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2
votes
3answers
41 views

Best resource to learn quadratic reciprocity?

I took a very basic intro to number theory course last semester. We learned about many of the standard topics (gcd, primes, cryptography, congrences, pythagorean triples, etc), but we never learned ...
1
vote
1answer
36 views

Logic of steps in proofs

Whenever I read a textbook or get help with a proof, I'm always suggested a very obscure and tricky step which usually helps me immediately solve the rest of the proof because everything else seems to ...
6
votes
2answers
94 views

How would you change math notation? [on hold]

Mathematical notation has been evolving for hundreds of years (really thousands, I guess, but most of the noteworthy examples seem to be only a few hundred years old). Sometimes we are stuck with old ...
1
vote
1answer
43 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
21
votes
13answers
631 views

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend ...
0
votes
0answers
21 views

Countable Baye's theorem?

Disclaimer: If this is a foolish question, I'm sorry.. this is the first time I've looked at probability theory in very many years, and have begun to re-read everything from scratch... Question: If ...
4
votes
4answers
262 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
0
votes
0answers
121 views

What is the Kadison-Singer Conjecture

I recently heard on the news that Adam Marcus, Daniel A Spielman, Nikhil Srivastava have won the George Pólya Prize for proving the 1959 Kadison-Singer Conjecture. So I'm curious...what is this ...
11
votes
6answers
586 views

Understanding mathematical texts

Please could you comment on following: I always wanted to know what mathematicians mean by "understanding a piece of mathematics". For example, I have just finished the second chapter from Rudin's ...
3
votes
0answers
76 views

Abstract algebraic geometry vs complex algebraic geometry

Sorry in advance if my question is not precise enough. I'm currently trying to study algebraic geometry on my own. I've started by trying to read Harsthorne and Liu's book. And i found it very ...
2
votes
1answer
30 views

Frequency computation

I generate pulses using microcontroller with following method: Set $A=0$ With frequency $F$ execute the following code: 2.1. Add $S$ to $A$ 2.2. If $A\ge N$ then produce a pulse and ...
1
vote
1answer
44 views

Other Interesting solutions to $a=bq+r$? [on hold]

The division algorithm says $a=bq+r$, with $r$ between $0$ and $b$. Are there interesting restrictions on $r$ using number-theoretic properties that make the equation $a=bq+r$ hold, or hold with ...
5
votes
5answers
365 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
1
vote
3answers
94 views

a second course in abstract algebra

I recently read an abstract algebra textbook, "A first course in abstract algebra" by John Fraleigh. I am interested in continuing to do some more self studying. What is a good book for a second ...
2
votes
1answer
63 views

What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
1
vote
1answer
84 views

Where to post discovered formulae? [on hold]

I have discovered an alternate formula for the Fibonacci sequence and I would like to find a way in which I can present this. Please could you give me suggestions on how I can go about posting this ...
6
votes
2answers
495 views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
10
votes
0answers
153 views

Are NSA Mathematicians second-rate? [on hold]

I recently read that the National Security Agency (NSA) is the single largest employer of mathematicians in the United States. Although spies and high-level secret government agents are glamorized in ...
0
votes
0answers
32 views

Koblitz - Are chapters III & IV independent of I & II

I am interested in learning about Modular forms and have heard many great things about Neal Koblitz's Introduction to Elliptic Curves and Modular Forms. However, Koblitz doesn't discuss modular forms ...
0
votes
0answers
34 views

Which Trigonometry Book is Recommended? [duplicate]

I'm taking trigonometry for this upcoming fall, and I want to get a good head start like I did with statistics a while back. I was recommended Cynthia Young' s Trigonometry book and Loney's book. ...
7
votes
7answers
208 views

Is $\mathbb{C}$ equal to $\mathbb{R}^2$?

Complex numbers are usually formally defined as pairs of real numbers. Although there are operations on $\mathbb{C}$, such as complex multiplication, which are not found in operations usually applied ...
8
votes
1answer
59 views

Parametrizing Walks on Sphere and Torus

This question is very underdeveloped, but I was wondering if there was a map from the sphere to the torus which preserves length of closed curves? I was just thinking about taking a walk on a ...
0
votes
0answers
54 views

What is way of getting good at math without math? [closed]

What is way of getting good at math without math? I know this question might seem ridiculous, but It have been said that listening to Mozart can improve your math and science skills. I do not ...
10
votes
1answer
179 views

Soviet Russian Mathematical Books

The introductory part of the book briefly describes the popularity of mathematics in Soviet Russia, touches on Russian mathematical circles and generally how Russian society took to mathematics in a ...
1
vote
3answers
346 views

I'm forgetting maths.. [closed]

I'm 14, and I'm the Math topper in my grade. But suddenly, I've started loosing confidence, and I've started forgetting maths.. I am looking for a good solution. (If it helps, I'm forgetting a bit of ...
3
votes
0answers
98 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
2
votes
1answer
88 views

The J programming language: is it useful for mathematics?

I just stumbled upon the J programming language, which has the description: J is particularly strong in the mathematical, statistical, and logical analysis of data. It is a powerful tool in ...
0
votes
3answers
146 views

How do we define equality in real numbers?

How do we define equality in real numbers? I know in logic we define equality by Leibniz's law. $$ \forall x \forall y[x=y \rightarrow \forall P(Px \leftrightarrow Py)] $$ But how do we define the ...
17
votes
12answers
2k views

Gap year to study math

This is a plan in its earliest and thus least concise stage, so either bear with me or don't read the following babble (I bolded some of the important stuff): I am a high school graduate who is about ...
3
votes
1answer
25 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
1
vote
3answers
54 views

Creating fractals through computers

What are some beginner softwares for creating fractals on computers?
2
votes
1answer
31 views

Introduction to nests

I've just read Chapter 7 of Alice in Numberland by Baylis and Haggarty, it's called "Nests - in which the rationals give birth to the reals and the scene is set for arithmetic in $\mathbb{R}$". ...
1
vote
0answers
47 views

Soft question (Etymology - Flatness)

Why where flat modules named "flat"? Is it because they are necessarily torsion free so in a "not convoluted" or circular like $\mathbb{Z}/n\mathbb{Z}$ is as a $\mathbb{Z}$-module?
1
vote
0answers
45 views

Should I use the minus sign when writing papers in characteristic two

Is there any consensus regarding whether one should write both "+" and "-" or only "+" when performing computations in characteristic two fields? To give some context, I am writing my thesis, which ...
7
votes
3answers
88 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
5
votes
1answer
75 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
4
votes
1answer
84 views

Lecture notes ready for $\LaTeX$

Are there on the internet lecture notes in calculus in .tex or .txt format, that is, ready to be edited/modified/re-used and compiled using $\LaTeX$? EDIT: now I am specifically asking for calculus, ...
0
votes
0answers
49 views

Complex analysis a good thing to take with algebra?

This is sort of a follow up to a question I asked awhile ago. I'm working out which courses to take my first semester of my second year of university. I've had linear algebra, calculus, and analysis ...
10
votes
4answers
835 views

How is addition different than multiplication?

Is there a fundamental difference in the things we call multiplication and those we call addition? In a field, both binary operations obey exactly the same rules (commutativity, associativity, ...
0
votes
0answers
39 views

Courses for Commutative Algebra

If I want to learn more about commutative algebra, which of the two courses will help me the most: coxeter groups or schubert calculus? Also, what kind of background is necessary these courses?
3
votes
0answers
24 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
2
votes
5answers
666 views

What should I use Latex or Microsoft Word Professional? [closed]

What should I use Latex or Microsoft Word Professional for writing mathematics papers and documents and notes and courses...?
0
votes
1answer
77 views

The fourth pillar of mathematics? (analysis, algebra, geometry and …) [closed]

Many universities claim that there are three general areas in mathematics: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. If ...
-3
votes
1answer
44 views

second derivative of discrete function

given function $y[n]$ what is the best way to define the second derivative? some background to the question: in linear systems we often sample a continuos signal to a discrete one with sample rate of ...
7
votes
3answers
717 views

Masters' thesis in group theory [closed]

I would like some ideas on topics in group theory which would be suitable for a masters' thesis. What sort of problems would be suitable for this level? Because it is at masters' level, no original ...
5
votes
0answers
73 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
0
votes
2answers
108 views

Open Problems for High School Students

I am a homeschooled rising senior in high school, and I would like to research an open problem in mathematics. I have taken a number of undergraduate-level mathematics courses, including ...
0
votes
1answer
30 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
1
vote
3answers
67 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
1
vote
2answers
110 views

Linear Algebra without Matrices

How far could one get in linear algebra without matrices? It seems like the more I learn, the less I actually use them, but most of the basic theorems and invariants that learned first -- and still ...