I want to calculate the irrational number, expressed by the following formula in gap:
3^(1/7). I've read through the related description here, but still can't figure out the trick. Will numbers like this appear in the computation of the character table and corresponding (unitary) representations?
P.S. Basically, I want to figure out the following question: For the computation of the character table and corresponding (unitary) representations, what is the minimum complete set of atomic irrational numbers used to express the results?
Regards,
HZ
You can't do that with GAP's standard cyclotomic numbers, as seventh roots of 3 are not cyclotomic. Indeed, suppose $r$ is such a root, i.e. a rot of the polynomial $f = x^7-3 \in \mathbb{Q}[x]$. Then $r$ is cyclotomic if and only if the field extension \mathbb{Q}[x] is a subfield of a cyclotomic field. By Kronecker-Weber this is equivalent to that field being an abelian extension, i.e., the Galois group is abelian. One can check that this is not the case here (the Galois group is a semidirect product of C_7 with C_6).
So, $r$ is not cyclotomic.
Related
It is more of a language question as I am working on translating a technical document into English. The current description is as below:
The value of this parameter can be an array of integers or an integer range. The integers start from 0. Examples: '0-2' and '1,2,3'.
My question is: does this description cause confusion to imply that the value must start from 0 which makes the second example look incorrect on first sight?
The OASIS Working Draft 01 for ALFA (alfa-for-xacml-v1.0-wd01) of 10 March 2015 says about constant values
3.15 Constant Values
Constant values can appear in the policy expressions. ALFA supports constants of type strings, integers,
doubles and Booleans directly. Strings are quoted with single or
double quotes. Integers consist of a number and optionally a minus
sign. Double consists of a number with a decimal dot and optionally a
minus sign. Booleans consist of the value true and false, without
quotes. Other datatypes are represented using a string followed by a
colon and the name of the datatype..
What that means is, you can use constant values like in that example (while report is the constant value):
target clause requestedType == "report"
But the thing is, once the ALFA files grow and you have written the constant value report all over, you might want to change the constant value into let's say my.company.attributes.medicalReport. In order to do that you have to find and replace all occurrences of the constant value.
Therefore (for the sake of avoiding redundancy) constants have been invented in other languages, where you define smth. like
const string REPORT_TYPE = "my.company.attributes.medicalReport"
or even more performant:
const integer REPORT_TYPE_ID = 3
or even more elegant:
const enum SUBJECT_TYPES { PATIENT, USER, EXAM, REPORT }
With those constants being defined I could write my target like:
target clause requestedType == REPORT_TYPE_ID
Does ALFA support constants or is there a way to "emulate" them (maybe a function that returns the desired value)?
Not yet! It is definitely a feature we want to have. We've had similar requests so stay tuned.
If I have a text string to be vectorized, how should I handle numbers inside it? Or if I feed a Neural Network with numbers and words, how can I keep the numbers as numbers?
I am planning on making a dictionary of all my words (as suggested here). In this case all strings will become arrays of numbers. How should I handle characters that are numbers? how to output a vector that does not mix the word index with the number character?
Does converting numbers to strings weakens the information i feed the network?
Expanding your discussion with #user1735003 - Lets consider both ways of representing numbers:
Treating it as string and considering it as another word and assign an ID to it when forming a dictionary. Or
Converting the numbers to actual words : '1' becomes 'one', '2' as 'two' and so on.
Does the second one change the context in anyway?. To verify it we can find similarity of two representations using word2vec. The scores will be high if they have similar context.
For example,
1 and one have a similarity score of 0.17, 2 and two have a similarity score of 0.23. They seem to suggest that the context of how they are used is totally different.
By treating the numbers as another word, you are not changing the
context but by doing any other transformation on those numbers, you
can't guarantee its for better. So, its better to leave it untouched and treat it as another word.
Note: Both word-2-vec and glove were trained by treating the numbers as strings (case 1).
The link you provide suggests that everything resulting from a .split(' ') is indexed -- words, but also numbers, possibly smileys, aso. (I would still take care of punctuation marks). Unless you have more prior knowledge about your data or your problem you could start with that.
EDIT
Example literally using your string and their code:
corpus = {'my car number 3'}
dictionary = {}
i = 1
for tweet in corpus:
for word in tweet.split(" "):
if word not in dictionary: dictionary[word] = i
i += 1
print(dictionary)
# {'my': 1, '3': 4, 'car': 2, 'number': 3}
The following paper can be helpful: http://people.csail.mit.edu/mcollins/6864/slides/bikel.pdf
Specifically, page 7.
Before they use an <unknown> tag they try to replace alphanumeric symbol combination with common pattern names tags, such as:
FourDigits (good for years)
I've tried to implement it and it gave great results.
I'm trying to write binary text files from a data frame in Julia using something along the lines of:
for x in RICT["$i"]["Sick"]
write(f9, convert(Int16, x ))
and everything works nicely except for when it comes to NA values. Missing values are treated as NA it seems, and I know that there are different ways of handling such values using the data frames package. Does anyone have any experience with these NAtypes? Should I convert the NAtypes to a more conventional type and then write them in? As always any help is much appreciated.
If you are writing a 16-byte integer value, there's no canonical representation of "blank", so you'd have to pick a special 16-byte integer value that represents NA. A common choice for this kind of thing is the smallest representable value – in this case typemin(Int16) == -32768. You can generalize this to other signed integer types.
For example:
code = '7777-5';
input = code.substring(0, 4); // Returns '7777'
checkdigit = f(input); // f() produces a checkdigit
assert.areEqual(code, input + "-" + checkdigit)
Is there a technical term for input used above?
Specifically I'm calculating checkdigits for ISBNs, but that shouldn't effect the answer.
Is "original number excluding the check digit" technical enough? :)
Actually, it's often the case, as in the link you posted, that the check digit or checksum ensures a property about the full input:
...[the check digit] must be such that the sum of all the ten digits, each multiplied by the integer weight, descending from 10 to 1, is a multiple of the number 11.
Thus, you'd check the full number and see if it meets this property.
It's "backwards" when you're initially generating the check digit. In that case, the function would be named generate_check_digit or similar, and I'd just name its parameter as "input".
Although I am not sure if there is a well-known specific technical term for the input, what LukeH suggested (message/data) seems common enough.
Wiki for checksum:
With this checksum, any transmission error that flips a single bit of the message, or an odd number of bits, will be detected as an incorrect checksum
Wiki for check digit:
A check digit is a form of redundancy check used for error detection, the decimal equivalent of a binary checksum. It consists of a single digit computed from the other digits in the message.