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8
votes
0answers
132 views

What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
8
votes
3answers
174 views

difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
0
votes
0answers
19 views

Limit of power function

I want to be sure if that is correct, Could you please point me to some good references for that kind of function : When $a \to +\infty$ $ x^a =\begin{cases} 1 & x = 1\\ \\ +\infty & x>1 ...
3
votes
1answer
121 views

A finite alternative to Hilbert's Hotel?

Here, I propose a finite alternative to Hilbert's hotel as the intuition for logically developing Dedekind's definition of infinite. I would like to know if this analogy entirely justifies the given ...
3
votes
1answer
72 views

Reference request: approximation theory

I am somewhat intrigued by the ideas behind approximation theory. So, I would like to know (1) what are some thorough clear reference books to get acquainted with approximation theory; (2) what are ...
2
votes
1answer
58 views

Good ways to sample $n$ identical and dependent random variables

I'm wondering if there's a good way to talk about sampling identical but dependent random variables where it's also easy to see how the distribution evolves as we move from $n$ random variables to ...
1
vote
0answers
62 views

Good community colleges for mathematics in US

Which community colleges in the US have the strongest mathematics programs ? After searching a bit on my own, I found North Seattle Community College - it seems to have a little stronger program ...
2
votes
2answers
87 views

Why does the arrow notation of categorical limit go from right to left?

Whilst studying some category theory, I was blindly using the notation $\lim_\leftarrow D$ for the limit of a diagram $D$ in some category $\mathcal{C}$ (as they are notated in MacLane, Awodey, my ...
5
votes
2answers
66 views

Nuances of the word “proposition” (versus “theorem”) in mathematical writing

In mathematical writing, the word "Proposition" is often used to label lesser theorems. However, I tend to feel that there's a further difference in the way the words "Proposition" and "Theorem" are ...
1
vote
0answers
77 views

How to deal with a lack of ability to apply ideas in math?

I am currently studying Theoretical Computer Science, but as a Computer Science student who does not have a formal background in mathematics, past A Level (High School), I find that the ideas I learn ...
2
votes
0answers
55 views

How do we know we have all subsets of a set?

The axiom schema of comprehension basically says that every definable subclass of a set is also a set. However, we know there are only countably many formulas in the language of set theory. So, this ...
0
votes
1answer
48 views

what does “closed subspace” in papers mean?

In many books and articles one finds sentences like this: "let $A$ be a closed subspace of ...". Now my question might be stupid, but I am always wondering what they mean by closed subspace? Is this ...
3
votes
0answers
46 views

A topic for seminar in finite group theory.

I am doing a course on "Structure of finite groups" and we have a choice of giving a presentation on topics related to the course, Course outline is as follows- I was thinking of O-Nan Scott ...
2
votes
2answers
54 views

What does “of independent interest” mean in math papers?

Often I come across phrases like "We believe theorem $X$ is of independent interest"... I know this a phrase specific to math since I can google the phrase "of independent interest" and get mostly ...
0
votes
4answers
115 views

what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
1
vote
1answer
33 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
3
votes
2answers
124 views

Remove sticker on the back of Rudin book [closed]

I bought Principles of Mathematical Analysis by Walter Rudin. It came with a sticker on the back (courtesy of McGraw-Hill Higher Education) with a barcode. I am having trouble with the exercise my ...
1
vote
2answers
136 views

why do you need to know topology to study differentional geometry

Why do I need to know topology to study differentional geometry? I just try to understand differentional geometry, but I am not sure why topology is needed for it. while I see that topology is an ...
0
votes
1answer
82 views

who, by doing what, can make major contributions (breakthrough/discoveries) in math research?

I am a Math Ph.D student, had already published two small articles. I want to ask more experienced mathematician a question. What kind of person, by doing what, can make major contributions ...
16
votes
5answers
1k views

speaking about math

How should one vocalise difficult terms, which have no short names, in a talk? Have you other hints for talking about maths to other people? Do I need all to mention, what I write down on the ...
3
votes
1answer
90 views

Except First year Abstract Algebra and commutative Algebra, what else do i need to start read Algebraic Geometry text?

Except First year Abstract Algebra and commutative Algebra text, what else do i need to read before start read Algebraic Geometry texts? I am refer to the beginning texts: "Algebraic geometry an ...
3
votes
1answer
51 views

What is a list of book that i need to read as a prerequisite before start reading “lectures of logic and set theory vol.1 by George Tourlakas”?

What is a list of formal textbook that i need to impose myself to read as a prerequisite before start reading a book called lectures of logic and set theory vol.1 by George Tourlakas? That book is ...
3
votes
0answers
61 views

Theorem cannot be proven directly

Can we ever prove a theorem cannot be proven directly (i.e. We must use contrapositive or prove by contradiction.)? Can we even rigorously defined whether a proof is direct or not? Example: I was ...
2
votes
1answer
51 views

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
62
votes
22answers
1k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
2
votes
4answers
41 views

Is the matrix that we get from diagonalization the only possible diagonal matrix that can be transformed from say matrix A?

Is the matrix that we get from diagonalization the only possible diagonal matrix that can be transformed from say matrix A? Assuming that A is diagonalizable? I think it is but I don't know how to ...
4
votes
4answers
144 views

Good books written by great mathematicians

I read many of Richard Fenynman's books and I found them both very entertaining and moving, showing the human side of a brilliant scientific mind. I recently read also a collection of P.A.M. Dirac's ...
3
votes
1answer
47 views

Self-learning Book recommendation for topics in ring-theory

I failed badly in my Internal examination in ring theory , and at any cost want to improve upon my grades in the final eamination,with a month and a half to go .... I haven't yet covered the below ...
0
votes
0answers
23 views

Any book on timeline of progress of Math concepts and applications

I was wondering if there is any book that chronicles the progress of Math over the centuries and also mentions about how/when applications of various theories were discovered/invented. I have been ...
5
votes
1answer
85 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions ...
0
votes
0answers
51 views

What's the origin of the word “mathematics”?

I'm curious to know what the word "mathematics" literally mean. I think this word originated from greek, and the word itself has a meaning. For example, the chinese&japanese vocabulary for ...
1
vote
0answers
29 views

Infinite strings and infinite theorems - Is there a theory on these stuffs?

I can have an alphabet $\mathcal{A}$, a set of axioms $\mathcal{X}$ which are finite strings of $\mathcal{A}$ and a set of rules $\mathcal{R}$. Every finite strings produced by applying a finite ...
5
votes
0answers
84 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
2
votes
2answers
110 views

Mathematicians average in student life but later became significant

What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians. Up until ...
3
votes
2answers
43 views

Why is convergence w.r.t $\mathcal L^p$-norm of a sequence $(f_n)$ of $\mathcal E-\mathcal B(\mathbb R)$-functions called “convergence in $p$-mean”?

In measure-theory, why is convergence with respect to the $\mathcal L^p$-norm of a sequence $(f_n)_{n \in \mathbb N}$ of $\mathcal E-\mathcal B(\mathbb R)$-measurable functions called "convergence in ...
12
votes
4answers
741 views

Bad at computations… but not math?

Very generally, the question I'm trying to ask is: Can someone be a good professional mathematician without being particularly good at (or even being - relatively - bad at) computations? More ...
2
votes
1answer
53 views

Order of a study

So i just recently had to drop two math courses, topology, math logic, because my math maturity wasn't up to the level needed to excel in them. I intend on taking them again, but not without first ...
5
votes
1answer
154 views

Second reading on set theory? Any recommendations?

I have in past six-ish months studied through the Herbert Enderton's Elements of set theory book. Up to the point the book is great,I loved most parts of it and learned almost everything up to the ...
0
votes
0answers
22 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
3
votes
3answers
57 views

What does inner product actually mean?

What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things that we discuss about ...
0
votes
2answers
65 views

Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
1
vote
0answers
26 views

(soft question) Kreyszig's Functional Analysis or Rudin's Real and complex analysis for an introduction into more advanced analysis?

I realise they are quite different in their approach and material covered, but they share the central stuff like normed/Banach/Hilbert spaces, Hahn-Banach theorem etc. Not really understanding what ...
2
votes
1answer
44 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} ...
6
votes
3answers
210 views

Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely ...
2
votes
1answer
46 views

Why can real variable methods take place of complex variable methods in harmonic analysis' research?

We know that one complex variable methods have been largely used in the early research of (real variable) harmonic analysis. But, as is known to me, there is much difficulty when mathematicians ...
7
votes
3answers
87 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
0
votes
0answers
18 views

Basic Topology: Armstrong

I am currently reading basic topology by Armstrong and he references "thickening" a tree. I am not sure what this means. Can anyone briefly explain?
3
votes
1answer
100 views

What is the incorrect proof by Euler that $\pi = 0$ (or something like that)?

I seem to remember a proof by Euler, involving infinite series, which was really complex (for a maths hobbyist). I believe it was sent in a letter to someone, and that it ended up with $\pi = 0$ or ...
7
votes
2answers
179 views

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
1
vote
0answers
67 views

Is High Maturity and Proficiency in Calculus and Linear algebra necessary for successful research?

Is it the case that most successful mathematicians have very high Maturity and Proficiency(do it without thinking) in Calculus and Linear algebra, both calculation part(double/line integrals, ...