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3
votes
0answers
57 views

Are there concepts in nonstandard analysis that is useful for a introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense Can someone elaborate on this ...
0
votes
0answers
10 views

Any difference in factoring product of two large regular primes or two large (same magnitude) irregular primes?

an RSA-like situation: If one multiplies two 300 digit regular primes to form a product, and another multiplies two 300 digit irregular prime to form a product, is there any way that a factorization ...
3
votes
1answer
32 views

Is there a mathematical distinction between “on” and “in”?

Is there any difference if I said a function on an interval or a function in an interval? or a vector field on a manifold versus a vector field in a manifold? The main reason is because some online ...
5
votes
1answer
88 views

Ideas for basic application of homotopy theory to homological algebra?

I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material ...
5
votes
0answers
67 views

What Constitutes a Pattern

Mathematics is often referred to as the "study of patterns." What I'm wondering is whether there is somehow a technical way to describe a pattern. For the length of this question let's assume that ...
2
votes
1answer
81 views

A categorical approach to algebraic geometry

I learned Algebraic Geometry in a geometrical viewpoint, e.g. Hartshorne's book. But I want to learn algebraic geometry categorically, for examples, i) Sheafification $\mathcal{F}^+$ of a presheaf ...
0
votes
0answers
26 views

What is the Laplace Transform? [duplicate]

What is the Laplace transform? I understand how to do it (taken differential equations), but my professor just kinda told us to accept that $ F(s) = \int_0^\infty e^{-st}f(t)dt $ is gospel and to ...
0
votes
1answer
32 views

Are there undescribable mathematical objects when we represent them as drawings?

It was argued at "Are there mathematical objects that have been proved to exist but cannot be described in words?" that there are mathematical objects that we can't describe in words because there are ...
1
vote
0answers
38 views

Exotic applications of Hilbert spaces?

So my final exam for an introductory course on Hilbert spaces is just a weeks away. I enjoyed the course, we covered the theory in enough detail to illustrate its richness and elegance. I'm aware of ...
8
votes
0answers
76 views

How to explain the topic of Fourier transform interactively? [closed]

This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of Fourier transform. The principal of the institution ask me to take lecture ...
14
votes
6answers
1k views

How to think of a set?

I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. ...
-6
votes
3answers
130 views

What is the significance of stuff like the “Pigeonhole Principle”? [closed]

Pigeonhole Principle if n items are put into m containers, with n > m, then at least one container must contain more than one item src I thoroughly read What is your favorite application of the ...
8
votes
0answers
163 views

Work of Ted Kaczynski

I hope this question is not too crazy sounding, but I was wondering if anyone is familiar with the work of Ted Kaczynski (or even has cited/used it before). After reading in Lars Ahlfors' Complex ...
3
votes
2answers
41 views

Intersections between a function and its Taylor polynomial

Suppose $D \subset \mathbb{R}$ is open, $f : D \to \mathbb{R}$ is a smooth (not necessarily real analytic) function, $x_0 \in D$, and $T_n$ is the degree $n$ Taylor polynomial of $f$ centered at ...
2
votes
3answers
104 views

How do I figure out my math aspirations? [closed]

I am really confused. Here's my present situation: I'm 18. I am about to start a computer science degree this August--going there only coz my parents want me to. I know most teachers for undergrads ...
21
votes
4answers
2k views

How can you show Godel's incompleteness theorem using mathematical symbols?

I want to get this as a tattoo as I love the role maths plays in the universe and the idea that the farthest reaches of what we can ever know, fall short of the limits of what is true, even in ...
4
votes
1answer
111 views

Why more than 3 dimensions in linear algebra?

This might seem a silly question, but I was wondering why mathematicians came out with more than 3 dimensions when studying vector spaces, matrices, etc. I cannot visualise more than 3 dimensions, so ...
8
votes
3answers
2k views

Why doesn't the Taylor series always converge?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
0
votes
2answers
103 views

How important is it to learn latex?

So I am currently in my senior year (Undergraduate mathematics). And I am thinking to continue on and do my masters and PHD in mathematics. I need to now how important it is to learn latex and how ...
2
votes
1answer
74 views

Researching in Mathematics

I am presently pursuing Engineering, but I want to make my career in the field of mathematics. How do I come to know of the specific topic in math in which I would like to research, in which I would ...
7
votes
5answers
861 views

What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
1
vote
1answer
73 views

Name for norm with property $\|x+y\|=\|x\|+\|y\|$.

Is there a name for a norm with property $\|x+y\|=\|x\|+\|y\|$ ? I found this, but it doesn't seem to answer my question.
21
votes
3answers
868 views

“Least trivial” function preserving rationality

Is there a "non-trivial" function $f(x,y)$ such that $$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$ An example of a "trivial" function would be $$f(x,y) = \begin{cases} 0 & x,y\in ...
1
vote
0answers
31 views

Prerequsites for working through the 2nd half of Gradient flows in metric spaces and in the spaces of probability measures

I apologize in advance if this question is too general, that is, not a request for a specific reference, but more of a request for a road map, perhaps from someone that knows the material and, in ...
4
votes
3answers
146 views

Books for learning basic to advanced math

I am aged 62 and now have lost all skill in mathematics. I would like to learn basic to advanced (primary school > university) math. I am not sure if there is any such book that could explain such a ...
0
votes
1answer
74 views

Is the infinitesimal approach good.

I have heard of an infinitesimal approach to calculus. Is it better than the normal approach or is it the other way around.
7
votes
1answer
107 views

Would the professor take me seriously? [closed]

I'm a pure mathematics undergraduate. I have been working since a long time, and I am finally done with my little book. It handles a particular subject (which I won't mention). It is a collection of ...
20
votes
4answers
595 views

Why do single-author math papers use “we” instead of “I”? [duplicate]

Even when math papers are authored by one person, you will see stuff like "We now prove so and so" or "In the following section, we show that...". How come papers with one author use "we" instead of ...
1
vote
0answers
22 views

Network theory and football?

I was reading the latest post on Azimuth, Network Theory in Turin, and I watched many of the lectures Baez posted on his site here. This might be a crazy question to ask considering it's not ...
0
votes
2answers
131 views

What is a good reference that connects calculus with differential geometry?

It seems that most texts on differential geometry books tend to take a quantum leap from calculus without refering the latter. Differentials suddenly becomes forms, functions suddenly becomes ...
1
vote
1answer
163 views

Impossible Math Riddle [duplicate]

Mathematician A asks Mathematician B to guess the age of his three sons. Mathematician A starts off by giving Mathematician B two clues. The two clues are: The product of their ages is 72. When you ...
4
votes
3answers
124 views

Obfuscated proofs [closed]

I am, just for fun, looking for long and complicated proofs for statements which can be proven rather easily and much faster. The proof itself still has to be correct however. While the proof should ...
28
votes
4answers
818 views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
0
votes
0answers
22 views

Find the amount of squares in the given picture [duplicate]

How many squares are in the above figure?
4
votes
2answers
668 views

Has the Collatz Conjecture been proven to be unprovable? [closed]

This paper, from a peer-reviewed journal, purports to prove that the Collatz Conjecture is unprovable. If it’s valid, why has it not received more attention? If it’s invalid, what is the flaw, and ...
7
votes
4answers
232 views

How to learn commutative algebra?

I want to learn commutative algebra from scratch. I was wondering, as you guys are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course ...
2
votes
0answers
37 views

What subjects properly belong in operations research as their “owning” discipline?

Warning: This is a soft question, hence I would make it a wiki-community post if I could. Operations Research involves a broad swath of disciplines, ranging from probability and statistics/stochastic ...
2
votes
1answer
41 views

Why Square Brackets for Expectation [duplicate]

I've often seen $\mathbb{E}[X]$ instead of $\mathbb{E}(X)$, but it seems variance is almost always $Var(X)$. E.g., Wikipedia for Expected Value and Variance. Is there a good mathematical reason for ...
2
votes
1answer
40 views

n points can be equidistant from each other only in dimensions $\ge n-1$?

2 points are from equal distance to each other in dimensions 1,2,3,... 3 points can be equidistant from each other in 2,3,... dimensions 4 points can be equidistant from each other only in ...
4
votes
4answers
277 views

How to not feel bad for doing math? [closed]

I have a MsC and want to take a PhD in algebraic topology. Probably very few people in the world will have any interest of my thesis. They will pay me for doing my hobby. Its the only job I can think ...
0
votes
0answers
29 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
0
votes
1answer
59 views

Learning math by analyzing/proving theorems?

Hello I want to learn mathematics. In order to do this I want to get familiar with formulas/theorems by taking one and just analyze it and try to manipulate it to understand it better. I wanted to ...
3
votes
1answer
69 views

Why are there so many conjectures in number theory and comparatively less in others?

My question is that : Why are there so many conjectures in elementary number theory and comparatively less in others? This is particularly weird because every topic in maths should have its equal ...
4
votes
0answers
58 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
1
vote
0answers
24 views

How to visualize orientation of 3d objects

The way I visualize orientations of $1$- and $2$-dimensional objects is by an ant walking along a path. For a $1$d object (like a line/ line segment/ etc), just place the ant on the line and confine ...
3
votes
2answers
82 views

Why are radians used in calculus. [duplicate]

Ok, please ignore my silliness. So, why do we use radians in calculus and why is it considered more scientific than degrees. And how did mathematicians know or prove that radians would work for all ...
4
votes
5answers
199 views

Which are the operations used in mathematics? [closed]

Everyone knows +,-,x,:,^. But I would really like to know which other operations exist, and what they do.
4
votes
1answer
301 views

What branch of Mathematics does the study of Algebraic/Transcendental Numbers lie in?

I've always been fascinated by polynomials, ever since first learning them in high school. I absolutely adore the notion of 'playing around with the coefficients' and watching what happens to the ...
6
votes
3answers
90 views

examples of non-abstract rings?

In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness? I ...
3
votes
1answer
77 views

Relationships Between Moduli Space and Objects They Parametrize

My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the geometry of the objects that the space parametrizes. As an ...