MAGMA : Automorphism Groups acting on Involutions - finite-group-theory

For example if we let
G:= Sym(6);
A:= AutomorphismGroup(G);
P:= PermutationGroup(A);
I:= {# i : i in G | Order(i) eq 2 #};
then how does one construct the G-set that has the P acting on I by the map the shows how the automorphisms permute the involutions?
I'm new to MAGMA but I have speant a lot of time reading the handbook lately and I have not found any way to do this. Does anyone have any helpful suggestions?

Related

Sentence-count function not returning total count

So i've been trying to create a sentence-count function which will cycle through the following 'story':
let story = 'Last weekend, I took literally the most beautiful bike ride of my life. The route is called "The 9W to Nyack" and it actually stretches all the way from Riverside Park in Manhattan to South Nyack, New Jersey. It\'s really an adventure from beginning to end! It is a 48 mile loop and it basically took me an entire day. I stopped at Riverbank State Park to take some extremely artsy photos. It was a short stop, though, because I had a really long way left to go. After a quick photo op at the very popular Little Red Lighthouse, I began my trek across the George Washington Bridge into New Jersey. The GW is actually very long - 4,760 feet! I was already very tired by the time I got to the other side. An hour later, I reached Greenbrook Nature Sanctuary, an extremely beautiful park along the coast of the Hudson. Something that was very surprising to me was that near the end of the route you actually cross back into New York! At this point, you are very close to the end.';
And I realise the problem I'm having but I cannot find a way around this. Basically I want my code to return a the total sCount below but seeing as I've returned my sCount after my loop, it's only adding and returning the one count as a total:
const sentenceTotal = (word) => {
let sCount = 0;
if (word[word.length-1] === "." || word[word.length-1] === "!" || word[word.length-1] === "?") {
sCount += 1;
};
return sCount;
};
// console.log(sentenceTotal(story)) returns '1'.
I've tried multiple ways around this, such as returning sentenceTotal(word) instead of sCount but console.log will just log the function name.
I can make it return the correct sCount total if I remove the function element of it, but that's not what I want.
I don't see any loop or iterator which would go through story to count the number of occurrences of ., ?, or !.
Having recently tackled "counting sentences" myself I know it is a non-trivial problem with many edge cases.
For a simple use-case though you can use split and a regular expression;
story.split(/[?!.]/).length
So you could wrap that in your function like so:
const sentenceTotal = (word) => {
return word.split(/[?.!]/).length
};
let story = 'Last weekend, I took literally the most beautiful bike ride of my life. The route is called "The 9W to Nyack" and it actually stretches all the way from Riverside Park in Manhattan to South Nyack, New Jersey. It\'s really an adventure from beginning to end! It is a 48 mile loop and it basically took me an entire day. I stopped at Riverbank State Park to take some extremely artsy photos. It was a short stop, though, because I had a really long way left to go. After a quick photo op at the very popular Little Red Lighthouse, I began my trek across the George Washington Bridge into New Jersey. The GW is actually very long - 4,760 feet! I was already very tired by the time I got to the other side. An hour later, I reached Greenbrook Nature Sanctuary, an extremely beautiful park along the coast of the Hudson. Something that was very surprising to me was that near the end of the route you actually cross back into New York! At this point, you are very close to the end.';
sentenceTotal(story)
=> 13
There a several strange things about you question so I'll do it in 3 steps :
First step : The syntax.
What you wrote is the assignement to a const of an anonymous variable. So what it does is :
Create a const name 'sentenceCount'
To this const, assign the anonymous function (words) => {...}
Now you have this : sentenceCount(words){...}
And that's all. Because what you wrote : ()=>{} is not the calling of a function, but the declaration of an anonym function, you should read this : https://www.w3schools.com/js/js_function_definition.asp
If you want a global total, you must have a global total variable(not constant) so that the total isn't lost. So :
let sCount = 0; //<-- have sCount as a global variable not a const
function isEndOfSentence(word) {
if (word[word.length-1] === "." || word[word.length-1] === "!" || word[word.length-1] === "?") {
sCount += 1;
};
};
If you are forbidden from using a global variable (and it's best to not do so), then you have to register the total as a return of your function and store the total in the calling 'CountWords(sentence)' function.
function isEndOfSentence(words) {...}
callingFunction(){
//decalaration
let total;
//...inside your loop
total += isEndOfSentence(currentWord)
}
The algorithm
Can you provide more context as how you use you function ?
If your goal is to count the words until there is a delimiter to mark the end of a sentence, your function will not be of great usage .
As it is written, your function will only ever be able to return 0 or 1. As it does the following :
The function is called.
It create a var called sCount and set it to 0
It increment or not sCount
It return sCount so 1 or 0
It's basically a 'isEndOfSentence' function that would return a boolean. It's usage should be in an algorithm like :
// var totalSentence = 0
// for each word
// if(isEndOfSentence(word))
// totalSentence + totalSentence = 1
// endfor
Also this comes back to just counting the punctuation to count the number of sentence.
The quick and small solution
Also I tried specifically to keep the program in an algorithm explicit form since I guess that's what you're dealing with.
But I feel that you wanted to write something small and with as little characters as possible so for your information, there are faster way of doing this with a tool called regex and the native JS 'split(separator)' function of a string.
A regex is a description of a string that it can match to and when used can return those match. And it can be used in JS to split a string:
story.split(/[?!.]/) //<-- will return an array of the sentences of your story.
story.split(/[?!.]/).length //<-- will return the number of element of the array of the sentences of your story, so the sentence count
That does what you wanted but with one line of code. But If you want to be smart about you problem, remember that I said
Also this comes back to just counting the punctuation to count the number of sentence.
So we'll just do that right ?
story.match(/(\.\.\.)|[.?!]/g).length
Have fun here ;) : https://regexr.com/
I hope that helps you ! Good luck !

Hovertool Bokeh "Cannot read property"

My problem is that in Chrome, when I have my cursor on my histogram hover my data, I have this error :
Uncaught TypeError: Cannot read property '0'
There is my code :
hist, edges = np.histogram(data,bins=3000)
plot = quad(
top=hist,
bottom=0,
left=edges[:-1],
right=edges[1:],
fill_color="#036564",
line_color="#033649",
tools="pan,wheel_zoom,box_zoom,reset, hover",
x_range=[-0.5,3.5],
plot_width=1100,
title="",
)
hover = plot.select(dict(type=HoverTool))
hover.tooltips = [('index','$index')]
resources = Resources("inline")
plot_script, plot_div = components(plot, resources)
html_script = mark_safe(encode_utf8(plot_script))
html_div = mark_safe(encode_utf8(plot_div))
figure()
return html_script, html_div
"data" is a array like this :
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.24,1,1.32,1,2,3]
I tried to add a "source" in the quad, changed the figure, changed my code for the one on the documentation but I still have my error.
For information, everything is working very well, except the hover tool.
Indeed, in the "hover box" I want another informations then just "index", but it's just for testing.
Thanks for reading !
Sorry I missed this earlier. You have uncovered a small bug with the hover tool that is particular to quad glyphs. In the mean time you can add hover.snap_to_data = False to get it to work.
Here is a GH issue you can track for the full solution:
https://github.com/bokeh/bokeh/issues/1644
A fix should be in the 0.7.1 release next Monday.
Also BTW, you are using a deprecated API. You should now write code like:
p = figure(...)
p.quad(...)

SML Issue with a function creating another function

Keep in mind, this is part of a homework assignment - so please, no direct answer. I just need some help in finding out the answer, so a link to a tutorial to help me understand the material would be great.
SML code:
datatype 'ingredient pizza =
Bottom
| Topping of ('ingredient * ('ingredient pizza));
datatype fish =
Anchovy
| Shark
| Tuna;
(* Testing Pizza Objects *)
val my_pizza1 = Topping(Tuna, Topping(Shark, Topping(Anchovy, Bottom)));
val my_pizza2 = Topping(Shark, Topping(Tuna, Topping(Anchovy, Bottom)));
val my_pizza3 = Topping(Anchovy, Topping(Shark, Topping(Tuna, Bottom)));
(* My Function Start *)
fun rem_ingredient Bottom = Bottom
| rem_ingredient(t) = fn(Topping(p)) => Topping(t, rem_ingredient(p))
| rem_ingredient(Topping(t,p)) = Topping(t, rem_ingredient(p));
(* My Function End *)
If I call the function rem_ingredient with 1 parameter
val rem_tuna = rem_ingredient Tuna;"
I should get a function that can then call
rem_tuna my_pizza3;
to remove Tuna from pizza3
If I call the same function with 2 parameters
rem_ingredient Tuna my_pizza2;
I should directly remove Tuna from the pizza2 object with the 2 parameters.
The Problem:
I keep getting the error: syntax error: replacing EQUALOP with DARROW on the 3rd constructor of rem_ingredient, I know I'm missing something that is probably obvious. We just start learning SML last week in Programming Languages and I'm still trying to understand it. Anyone pointing me in direction would be appreciated.
Again, no direct answer please, I want to learn the material, but I'm not sure what I'm trying to fix.
To get rid of the syntax error you need to put parentheses around the fn expression (since the following | pattern is otherwise taken to be part of the fn).
However, that is not your real problem here. The function as written does not have a consistent type, because the second case returns a function while the others do not.

Mathematica 9 - function definition problems

Please excuse the beginner question. I couldn't find an appropriate answer in any Mathematica tutorial.
I am confused why a definition as a function or a definition in terms of a simple replacement produce different results. Consider this example (Mathematica 9 code):
In[397]:= ClearAll["Global`*"]
In[398]:= Test := 3 c^2 + d^4
In[399]:= v[f_] := D[f, c]
In[400]:= v[Test]
Out[400]= 6 c
The first definition of this simple derivative function "v" acting on a variable is fine. Defining a replacement Test = ... to replace the variable produces the expected result (It derives 3c^2+d^4 with respect to c and answers 6c).
However if I define a function instead of a simple replacement this does not work:
In[401]:= TestFunction[a_, b_] := 3 a^2 + b^4
In[403]:= vFunction[f_[a_, b_]] := D[f[a, b], a]
In[405]:= vFunction[TestFunction[a, b]]
Out[405]= \!\(
\*SubscriptBox[\(\[PartialD]\), \(3\
\*SuperscriptBox[\(a\), \(2\)]\)]\((3\
\*SuperscriptBox[\(a\), \(2\)] +
\*SuperscriptBox[\(b\), \(4\)])\)\)
Why is that? I am risking to look like a moron here, but please enlighten me!
For your convenience, I uploaded a copy of my workbook here
Thanks a lot,
Michael
Do this instead
vFunction[f_,a_,b_]:=D[f[a,b],a];
and when you need derivatives simply use vFunction[TestFunction,a,b] to get it.
When you write down f[x], it means the evaluated value of f with argument value x. So, f[x] is technically not a function anymore. What you want as the argument of vFunction[] is the function TestFunction, not the evaluated value.

Octave and multiple Bode plots

I'm teaching myself Octave and as a motivational exercise am attempting to create some Bode plots. I'd like to create a plot that has multiple curves for different values of a parameter in a transfer function, for example the time constant of a simple RC filter. I'm trying to do it as follows:
tau = [1,2,3]
for i = tau
g(i) = tf(1,[tau(i),1])
endfor
bode(g(1),g(2),g(3))
But it doesn't work, I get the error
error: octave_base_value::imag (): wrong type argument `struct'
However, it works fine if there are not multiple arguments to the bode command and the last line is simply:
bode(g(1))
Any advice as to where I've gone wrong would be appreciated - is there a better way to do what I want to do?
I was able to do it with the following sequence (with octave 3.2.4 on debian):
bode(g(1))
set (findobj (gcf, "type", "axes"), "nextplot", "add")
bode(g(2))
bode(g(3))
The second command is similar to hold on but it works when there are subplots; I found it here.
Using your own code:
subplot(211), hold on
subplot(212), hold on
tau = [1,2,3]
for i = 1:length(tau),
g(i) = tf(1,[tau(i),1]);
bode(g(i))
endfor
The problem with this solution is that you cannot identify a specific plot. You cannot access figure properties through bode() function directly.
Here then a plausible solution to bring you colorful plots:
colorsplot = ["b","m","g"]
tau = [1,2,3]
g = tf(1,[tau(1),1]);
[mag, ph, w] = bode(g);
subplot(211), semilogx(w,20*log(mag)), hold on
subplot(212), semilogx(w,ph), hold on
for i = 2:length(tau),
g = tf(1,[tau(i),1]);
[mag, ph, waux] = bode(g,w);
subplot(211), semilogx(w,20*log(mag),colorsplot(i))
subplot(212), semilogx(w,ph,colorsplot(i))
endfor