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Old 2021-05-19, 13:10   #23
drkirkby
 
"David Kirkby"
Jan 2021
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I don't see anyone mentioned

https://ocw.mit.edu/index.htm
which covers all topics.

There's a number theory course on YouTube
https://www.youtube.com/watch?v=19SW3P_PRHQ
based around a Dover book. I have not watched much of the video. As the author points out, Dover books are quite reasonably priced. I have bought the book, and intend watching the video, using that in conjunction with the book.

The Open University put a bit of a course on number theory on their website.

https://www.open.edu/openlearn/scien...tion-0?intro=1
However, I think the amount is pretty small compared to other resources mentioned where people have put whole books, or several chapters of books freely available. As such, the OU course might not be worth bothering with.

Dave (engineering graduate).
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Old 2021-05-25, 12:21   #24
Nick
 
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Thanks to Mike for the following tip:
https://artofproblemsolving.com/arti...les/SatoNT.pdf

Slight caveat on this one: the puzzles are good but the theory is not always completely precise or optimal.
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Old 2021-05-25, 13:06   #25
drkirkby
 
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Quote:
Originally Posted by Nick View Post
Slight caveat on this one: the puzzles are good but the theory is not always completely precise or optimal.
You got me thinking, that perhaps a forum that is devoted to comments/errata about number theory text books would be useful. One thread per text book.

Dave
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Old 2021-07-26, 08:29   #26
Nick
 
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For anyone looking for a fun bit of summer reading:
The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

I think the above link is open access, but if you have any trouble there is also an arXiv version here.
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Old 2021-08-02, 04:12   #27
jwaltos
 
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Quote:
Originally Posted by Nick View Post
For anyone looking for a fun bit of summer reading..
Hi Nick,
I haven't been doing any math for the last several months with the exception of occasional sporadic jottings of approaches to solving a multivariate equation of degree 4. ie X^4*(a*b*c*d*e+...)+X^3*(b^2*d*e...)+..+X^0*(c*d..). The variables (there are up to five with their associated combinations) within the brackets where each variable does not exceed degree 2 but they may all be of degree 2.

Do you know of any math books, notes or approaches that apply to solving such expressions? I have a few that apply but I'm wondering about some other clever ideas that may exist. I've considered operational calculus, umbral calculus, trig. substitutions, iterative programming... Since this is a fourth degree form, I was also considering moments of inertia, chemical equations.. any thoughts you may have would be appreciated.

Just to provide some context, all of the variables exist as part of a Cartesian coordinate frame (with a slight modification). The expression exists under a radical so it represents solution(s) which may be +ve/-ve or complex...algebraic in any case. The variables composing the expression were combined from several merged general equations. There are some interesting facets regarding the interpretation of this degree four equation and I would like to obtain as broad a perspective as possible by encapsulating it via mathematical theory and utility. The coordinate frame mentioned previously represents the scaffolding for actual measurable figures. Further, that coordinate frame exists in conjunction with another required coordinate frame which has its own special features, some of which extend directly into lattice theory. Perhaps this context helps or perhaps not but at the very least I hope it provides you with an idea that a solid theoretical foundation exists within some elementary mathematics. Without being disingenuous, I'd like the final construct to display some ingenuity when the crank is turned and meaningful numbers are either turned out in one direction or meaningful concepts are turned out in the reverse direction...yes, there is also some thermodynamic reasoning involved with a 50/50 probability of some quantum logic involved as well.

Last fiddled with by jwaltos on 2021-08-02 at 05:09
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Old 2021-08-02, 09:39   #28
Nick
 
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The short answer is that if you are working with a single polynomial equation of degree
at most 4 over the complex numbers, then you can choose any values you like for
all variables except one, and then find the possible values of the remaining variable
by the historical methods passed down to us by Cardano and Ferrari.

To gain more insight into such problems means delving into Algebraic Geometry,
which is a vast and fascinating area of modern mathematics. It takes time to master,
however: where Differential Geometry relies on Calculus (which everyone knows already),
Algebraic Geometry relies on Commutative Algebra (which is new to most people).
A good introduction is Miles Reid's Undergraduate Algebraic Geometry.
For the number theory point of view, the book Rational Points on Elliptic Curves
is a good place to start (but won't cover quartic equations, of course).
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Old 2021-08-04, 01:30   #29
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Quote:
Originally Posted by Nick View Post
The short answer is that if you are working with a single polynomial equation of degree
at most 4 over the complex numbers, then you can choose any values you like ..
Thanks for responding Nick. Unfortunately it didn't really help me out. I'm a little more advanced than the books you recommended (or at least think I am) and where you state "you can choose any values you like" that is the crux of the matter. There must be a guiding principle which must limit my options (free will concept should not be in play) to specific numeric values of the parameters related to the value(s) I'm seeking. I'm looking initially at "real world" applications, here is mathstackexchange link: https://math.stackexchange.com/quest...rtic-equations

Theoretically any univariate polynomial can be considered as a base number system. Consider the single parameter as 10 and simplify the coefficients to represent a base 10 number..and so on. Multivariate systems..same deal. Extending this concept within base 10..consider digit sums and the picene hydrocarbon. Then consider the inverse tessellation of the same. Then consider the geometric form as an infinite sheet or surface...and so on down this particular rabbit hole.

Considering the difference of squares, sum of squares, primes that are and aren't the sum of two squares geometrically, such relations provide some simple figures. Embellishing these figures properly leads to reciprocity and a few other gems. Given that the general quartic can be solved via radicals is a particularly interesting limit point.

In any case, dimensional analysis and measurable objects/influences is what I'm looking at presently. Prior to this I was examining the general cubic parabola from which I have since elucidated some interesting facts.

Just to keep in line (pun intended) with the purpose of this thread here are a few "light read" books containing some of the above ideas:
Geometry and the Imagination, Hilbert and the other guy.
Explorations in Mathematical Physics, An Elegant Language...,Koks
Curves for the Mathematically Curious, Havil
Proofs from the Book, 6th Edition

..and not so light..
https://scholarworks.umb.edu/cgi/vie...masters_theses
https://uma.ensta-paris.fr/files/pub...imon-Petit.pdf

Last fiddled with by jwaltos on 2021-08-04 at 02:05
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Old 2021-08-04, 09:54   #30
Nick
 
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Quote:
Originally Posted by jwaltos View Post
Thanks for responding Nick. Unfortunately it didn't really help me out...
Well you have a very individual perspective so any suggestions are bound to be a bit hit-and-miss!

Mathematically, quartic polynomials tend to arise from 4-dimensional spaces,
for example as the determinant of a 4x4 matrix. And a key difference between those and
3-dimensional spaces is that 4 is even - yes, it sounds trivial, but it has deeper consequences.
Calculate the volume of the unit ball in n dimensions, for example, and you will see one pattern
for the odd dimensions and another pattern for the even ones.
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Old 2021-08-29, 07:55   #31
pinhodecarlos
 
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Ok Nick. I’m home on A/L and I’ve already packed my Piskunov books. I will re study them but I want then to think about next step. I always wanted to take a math degree but to get a job it would be better to have a engineer degree. Need to go back to my dream. Please I want your guidance.
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Old 2021-09-09, 09:50   #32
Nick
 
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Quote:
Originally Posted by pinhodecarlos View Post
Ok Nick. I’m home on A/L and I’ve already packed my Piskunov books. I will re study them but I want then to think about next step. I always wanted to take a math degree but to get a job it would be better to have a engineer degree. Need to go back to my dream. Please I want your guidance.
Apologies for the late reaction: we have been away on holiday and I have only now seen your post.

University-level mathematics for mathematicians differs from mathematics for engineers or natural scientists
in its emphasis on abstraction and rigour. So alongside Piskunov it would be a good idea to read
"A Companion to Analysis" by T. W. Körner. Chapter 1 is available free here.

Similarly, you probablly have a good knowledge of linear algebra from an applied point of view, but may gain
something from a more abstract perspective. Prof. Ronald van Luijk has written his lecture notes in English
and made them freely available here, with follow-up notes by him and Michael Stoll available here.

There are various good books on the theory of groups, for example the book "Groups and Symmetry" by M. A. Armstrong.
This is where you start to get into the mathematics that engineers do not usually see!

If you are looking for a broad, accessible overview of modern pure mathematics, then the "Princeton Companion to Mathematics"
edited by Prof. Tim Gowers is worth dipping into.
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