re: Atomic orbitals

Hi!

The magnetic quantum number is important because it tells us about the orbital that an electron occupies and also identifies the total number of orbitals there are along with their position within a subshell. For example, the p-subshell, whose angular momentum number (l) is 1, has three possible orientations of orbitals, since there are three values for the magnetic quantum number (-1, 0, and 1). The d-subshell, however, has an angular momentum number of 2, and therefore has 5 possible orientations of orbitals give by the values of the magnetic quantum number (-2, -1, 0, 1, 2). Overall the magnetic quantum number helps break down the subshell further by defining the number of orbitals within the subshells that electrons can fill.

Hope this helps!

Kimberly Santander3L
Posts: 16
Joined: Fri Sep 25, 2015 3:00 am
Been upvoted: 1 time

### Re: Atomic orbitals

A magnetic quantum number tells us the orbital an electron is occupying. It provides us with the amount of orbitals the electron has and the sub-shells.

Justin Lai 1C
Posts: 50
Joined: Fri Sep 29, 2017 7:04 am

### Re: Atomic orbitals  [ENDORSED]

It's also interesting because s has a magnetic quantum number of 0. And there's only one orbital for s because it is the orbital with symmetrical e- density. However p would have magnetic quantum numbers of -1, 0, and 1 because it has orbitals in the orientation of xy, yz. xz that line up on those nodal planes.


text:
# Functional Programming

So you have decided to learn functional programming, great! You visit the first website, read a book, watch a video, and maybe listen to a podcast; when suddenly you realize that functional programming is a completely different kind of programming. That it is a fundamental change in the way you see and interact with computer programs. And maybe it’ll even require you to almost forget about everything related to computer programs!

But what do we know, as programmers, so that we forget? I’ll assume that you’re coming from an object-oriented background, and if so, we can summarize the basic components of almost any program -that is, on a higher level- in the following chart:

So instead of forgetting about everything we know, how about we rebuild our idea on programming and what it means, and how it is constructed.

## Three Elementary Structures

In 1966 a paper by Corrado Böhm and Giuseppe Jacopini, 2 great computer scientists, introduced the Böhm–Jacopini theorem [1] or the structured programming theorem. This theorem states that all programs can be constructed from just three structures: sequence, selection, and iteration.

So based on that, we definitely will have to have these three elementary structures in any functional program. Of course, we’ll also have data types; strings, integers, and booleans are all fundamental in the way we develop programs. Add to this the more advanced data structures like arrays, trees, hashes, and so on.

Turns out the only level of the programming pyramid that we need to rebuild, are the ideas that were brought by the Object-Oriented Programming paradigm. We need to find a more “functional” approach to doing programming.

## Mathematics and Abstractions

One of the best places to go to when you want to look for a new paradigm of programming is Mathematicians! Mathematics and Computer Science share the same characteristics in problem-solving techniques, formulating and modeling problems, and a lot more. But most importantly: Abstractions.

You might remember that moment when you were trying to build this complex system, and you’re having this abstraction of the system as a model, and then this other abstraction and the other and so on, all that combined together make up this functional goal of the system. Turns out, mathematicians share the same problem as us! They’re also trying to keep in their mind an abstraction over the other to derive this “functional” goal of a mathematical model. So, what did mathematicians use that we might find useful? Functions.

## Mathematical Functions

If you are a programmer you have probably heard of functions (maybe methods, routines, etc.), but mathematical functions are not the same. In mathematics, a function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output [2]. We can view a function as a machine that can take an object and turn it into (or map it to) a different object, this is called the function machine metaphor [3].

Mathematical functions are different than programming methods/routines because of a very important thing and that is the mechanism of how they work. A simple ruby method can do a lot of things including updating the database, creating new files, sending emails. A mathematical function on the other side does only one thing and one thing only. Every input is mapped into the exact same output each time that function is called. So it’s not like f(x) is gonna be 24 today, and then 3 days later it’s 77.

So what if we borrow these ideas from mathematics and apply it to programming, in hope that our functions (programs) are easier to read and to write? There are a couple of rules and conditions that we have to follow in order for our functions to be “pure”.

## Rule #1: Functions as values

A fundamental rule is that those mathematical functions are considered to be “values”. This is what we call a 1st class function [4], and they’re actually in many programming languages (that are not even purely functional). If you look for example on this ruby code:

fn = lambda { |x| x + 2 }

puts fn.(2) # = 4

As you can see, we defined a variable fn that holds a function that takes an input x and returns x + 2. Nowadays, if you look over the documentation of almost all programming languages, you’ll find the idea of pure functions developed and utilized. A function is a value that can be defined and passed as an argument and manipulated just like other variables.

# A function add that takes a function fn
# and a number a as parameters, applies
# fn to a and adds the result of
# fn(a) to a again.
fn.(a) + a
end

fn = lambda { |x| x + 2 }

puts result # = 2 + 2 + 2 = 6

## Rule #2: No Side Effects

The second most important rule for our functions to behave purely is to behave exactly like mathematical functions do: Take an input, produce an output. That is, without any side effects.

But, what are side effects?

# A side effect
File.open("log.txt", "w") { |f| f.write "#{Time.now} - User logged in\n" }

Side effects are basically the change of the “outer” state of the function. A change that is outside the scope of the function’s inputs and outputs. Anything that deals with IO, or any external resource is considered to be a side effect. But what are our programs other than side effects?

So we need this bridge that connects our functions with the actual events and side effects that we want to do. The solution to this problem has been implemented differently in different programming languages, and we’ll take a Clojure look at that.

There’s also a huge discussion nowadays on reactive functional programming (bonus: Elm [5] is a compellingly beautiful example of that), where they deal with a lot of state change and UI elements in a functional way.

## A Bridge to Side Effects

### Atoms

An example on this bridge that the programming language Clojure has taken is what they call an Atom. An atom is the closest thing that someone can find to a mutable state. It is Clojure’s way of bridging the clean organized functional approach with the messy world of side effects.

Here’s how we define an atom in Clojure:

(def x (atom 0))

And then to change the value of this atom, you just throw it into a function that applies the function on that atom, and returns the value of the applied function as the new value of the atom. For example here’s how you can increment an atom:

(swap! x inc)

What this does is that it takes x and applies inc(x) and swaps the current value of x with the returned value of the applied function.

Now the beautiful thing happens when two threads are trying to update x at the same time.

Here, inc(x) and dec(x) both come at the same time, and they see the current value of the atom X. What happens is that this atom runs the function and notices that the value of X has changed and what happens? It runs the function that came last again.

### Agents and Actors

Okay, what if I want to update a file or a database or something that is completely outside the scope of the program I’m dealing with? Here comes Clojure’s agents.

In that you basically send these side-effects into a queue specified for doing these tasks, and all of them run synchronously. The topic of agents is huge and shall be discussed in a separate article.

## Rule #3: Immutability

Based on the previously mentioned rule #2, our functions should behave purely and with no side-effects. But what if our variables are mutable? That is, its state can change. You find yourself back at the problem of side-effects! So to ensure that side-effects are completely demolished we need one more important rule: Variable Immutability.

x = [1, 2, 3]
y = f(x)
# what is that value of x at this point? x!

Notice that even if you make y = f(g(h(...(x))) , the value of x is still going to be the same. Nothing is changed! f(x) takes a copy of x and returns a result based on this copy. You don’t have to figure out why x’s value is being weird when it is clearly defined in front of your eyes.

But a problem arises here. y = f(x) + g(x) - h(x) / .... + f2(x), if I define y to be the result of this huge chain of functions over functions over functions, the compiler will make this copy over copy over copy. What if x was a 1-million element list instead of 3? And with every function, we make a new copy? That doesn’t sound very memory friendly.

Turns out functional languages deal with this problem in a very beautiful way: Lazy data structures.

### Lazy Data Structures

Let’s have a look at this Clojure code:

;; (0, 1, 2, 3, 4, ...)
(def integers (range))

This defines an infinite list of all integer numbers in a list called integers. Of course not the infinite list of integers will be saved in your memory, but instead only what you want to evaluate, gets evaluated.

;; gets first 10 integers and maps an
;; increment function on all of them.
(def first-ten-integers (take 10 integers))
(def incremented (map inc first-ten-integers))

The way lazy (or persistent) data structures are defined in memory is a very complex and interesting topic that will be discussed in the next article.

## Conclusion

So taking the ideas of mathematicians into programming wasn’t very bad after all. We’ll still use the same fundamental concepts of programming, but we’ll only change the higher level way of modelling our programs. We added some rules in hope that our programs are cleaner, and more concise. But functional programming isn’t this magical box that will make all your dreams come true. You’ll still find yourself making errors, you’ll spend a couple of minutes looking for a missing bracket or to fix a simple bug. It is just a cleaner, more organized way of thinking about computer programs.

## References:

[1] - Flow diagrams, turing machines and languages with only two formation rules: https://dl.acm.org/doi/10.1145/355592.365646#CIT

[2] - Function: https://mathinsight.org/definition/function

[3] - Function Machine: https://mathinsight.org/function_machine

[4] - What are first-class functions? https://lispcast.com/what-are-first-class-functions/

[5] - Elm Programming Language: https://elm-lang.org/

This article was heavily based on the work of Bob Martin and his book Clean Architecture: https://blog.cleancoder.com/uncle-bob/2012/08/13/the-clean-architecture.html and Russ Olsen’s talk Functional Programming in 40 minutes https://www.youtube.com/watch?v=0if71HOyVjY.