The solutions to certain quintics (or even higher order equations) with integer coefficients are not expressible using finite nested radicals. I believe these are called 'unsovlable' quintics.
But the solutions to these equations are still by definition algebraic numbers.
Edit: For people saying that numbers expressible in terms of finite terms of radicals should be called 'FOO' and thus what I want should be called non-FOO, non-FOO would also include the transcenyoudental numbers, something I explicitly don't want. So I guess you would have to call them non-FOO algebraic numbers.
For example, x + 5 = 10. We subtract 5 from 10 and also subtract 5 from itself, and the answer will give x = 5. Why do we have to do it on both sides? Why can't we just subtract 5 from 10?
I solved my earlier problem, but have a new one. I have COVID-19 data for my state, with one column per county as well as a total:
total_infected <-
infected_df %>%
arrange(Date) %>%
filter(Date >= as.Date("2020-03-06"))
# A tibble: 112 x 100
Date Total `ERROR!!!` Anderson Bedford Benton Bledsoe
<dttm> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2020-03-06 00:00:00 1 0 0 0 0 0
2 2020-03-07 00:00:00 1 0 0 0 0 0
3 2020-03-08 00:00:00 3 0 0 0 0 0
4 2020-03-09 00:00:00 4 0 0 0 0 0
5 2020-03-10 00:00:00 7 0 0 0 0 0
6 2020-03-11 00:00:00 9 0 0 0 0 0
7 2020-03-12 00:00:00 18 0 0 0 0 0
8 2020-03-13 00:00:00 26 0 0 0 0 0
9 2020-03-14 00:00:00 32 0 0 0 0 0
10 2020-03-15 00:00:00 39 0 0 0 0 0
# β¦ with 102 more rows, and 93 more variables:
I have similar tibbles for total/new infected, total/new recovered, total/new dead.
I would like to compute total sick per county per date, using the formula:
total_sick = total_infected - total_recovered - total_deaths
How do I take my three tibbles and do simple +/- operations across entire tibbles like that?
I also have a tibble with population data per county:
> pop_2018
# A tibble: 1 x 96
Total Anderson Bedford Benton Bledsoe Blount
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 6.65e6 75775 47558 16112 14602 128443
# β¦ with 90 more variables:
How would I compute total_infected per capita? That should be:
total_infected / population
But the actual R syntax eludes me. Any suggestions are greatly appreciated.
If they exist, are they useful in any way? If they don't, or more specifically, if nobody wanted to think about them, why is it so?
EDIT: Thanks for all the answers! Even if I hardly understand the formal concepts behind the examples, I got an insight to this topic and hope that I will be able to understand it in the nearer future.
Which algebraic operation to get from left to right?
https://preview.redd.it/rhjduhqrt6921.png?width=170&format=png&auto=webp&s=3a89c6c06b540ed61b1bc43e43c53de0fd358f4f
I donβt fully understand why, but for some reason people are shown quite an interest to my previous post (https://www.reddit.com/r/math/comments/abllub/is_there_a_way_to_generalize_this_function_for/ ) so I feel an ought to explain what it was really about and probably initiate more general discussion about the idea I had. At first I must warn you that I am not a mathematician, so feel free point on any of my mistake or misunderstanding if there would any.
Iβve been working on generative algorithm that have a peace of code where I define several operations of different orders and I thought that there should exist some elegant function that could generate an operation over two numbers of a given order n. That lead me to wikipedia page of hyperoperations and specifically Albert Bennettβs definition of them that looks like exactly what I was looking for.
On Wiki page there is recursive definition of it, but playing with it a little bit I found out that there is more general definition that constructs as following:
At first we should define Bennett function Ξ²^(n) like:
https://preview.redd.it/v34uyoesr2821.png?width=287&format=png&auto=webp&s=1115f5828b6b331654373ac91bf3d55bf85a2358
And then define Bennett operation B^(n) like:
https://preview.redd.it/p06ui6kag2821.png?width=394&format=png&auto=webp&s=23c1f2117c0c0d04642b8544a05cadc20035b34d
For any n from β€.
Now we can play around with it, and first thing that I notice was an existence of zero and negative elements for any B^(n) defined as:
Z^(n) = Ξ²^(n) (0)
N^(n) = Ξ²^(n) (iΟ)
These elements have following properties
B^(n) (x, Z^(n)) = x
B^(n-1) (x, B^(n) (x, N^(n))) = Z^(n-1)
So, Z^(n) acts as identity element over operation B^(n) and N^(n) generate Inverse element for given x over operation B^(n-1)
Here is table of zeros and negatives for some operations:
| n | As Bennett operation | Corresponding expression | N^(n) | Z^(n) |
|---|---|---|---|---|
| -1 | B^(-1)(a, b) | ln(e^(a) + e^(b)) | ln(iΟ) | ln(0) |
| 0 | B^(0) (a, b) | a + b | iΟ | 0 |
| 1 | B^(1) (a, b) | a * b | -1 | 1 |
| 2 | B^(2) (a, b) | a^(ln(b)) | e ^(-1) | e |
| 3 | B^(3) (a, b) | exp(a^(ln(ln(b)))) | e^(1/e) | e^(e) |
Note: it ok that there is no solution for ln(0) because we still can use it as zero element like so
B^(-1)(x, B^(0)(x, N^(0))) = Z^(-1) β ln(e^(x) + e^(x + iΟ)) = ln(e^(x) \
... keep reading on reddit β‘There are lots os interesting structures defined by a set and one or more binary operations, like Monoids, Groups and Rings.
Can you define any unique and/or interesting algebraic structures using ternary operations?
I know this has been asked on this very sub and on other forums and honest I've probably read some 25 threads regarding this topic over the past couple of years, but I can never get a satisfactory answer. I've asked my (community college) teachers, where I completed my calculus series and didn't get a good answer either.
What's the deal with multiplying and dividing with dy, dx, etc? So many texts, videos, and instructors make strange comments, and ominous warnings regarding these maneuvers, ie, when multiplying and dividing by them as if they were variables. Some of those videos which are posted on this sub and are mentioned to be of good quality, like 3blue1brown's "essence of calculus", always have explanations that involve moving these around willy nilly, and Grant even goes out of his way to mention that dx is a finite, non zero value, which seems to allow for this. But there are always commenters in the videos that recoil in horror.
I really want to understand what's the deal with dx. My math knowledge is limited to calc 1-3 and diff eqs, but with As in all and I generally have striven to really understand as much as could. I know that lower math classes do not stress proofs and I do not remember everything as some of that stuff just never gets used, (I'm not a math major) but I'm just trying to demonstrate my ability and willingness to learn. I think if I were given a straightforward answer, without the assumption that I understand any higher level maths, I could understand this (baffling) concept. How do differential forms tie in? Is dx always dx? Why can I or can't I multiply/divide with these terms? When can I? Why don't teachers have a good answer? Is this something I should be banging my head over?
To the left of it, that is..
I'm studying abstract algebra (mostly groups) at the moment and was wondering whether there was research into abstract algebraic objects that generalise numbers under the operations of addition, multiplication and taking powers? Obviously you can take integer powers within any ring (or any group depending on your notation), but I am looking for structures that have two commutative operations, the second of which is distributive over the first, and a third not necessarily commutative one which is distributive over the second one. Let me know if there's anything I need to specify, am interested to hear your replies!
This is a really basic question. However, I don't think it's very well defined (Wikipedia says it's the traditional operations which is rather narrow!) and we need a better way of defining stuff like this.
Hello
Could someone please explain me how they got to the formula of doppler effect on line 3? I would like to know which algebraic operations they did to get from line 2 to the formula on line 3.
http://imgur.com/Z7BjO82
I've been looking for a while, and still don't know
Actually this problem came when writing code for a grid based display of text. A certain 'remainder' value I end up with assigning the following algebraic formula:
(x - y % x) % x
Where '%' is the remainder operator in C++. If remainder had the same distributive properties as multiplication it could be rewritten as
x%x - y%(x%x) -or- (x%x)%(1 - y)
which simplifies to
-(y%0) -or- 0
Which is undefined in the first case. Is there a way to tell if what I've come up with is 'well formed', or if there is a better way of writing it besides the fact that it seems to work for what I need?
The Cayley-Dixon algebras are by far the most important algebras used today, and because reals have the multiplicative identity, the number of imaginary parts always has to be 2^n - 1 for n >= 0 (0 for reals, 1 imaginary part for complex numbers, 3 for quaternions, 7 for octonions, etc.). But that's an axiomatic assumption that any number's multiplication has to be binary. With a non-commutative ternary operator, where under the reals multiplication is commutative to make the appearance of binary multiplication, an algebra could "divide" with 1 real part and 5 non-real parts.
The first is a list of texts I felt were really useful, with comments about what is useful to use them for. Some have good exercises, others good accessibility, others good for dipping into when encountering a research problem.
The second is a list of online resources which are useful to use. This could particularly be interesting for students outside of operator algebras, although I included a few online groups and discord servers that have a good active discussion for research.
I am looking to create a small, templated linear algebra library as a personal project and have recently discovered BLAS and libraries such as OpenBLAS which make use of SIMD instructions to perform vector operations faster.
My question is, should I be using these libraries to compute e.g. dot products, matrix products etc.? Is that there ultimate intended use? If so, is OpenBLAS a reasonable choice for this? Or should I use something like usimd to compute these using SIMD instructions directly myself?
Also, do libraries like Eigen and GLM use BLAS under the hood, or do they directly go to SIMD instructions?
How do I put the following matrix into a tridiagonal one using partial pivoting and row operations
| 0 | 1 | 3 |
|---|---|---|
| -1 | 2 | 8 |
| 2 | -1 | 5 |
First I swapped rows 1 and 3 to get the largest absolute pivot/diagonal entry
| 2 | -1 | 5 |
|---|---|---|
| -1 | 2 | 8 |
| 0 | 1 | 3 |
Then I took R2 <-- 2R2 + R1
| 2 | -1 | 5 |
|---|---|---|
| 0 | 3 | 21 |
| 0 | 1 | 3 |
I'm unsure how to get that 21 in the bottom position in the last column as it will effect the diagonal entry in column 2. Unless I am doing this wrong, any help is appreciated.
How could I write an algorithm that compares the time and space complexity of mathematical operations? specifically, Calculus vs Linear algebra?
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