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3
votes
3answers
104 views

Effective Methods of Studying in different areas of Math

I apologize if this question isn't appropriate for this site, but I am looking for advice that I think other math students might be better able to give me. I am an undergraduate math major entering ...
1
vote
1answer
32 views

Introductory book on Riemann Surfaces

What is a good textbook (or set of online lecture notes) to start learning about Riemann surfaces? They were briefly mentioned in a complex analysis course I am taking, and seem like a very ...
-1
votes
2answers
52 views

$a,x,t$ are non-zero integers. $x^2+2ax+at$ is a square for what value(s) of $t$?

For what values of $t$ does the diophantine equation: $$x^2+2xa+at$$ is a square? For me, the obvious solution is: $$t=a$$ However, by assuming that there exists an integer $r$ such that: ...
2
votes
1answer
46 views

What's the point with the Gelfand–Naimark theorem?

Does anybody can explain me in plain english what's the real point with the Gelfand–Naimark Theorem. I know it's crucial, but I think I'm missing how much it's crucial.
2
votes
1answer
44 views

Is there any operation that really exists beyond just a Rotation and a Reflection?

I'm looking for fundamental operations between numbers. Kind of really really essential operations which can be done on $\mathbb N$ and $\mathbb Z$, following Kroenecker motto "Naturals where invented ...
1
vote
0answers
16 views

Intuitive idea of Green's functions

We use Green's functions in differential equations. I also see them in electromagnetism in physics. I would like to know the intuitive idea behind their usage/existence and why they are useful. It ...
5
votes
3answers
35 views

Can we, in a certain way, quantify the measure of non-differentiability of functions that are continuous everywhere but differentiable nowhere?

I am not sure how to ask this question because it seems to me that my thoughts on this topic are not clear enough, but I will give it a try. What, really, do I want to know? Well, I would like to ...
0
votes
2answers
65 views

(Soft Question) Is it bad to use Sage built in functions instead of creating my own?

I've been doing Project-Euler just as a way to increase my competency in computer science. I'm currently a Pure and Applied Math major who recently adopted computer science as a minor in order to ...
4
votes
2answers
99 views

What does the case of $\operatorname{Spec}C^{\infty}(M)$ tell us about the relavance of scheme theory to general rings?

I guess a lot of people with previous exposure to differential geometry have had this naive question pop out in their mind when studying schemes for the first time. The category of compact smooth ...
0
votes
0answers
50 views

Learning multivariable/vector calculus through guided discovery

I am asking this question as a question similar to what has been asked previously for other topics as well as math in general. But I'd like to ask for text references specifically in the domain of ...
28
votes
10answers
10k views

How do I explain the Königsberg Bridge problem to a child?

I am going to demonstrate the Königsberg seven bridge problem in a science exhibition. I am also going to use a model for a more visual representation of the problem. Now, how do I explain this (the ...
5
votes
1answer
81 views

Reference Request for Topics in Group Theory

I'm taking an honors-level algebra class and we're getting into group actions, Sylow subgroups and semidirect products. These topics are fairly intimidating, but the lengthier discussions in the book ...
6
votes
1answer
45 views

What is the Benefit If a Stochastic Process is a Submartingale?

Suppose if I have a stochastic process which is a submartingale. What is the "practical" benefit from this property? I have a roughly idea that this submartingale property suggests a favorable ...
0
votes
0answers
27 views

Best source on learning how to do limits [duplicate]

I am just curious if there is some website or book comprehensively covering theme of counting limits of functions. I am interested in tricky limits, not an introductory ones.
3
votes
2answers
49 views

Find an example of a non trivial Think of a number game

Everyone knows the game where someone ask you to think of a positive integer, then ask you to do some elementary mental operations and finally predict the result. Example: think of a number $n$, ...
4
votes
2answers
167 views

Definitions of “linearity” across branches of mathematics or levels of math education

Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, ...
7
votes
2answers
306 views

Is it neccessary to study things earlier? [closed]

I got confirmed from a graduate school starting from next year and I will major algebraic geometry. Until now, I have never thought that I study little things than others with my age. However, I ...
2
votes
0answers
47 views

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$?

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$? What of the map on spectra induced by the inclusion $i : k[f_1, \ldots, f_n] \to ...
27
votes
5answers
1k views

What does it mean when two Groups are isomorphic?

I'm not asking for the formal definition I know it. An isomorphism is a bijective homomorphism. In my book it's indicated many times when two groups are isomorphic, and I don't understand what's the ...
2
votes
2answers
55 views

What is a intuitive way to look at substitution?

Substitution can be a very powerful tool in mathematics sometimes simplifying problems, as an example: $\lim\limits_{n \to \infty} {(1-\frac{1}{n})}^n = {(1+\frac{1}{n})}^{-n} = {(1+\frac{1}{n})}^{n ...
1
vote
2answers
62 views

Motivation for Covering Spaces

Once again it seems like I am introduced to a concept with very little idea why I should care. According to Munkres, it's to find fundamental groups that are not trivial, but it's not enough to not ...
2
votes
2answers
178 views

Theoretical and applied math help - what and why? [closed]

(Edit: If you wish to skip the prologue, you may go straight to the questions in the last few paragraphs.) I'm not very far ahead at the moment (going to begin my undergraduate years after next ...
1
vote
1answer
108 views

Humorous mathematical essays

Even though there are plenty examples of mathematical jokes, the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an essay makes you smile with a nice pun ...
5
votes
5answers
156 views

what is a valid mathematical proof?

from what i have seen in my experience with math we can say that a valid proof is one that uses some form of logic (usually predicate logic) and uses logical rules of deduction and axioms or ...
1
vote
3answers
106 views

Math philosophy:about arithmetic operations and equality [closed]

This is some philosophic stuff about arithmetical operations that bothers me in some sense. During first grades, pupils are taught to perceive the arithmetical operations as "a process with a result". ...
7
votes
3answers
482 views

How does aptitude at solving Olympiad problems relate to success at further mathematical studies? [duplicate]

I spent last 6 years mostly practising my problem solving skills so I do well in my national Math Olympiad. Out of curiosity I did some reading on basics of what undergraduate students are taught - ...
1
vote
1answer
83 views

How to explain aspects of derivatives to little brother?

So as the title suggests, my $12$ year old little brother loves math. Since he is a bright kid, I started teaching him derivatives. The issue is he always keeps asking weird questions that are above ...
12
votes
5answers
1k views

Mathematicians who overcame academic failure to achieve success [closed]

Does anyone have any story of mathematicians who overcame "academic failure" or setbacks to achieve success later as a result of their perseverance? This is a soft question, that hopefully can inspire ...
18
votes
3answers
1k views

Understand it or burn [closed]

I have a big problem when I study math,namely when I am stuck on some problem I can't go on and I won't keep studying other material till I haven't solved the problem I've been stuck on. I am ...
1
vote
1answer
62 views

Nonexistence of conjugate points $\Rightarrow$ a geodesic is minimizing

Motivation: I am trying to prove a certian geodesic is minimizing. The only generic tool I know for doing that is the fact the a gedoesic is minimizing as long as it stays in a normal neighbourhood of ...
10
votes
5answers
319 views

What are some of the best book in mathematics for general reader? [closed]

I am preparing a list for my department library, consisting books of mathematics for general readers. I've included The men of mathematics by Bell, Fermat's last theorem by S.Singh , The man who knew ...
3
votes
1answer
78 views

The Connotation of an “Algebra”

I am studying some real analysis as we speak, currently on the topic of function spaces. In particular, this section of my book is discussing function algebras. I have heard of other types of ...
10
votes
7answers
703 views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
2
votes
0answers
50 views

What makes a good mathematical poster? [closed]

During my second year of university I have done some assignments and some papers, but this is my first time I have to make a mathematical poster. What could make my poster stand up. So up to now I ...
4
votes
0answers
55 views

Good Source For Mathematics News [closed]

Is there a good news source that specifically covers the latest results and findings in pure mathematics which is not an academic journal?
0
votes
1answer
84 views

Is there a Hilbert's list for $21$st century?

David Hilbert gave his famous list of $23$ unsolved problems presented in Paris in $1900$. Most of the problems are fully/partially resolved, some are still open (RH etc) and some are impossible to ...
1
vote
1answer
65 views

How to find old papers on a particular topic?

When learning mathematics, I think it is better to learn the motivation and the purpose of a particular maths idea. I believe such things are best found in the original paper which proposes the ...
1
vote
0answers
31 views

Higher Order Estimation Errors

I well know estimation measure is the so called minimum mean square error (MMSE) defined as: \begin{align} E[|W-\hat{W}(V)|^2] \end{align} where $W$ is a random variable (that we want to estimate) and ...
16
votes
4answers
333 views

Graph Isomorphism for non-mathematician

Question: What is a good explanation of the Graph Isomorphism problem, which would be understandable - and hopefully exciting - to a person who has minimal exposition to mathematics? Note that I do ...
2
votes
2answers
80 views

Elementary Geometry Problems, which require Non-geometric Tools

In number theory, many problems are easy to state but difficult to solve within number theory. Are there Elementary (solved and unsolved) problems from Euclid's geometry, which are difficult to solve ...
1
vote
0answers
67 views

information theory literature beyond Cover and Thomas

Can you recommend me some literature for information theory that goes beyond the book of Cover and Thomas? I know that this is a very broad question and therefore I would be happy about any suggestion ...
0
votes
0answers
15 views

Reference for differences between Euclidean Geometry and Related Terminologies

The three terms, I want to consider, related to Euclidean geometry/space are Euclidean Geometry, Affine Euclidean Geometry, and Linear (normed) Euclidean Geometry. As long as I gone through ...
-3
votes
1answer
58 views

Is this construction algebraically closed?

On the tetration forum Tommy1729 proposed a new kind of number : http://math.eretrandre.org/tetrationforum/showthread.php?tid=1036 Too avoid deletion or changes of that post , I copy it here : ...
5
votes
3answers
123 views

GAP Editor with Syntax Highlighting for Windows

Can anybody recommend a code editor for Windows which has GAP Syntax Highlighting? Thank you
1
vote
1answer
34 views

Difference between Euclidean Space and $\mathbb{R}^n$ other than origin?

Often it is said that Euclidean $n$-dimensional space over $\mathbb{R}$ is different from $\mathbb{R}^n$, because in Euclidean space, all points are equivalent, in $\mathbb{R}^n$, there is origin. ...
3
votes
0answers
47 views

reference request for a book on high dimensional probability and data analysis written for mathematicians

I hope someone can help with this. I am a statistician looking for a good book on high dimensional probability and data analysis. Basically I am looking for the equivalent of Terry Tao's 2 volume set ...
1
vote
1answer
28 views

Simple examples of applications of converse, contrapositive and inverse used in mathematical proofs rather than logic.

While learning simple logic in high school, I remember learning about converse, contrapositive and inverse (maybe some others as well). Yet, I don't seem to recall their usage for proofs (only ...
3
votes
2answers
232 views

Why we still diagonalize compact operators even tho we lack invertibility.

We know that any compact symmetric operator on a Hilbert space, has a orthogonal base of eigenvectors. But we also know that $0$ is in the spectrum if $X$ is infinite dimensional, which makes the ...
0
votes
1answer
113 views

On the resemblance of Fubini and integration by parts.

Suppose we have an expression of the follwoing form; $\int_a^b fg $, and want to understand it. By integration by parts we have the following $\int_a^b fg = -\int_a^b Fg^{'} + Fg \mid$. But $\int_a^b ...
2
votes
1answer
89 views

Who determines if a mathematical proof is valid? [closed]

I'm studying mathematics and as you all know the most important things in mathematics are proofs. My question is, who determines if a proof that someone invents ...