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8
votes
1answer
174 views

How much time does a great mathematician take to solve an extreme problem?

I really love math, and I can spend hours, days or even years to solve a really simple problem if I can't do it. However, there are certain problems, which I am not able to solve in an hour or so. It ...
-2
votes
0answers
15 views

Excercises on continuity (multivariable)

I hope this kind of question is appropriate for math.stackexchange. I would like to train excercises where I have to examine the continuity of functions that contain more than one variable. Are ...
14
votes
3answers
282 views

Axiom of Choice: Where is falsity in argument and understanding to put is as Axiom?

In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\...
1
vote
1answer
48 views

Real life illustration of the fact that rationals have measure zero

I wonder if there's any real world phenomenon that reflects the mathematical fact that $\Bbb Q^k$ has Lebesgue measure zero in $\Bbb R^k$, or put another way, the likelihood that we get a rational ...
6
votes
2answers
172 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
28
votes
1answer
660 views
+50

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
5
votes
0answers
43 views

Demystifying the tensor product

It seems to me, through my mathematical immaturity, that the tensor product seems to beg for more well-definition. I am working in vector spaces (so we always have a free module) and here is what my ...
0
votes
0answers
25 views

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...
1
vote
0answers
38 views

Recent advancement in Haar measure

From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ...
3
votes
0answers
61 views

Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. I guess what I am trying to ask is that "does everything an analyst could ever care about have ...
-4
votes
0answers
19 views

Reference and Textbooks for the given topics [on hold]

This is my post in Maths Stack Exchange. I am a student of BS-MS course .I want to get some recommendations in books. I will be giving different competitive exams also. I need one textbook and one ...
6
votes
0answers
142 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
0
votes
0answers
20 views

Techniques to turn expressions involving integer roots into polynomials by substitution?

Inspired by this question involving an Equation for a Torus How to find a parametrization for a torus? I started wondering if there is some systematic approach to do substitutions to make equations ...
3
votes
0answers
25 views

Is there a searchable database of mathematical objects that you can search by property?

For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function....
7
votes
2answers
160 views

Gödel's ontological proof and “modal collapses”

Recent findings on Gödel's ontological argument allowed to ultimately establish a couple of things: Gödel's original axiomata are inconsistent Scott's variation instead is consistent Scott's axioms ...
46
votes
13answers
7k views

List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
3
votes
1answer
55 views

Unsure on which sources to choose related to Calculus

I tried to get into Spivak's Calculus only to find that I've never been taught the type of Math presented there. First chapters talk about the properties of numbers, then mathematical induction, ...
0
votes
0answers
64 views

Phd topic in Field / Galois theory [on hold]

I just finished Masters degree in CS (Machine Learning) and I'm thinking of doing a Phd in Mathematics (I have first degree in math). I'm mainly interested in Finite Fields and Galois theory. What ...
0
votes
0answers
28 views

Calculus & Analytic Geometry VS Vector Calculus

This question may be applicable for Academia SE, however this is strictly math-oriented and requires math whizzes' opinions. I intend to go to a tech institute to get a BS majoring in Computer ...
-1
votes
1answer
59 views

Why is the notation for irrational number not mainstream? [on hold]

In everywhere you see the symbol for the set of rational number as $\mathbb{Q}$ However, to find actual symbol to denote the set of irrational number is difficult. Most people usually denote it as $\...
33
votes
4answers
8k views

Are We Teaching Pre-Calc Wrong?

It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that ...
8
votes
4answers
111 views

Are there any tricks to remembering proofs of mathematical theorems?

Is there a way to quickly and thoroughly remember theorems? For example, proofs of the mean value theorem, or Rolle's theorem. Having to remember all of them off by heart has been quite tedious. ...
3
votes
1answer
111 views

Old books on calculus

I'd like to know if there are other old books of the same level of the classic and well-known books like Apostol, Courant, Spivak and Hardy.
3
votes
0answers
42 views

How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
1
vote
0answers
39 views

Is it possible to study Lie algebras without knowing too much of representation theory?

There's a course on Lie Groups that I'd like to take, but it seems that for various reasons it's a good idea to take Lie algebras along with it. But after having a brief look at the contents of the ...
1
vote
2answers
35 views

Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
3
votes
3answers
64 views

Topology: is it ever good to write $x \in U \in \mathfrak{T}$

Sometimes I come across a sentence in my topology book that says, let $U$ be an open set that contains $x$ I can't help but write it down as: Let $$x \in U \in \mathfrak{T}$$ Is it good ...
2
votes
0answers
34 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
0
votes
0answers
10 views

References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics: Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is ...
6
votes
2answers
403 views

What are some mathematical problems which have been forgotten?

As mathematicians continue to study mathematics, often times they run into a problem which takes a considerable amount of effort to solve. For instance, trying to factor polynomials has lead to a ...
632
votes
162answers
39k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament,...
73
votes
18answers
30k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
3
votes
4answers
100 views

Is there an alternative way to represent the $\operatorname{diag}$ function?

In optimization, it is common to see the so called $\operatorname{diag}$ function Given a vector $x \in \mathbb{R}^n$, $\operatorname{diag}(x)$ = $n \times n$ diagonal matrix with components of $x$ ...
2
votes
1answer
73 views

Instructive video content for High School kids?

I need some math Youtube channels (or any other visual media, movies maybe...) that I can recommend to High School students, not solely as a method of learning math but more to illustrate the beauty ...
3
votes
0answers
45 views

Algebraic Geometry Project ideas related to Computer Science

I am a Computer Science Undergrad student with an interest towards Algebraic Geometry.I have just recently started and am currently reading Miles Reids' Undergraduate Algebraic Geometry(I have read ...
1
vote
1answer
59 views

Can someone come up with a better way to write $V = \operatorname{diag}(x_1,x_2)(Y-\mathbf{1}X^TY)$

$\newcommand{\diag}{\operatorname{diag}}$Let $X = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $Y= \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ I have a vector: $$V = \begin{bmatrix} x_1(y_1 - \...
2
votes
2answers
50 views

Finding counterexamples in elementary set theory.

I had the following two problems: Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for ...
0
votes
2answers
87 views

Why weren't “degrees” replaced with a more intuitive angle measure?

$\bf History$ It is speculated that the seemingly arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$ and $60 \times 6 = 360$. ...
3
votes
3answers
75 views

Is there a mathematical reason why rotation in the counterclockwise direction positive and clockwise rotation negative?

This inquiry has recently come to me in my study of trigonometry and the unit circle. It was said right from the very start that counterclockwise rotation were positive while clockwise rotations are ...
1
vote
2answers
202 views

Is there a reason for the existence of prime numbers? [closed]

Is there a reason for the existence of prime numbers, or is there a reason that some numbers are prime numbers, but others are not? Does the number theory have any answers or at least ideas about ...
2
votes
2answers
262 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
1
vote
0answers
39 views

What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
4
votes
1answer
38 views

Is it possible to express an integral equation inside of a convolution

Given $$u(t) = \int_0^t y(\tau) d\tau$$ Consider a convolution type of integral $$W = \int_0^t\lambda^{t-\tau}y(\tau) d\tau$$ $\lambda$ a positive real number Is it possible to write $W = f(u(...
2
votes
0answers
77 views

Can Local Martingales be characterized only using their FV process and BM?

See Theorem 1 here. Theorem 1 Any continuous local martingale $X$ with $X_0 = 0$ is a continuous time-change of standard Brownian motion (possibly under enlargement of the probability space). ...
1
vote
2answers
122 views

Mathematical formula representing human voice

I'm only confident that my question is reasonable, but I'm not confident that it's reasonable to be asked here. I am a physics undergraduate, and up to my knowledge, any sound, no matter what it is ...
11
votes
3answers
2k views

Why has the Perfect cuboid problem not been solved yet?

Why hasn't Perfect Cuboid Problem been solved yet, whereas (possibly) more nontrivial ones such as FLT and Sphere packing have been solved? I understand that calling some problems more nontrivial ...
3
votes
0answers
59 views

Monads in monoids

This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question ...