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3
votes
2answers
40 views

What would be an effective way to learn group theory on my own?

I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it. I want to study as much as I can on my own, as my course doesn't have group ...
0
votes
0answers
27 views

Some notes on $D_n, S_n$ and $A_n$

http://www.stat.uchicago.edu/~lekheng/courses/repth/sol2.pdf In these solutions it refers to See "Some notes on $D_n, S_n$ and $A_n$". Does anyone know where these notes can be found? They sound ...
2
votes
4answers
83 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
2
votes
0answers
44 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
4
votes
0answers
58 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
0
votes
3answers
29 views

Looking for good intro book on differential equations

I am looking for a good book to study ordinary differential equations. My background is that I have successfully completed calculus 1 through 3. So this included derivatives and integrals, ...
0
votes
1answer
33 views

How to evaluate a squared sum?

I need (the name of) the formula to evaluate $(x_1 + x_2 + ... + x_n ) ^2$ . I know the question is not very interesting, but I am stuck and WolframAlpha also doesn't get my input. Thanks in advance
-4
votes
0answers
47 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
2
votes
2answers
89 views

Why are quadrants defined the way they are?

I was thinking about planes and things, and suddenly wondered why quadrants are defined the way they are, the first on the top-right, and so on. I wonder if this gives us any benefit, or if any ...
36
votes
2answers
1k views

What is this pattern called?

Back-Story I became interested in the patterns in multiplication tables for different base number systems a while ago. Specifically, the pattern made by the last digit of each number in the ...
4
votes
7answers
78 views

Is a pattern proof?

Let's say I want a formula that takes any number and makes it into 170, and I come up with a formula that I think does it. If I plug 1 into it, 2 into it, 3 into it, etc. up to a pretty large number ...
2
votes
0answers
38 views

Textbook +reference book in complex analysis

Which book can be used as an introductory textbook in complex analysis? I have some choices (more suggestions are welcomed) Marsden & Hoffman J.B. Conway Ahlfors Palka Lang Stein & ...
4
votes
1answer
93 views

Where is Cauchy's wrong proof?

Allegedly, Cauchy mistakingly "proved" that pointwise convergence of continuous functions is continuous. I saw this somewhere in a book, and it is also in wikipedia: Uniform convergence. In his ...
0
votes
1answer
31 views

Which discrete mathematics book do you think is better between Epp's and Rosen's for a clueless self-learner?

I am a programmer, and I want to become a machine learning researcher and a good software engineer. I dabbled with calculus, linear algebra, and real analysis for a few months when I was enrolled in a ...
1
vote
0answers
27 views

Referencing a Theorem in a Paper?

I'm making the final edits for a paper and I have a question more about the etiquette. I want to use a theorem from another paper and its obvious I have to cite it, but do I need to prove it too? I ...
0
votes
0answers
29 views

What are some genuine ways to define the derivative of a fractal?

Seeing the success of applying measure theory to generalize integration to fractals, I wonder whether or not there is a method to generalize the derivative to a fractal. Most courses start off fractal ...
2
votes
0answers
54 views

What is the point of basis vectors?

Why do we even bother with basis vectors? Why don't we just notate an element $x$ of an $n$-dimensional vector space $V$ as an ordered set $(x_1,x_2,...,x_n)$ and go from there?
-1
votes
1answer
147 views

Who are the big names in Mathematics nowadays? [on hold]

The questions I pose are: Who are the big names in mathematics now? What branch do they study, what big problems are they looking at? I know of Wiles and Perelman (and I don't even know if those are ...
-1
votes
0answers
49 views

Why do some mathematics professors teach more/less courses than others? [migrated]

Not sure if this belongs on the Academia site, but since I'm a math major and the question is based solely on my observation of mathematics professors, I figured this site would be best. At my ...
8
votes
2answers
87 views

How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
2
votes
1answer
43 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
3
votes
0answers
33 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
3
votes
0answers
38 views

What is the densest, most opaque way of saying two odd numbers add up to an even number? [closed]

Here's what I've come up with: In $\mathbb Z$, any pairwise sum of elements in the $\langle 2 \rangle + 1$ coset is in $\langle 2 \rangle$. But there's got to be a way to make this even briefer, ...
0
votes
0answers
49 views

Which version of MIT Single Variable Calculus should one take? [on hold]

MIT offers nine versions of the single variable calculus course. five with course number 18.01..., and four versions under the course number heading "Supplemental." 1) Which do you recommend to ...
0
votes
3answers
35 views

Proving limits via episilon-delta definition vs. algebraic manipulation.

In real analysis textbooks, limits are proved using the epsilon-delta definition directly. However, at some point, limits start being solved using algebraic manipulation. For example: $$\lim_{x \to ...
2
votes
1answer
85 views

Amazing integrals and how is solved it [closed]

There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral ...
1
vote
1answer
26 views

SDE Modeling: Ito vs. Stratonovich

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. ...
0
votes
2answers
20 views

Given the solution to some differential equation, is the original equation necessarily unique?

For example, if I have the fundamental solution set $\{x^2\}$, such that $y(x)=Cx^2$ is the solution to some unknown differential equation, is it guaranteed that only one such equation exists with ...
1
vote
2answers
21 views

Confusion about division in rates.

I would really appreciate help with this because it's been driving me insane for a while now... I understand what "per" means in "$x$ kilometers per $y$ hours". What I don't understand is how to make ...
5
votes
1answer
67 views

Could the real numbers have been invented without the natural numbers

The real numbers are constructed from the rational numbers which are constructed from the integers which, in turn, are constructed from the natural numbers. But if we had no notion of the natural ...
-2
votes
2answers
43 views

Do you have any good iOS app suggestions for taking notes? [closed]

Thanks for stopping by my thread. I'm an engineering student with an avid interest in Maths. I hope to do some kind of research in Maths someday. I really enjoy Maths and am doing a lot of reading so ...
-4
votes
0answers
72 views

Math: A discovery or a creation? [closed]

I am just curious as to what math is at it's very basic state. Is math something that humans have invented? Or is it more of a discovery? Or possibly something completely different. If it is something ...
3
votes
3answers
129 views

What is it like to understand complicated/advanced mathematics?

Whenever I see very complex equations, they look, in a way, beautiful even though I don't understand them. This was directly taken from another question: "- Definition 1 - Given an open subset ...
2
votes
0answers
50 views

Does Fractional Calculus have a real connection with Fractals? (or is it just an extra variable trick)

The fractional derivative and integral (operators that let you differentiate or integrate a fractional number of times) have drawn a lot of attention from people outside the field. Yet, after reading ...
0
votes
0answers
39 views

intuition about calculating partial sums of series

The partial sums $$1 + 2 + 3 + \cdots + n$$ of the simple arithmetic progression can be calculated by reordering and adding. The partial sums $$1 + \frac{1}{2} + \frac{1}{4} + \cdots + ...
0
votes
0answers
16 views

Stochastic integral and usual integral addition

Let's say I have two processes and I would like to say something about their sum. In the case of deterministic functions, $\int f(t)dt + \int g(t)dt = \int f(t)+g(t)dt $, and I can then possibly say ...
10
votes
0answers
99 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
4
votes
3answers
33 views

Odd and Even Parity in Proofs

The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points. For example, ...
1
vote
0answers
26 views

Seeking after notation for two objects equal up to a constant

Sometimes we want to express that two objects are equal up to a constant but there is no need to keep writing out the constant or constants. For example, often times the constant or constants involved ...
2
votes
0answers
27 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
6
votes
2answers
76 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
0
votes
0answers
29 views

What should I study, if I want to learn about higher dimensional spaces and objects? Also, what resources should I obtain?

I am becoming interesting in learning about higher dimensions. What are subjects I could study, and what are some good resources for those subjects?
4
votes
0answers
57 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
21
votes
0answers
239 views

When are two proofs “the same”?

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude ...
2
votes
2answers
74 views

How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily?

How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is ...
6
votes
2answers
238 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
2
votes
0answers
106 views

Is Euclidean geometry really a “dead” subject? If so, why? [closed]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
1
vote
0answers
17 views

Application of theorems 'about free groups'

What consequences have theorem Any nonzero subgroup of free group is free or some another similar theorems? P.S. especially not-group-theretic applications.
2
votes
0answers
67 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
1
vote
0answers
35 views

Intuition for Entropy over Fractals

Is there intuition for "mathematical" entropy. I know that physical entropy tracks the order in a dynamical system, for thermodynamics. As entropy goes up, general randomness and disorder goes up. ...