Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

8
votes
1answer
81 views

Largest commutative diagram actually used

This is a very soft-question so feel free to close it if doesn't fit. I was just wondering what is the largest commutative diagram that you have encountered? Here I am only counting commutative ...
2
votes
0answers
25 views

What is really being asked by “Prove that $S^1 ⊂ R^2$ is a sub manifold”?

I'm self-studying smooth manifolds, and there is some terminology that bothers me a lot. In a lot of books, or homework questions that I looked, there are statements such as Prove that $S^1 ⊂ R^2$...
2
votes
0answers
54 views

Dynamical systems from an algebraic perspective

I am interested in learning more about dynamical systems. Most of my background is algebraic, specifically in group theory/geometric group theory. I was wondering if anyone knew of a reference that ...
2
votes
0answers
38 views

Career Advice: Where to apply Analysis in Industry?

$\textbf{Background:}$ I am a fourth-year undergraduate at a major university in the US, and I will be staying for a fifth year before proceeding to pursue a masters degree. I have taken courses in ...
2
votes
0answers
27 views

A measure theoretic Lipschitz condition

Let $f$ be a measurable function satisfying following condition: for every $\epsilon$, we have \begin{equation*} \limsup_{\delta \to 0} \bigg\{ \frac{1}{\delta^N} \mathcal L^{2N} \Big( \Big\{ (x,y) \...
1
vote
0answers
54 views

Is category theory more a field of study or a viewpoint on mathematics?

I heard a bit about Category Theory, especially when studying topology, although I never really studied it. I am quite curious about it, in particular I would like if someone could better explain: -...
0
votes
1answer
28 views

Is there any gamified software for learning advanced mathematics?

I have been thinking for a while that new ed-tech software, such as Babbel, could be applied to higher level mathematics. Although visualisations of mathematical forms, akin to diagrams, are often ...
8
votes
2answers
218 views

Why intuitively do the quaternions satisfy the mixture of geometric and algebraic properties that they do?

[I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may restore it.] ...
3
votes
0answers
27 views
+50

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
1
vote
0answers
43 views

Replacing differentiation with anti-integration?

When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of ...
1
vote
1answer
44 views

Which one is the correct notation?

I know this notation is correct: $a_1,a_2, a_3,\cdots,a_n=\left\{a_k\right\}_{k=1}^n$ Now, we have a function $f(n)$. I want to write this sequence in correct notation: $\left\{ f(1),f(2),f(3),\...
0
votes
0answers
23 views

Group theory : representation Matrix of generating element for families of functions?

Assume I have some sampling of a function $f(t)$ at points $t$: $$f(t_k) = d_k, \forall k \in \{1,\cdots,n\}$$ Assume we have vectors $\bf v_k$ which can be functions of $x_k$ and $d_k$, can we find ...
2
votes
1answer
16 views

Coarea-like formula for BV function (not its derivative)

Let $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Do we also have that $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ holds?
2
votes
1answer
22 views

On the function $\chi_{\{x \le F(y)\}}(x,y)$ where $F$ is Lipschitz

Let $F:\mathbb{R} \to \mathbb{R}$ be a $L$-Lipschitz function. Consider the function $$G(x,y) = \chi_{\{x \le F(y)\}}(x,y),$$ where $\chi$ is the indicator function. How can I plot this function ...
1
vote
0answers
48 views

Studying Differential Equations without Physics Experience [on hold]

As I get closer to becoming a graduate student, I'm trying to figure out what my interests in math are and which one I would like to pursue and do research in. Recently, I've been thinking about going ...
4
votes
0answers
51 views

History of letter $h$ in the formulation of differentiability

$f$ is differentiable in $x$ if $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ exists. Why is the letter $h$ typically used here, what it the history of this? I feel that $\delta$ or $\Delta$ for ...
1
vote
0answers
42 views

Visually suggestive way to present a finite group

When studying about groups, we can often grasp the structure of a small group "internally". For instance, we can "see" that $\mathbb Z/3\mathbb Z$ as a three-fold symmetric shape;more complicated ...
3
votes
1answer
132 views

Ars longa, vita brevis.

There's little use studying mathematics without actually doing mathematics. There is a plethora of exercises in any textbook worth its salt. I suppose those with some business in looking up something ...
-1
votes
0answers
50 views

Why is the subject of math overall considered an essential for compulsory education/non-scientific circles/society overall? [closed]

It would seem that math is pushed both scientifically and culturally as some sort of plethora of necessity. What really does math do for the layperson? Aside from basic counting and stuff, I can't ...
1
vote
0answers
24 views

Why is L an interesting random time for a Brownian motion?

Let $B$ be a Brownian motion and define $L=\sup \{ t \leq 1 : B_t = 0 \}$. My question is: Why is $L$ an interesting random time? Durrett's probability book proves something about the ...
2
votes
0answers
34 views

Does there exist an opposite to curve length in integral calculus, “radius length”?

Consider the formula for curve (arc-) length in integral calculus: $$\int_a^b\sqrt{1+\left(\frac{\partial y}{\partial x}\right)^2}dx$$ What would the geometrical opposite thing to measure be? Some ...
1
vote
0answers
69 views

Very basic question about ordinary differential equations.

Solve $$\frac{dy}{dx}=\frac{1}{x}$$ $$\int \frac{dy}{dx} dx = \int \frac{1}{x} dx$$ $$y = \log(|x|) + C$$ Is this solution right? I don't think this solution is right. I think the following ...
0
votes
0answers
26 views

List of extension theorems

As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example, extending ...
0
votes
1answer
38 views

Infinite sums and squares

So, I'm sure this has been thought of and said before but I'm curious. So $\sum \frac{1}{2^n}$ can be thought of by filling up a square. First we color in a whole square and then we draw a second ...
1
vote
0answers
98 views

Motivation of the von Neumann definition of ordinals

The von Neumann ordinals are defined in such a way that each ordinal is exactly the set of all smaller ordinals. I am wondering about the origin/motivation for this definition of ordinals (that is, ...
4
votes
1answer
93 views

Is using the “contains” symbol $\ni$ frowned upon?

I am currently taking discrete math, and we have been learning several math symbols that we have used in our proof-writing assignments. Obviously, we have discussed the $\in$ symbol for inclusion in a ...
6
votes
0answers
36 views
+50

Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
2
votes
2answers
98 views

Looking for a different type of Linear Algebra book

Are there any good linear algebra books with lots of (mathematical, preferably algebraic or geometric-flavored) applications? E.g. I'm not so interested in the typical engineering-style applications ...
3
votes
0answers
86 views

Is Fractional Calculus an important research topic “in pure mathematics” today?

Being a potential graduate student, I would like to know if fractional calculus is an actively developing research topic in the area of pure mathematics today.
0
votes
0answers
8 views

How to find basis of functions for best representation of function described in several different truncated ON systems?

I have two ON bases where I to the best $L_2$ precision possible represent the same function. Let us say that these have $n_1,n_2$ basis functions respectively for which we know the coefficient ...
0
votes
0answers
15 views

Can every separable complex Banach space be given the structure of a Banach algebra?

To be more precise, if $E$ is a separable complex Banach space, is there a Banach algebra $A$ for which $E$ and $A$ are either isometric or isomorphic as Banach spaces (linearly homeomorphic.) I ...
-1
votes
0answers
30 views

what is the story with the burning number of a graph(soft question) [closed]

I know the definition of the burning number of a graph. I saw it here: It is the minimum number of steps required for the burning process of a graph. The burning process can be defined as follow: We ...
1
vote
2answers
35 views

Why are functions as a type of relations special? [duplicate]

I.E. Why are relations where each member of the domain having one and only one mapping to a member in the range so important? I was wondering since most of mathematics that I'm familiar with (high ...
4
votes
2answers
132 views

Real Examples of Misleading Statistics

I need to give a presentation to a group of students on Tuesday about why one needs to be careful when examining statistics or mathematical results in the media or online. In his book How Not To Be ...
3
votes
1answer
32 views

Does the inverse function theorem provide a path to interpretation of more general infinitesimal quotients?

Let us first bring up inverse function theorem : If $y=f(x)$ and if $f'(x)$ exists at some point $x=a$, then exists in some neighborhood of $(a,f(a))$ an inverse function $f^{-1}(x)$ which around ...
1
vote
1answer
55 views

Most important non-elementary functions in Math

I am looking for non-elementary function that is Well behaved over $\mathbb{R}$ and $\mathbb{C}$ like piecewise smooth, not like Weierstrass function Ubiquitous and well studied Simple to compute ...
0
votes
0answers
58 views

Two simple questions regarding typesetting maths (in LaTeX)

The first question I have regards endomorphisms, especially in differential geometry. Everyone who had worked a little in (complex) differential geometry will know that it is very common to write the ...
1
vote
1answer
49 views

How to expand a vector to its monomial basis?

What exactly is monomial basis? Consider a vector of $n$ dimension. How can I write all monomials up-to a order say $k$ of this vector? If I manged to write the monomials of the vector up-to order $...
1
vote
1answer
52 views

Reviewing Advanced Calculus in $\mathbb{R}^n$ Analysis

I’ve taken a few courses in advanced calculus and real analysis (adv calc 1, metric spaces, normed spaces, lebesgue integration), but I’m realizing that my advanced calculus/R^n analysis is not as ...
1
vote
0answers
71 views

Why is the derivative called derivative? [duplicate]

What is the historical reason for this term? Derivative represents slope function of curve. I am not getting why this term make sense?
15
votes
3answers
3k views

Was there ever an axiom rendered a theorem?

In the history of mathematics, are there notable examples of theorems which have been first considered axioms? Alternatively, was there any statement first considered an axiom that later has been ...
1
vote
0answers
50 views

Does this inequality have a name?

$$\int_{B_R} f \leq C \int_{B_R} \left(\frac{1}{|B(x,r)|}\int_{B(x,r)} f(y) dy \right)$$ where $f \geq 0$, $R \geq 0$, and $B_R \subset \mathbb{R}^d$ be a ball of radius $R$. The above should hold ...
1
vote
1answer
36 views

Soft: A game where players use an arbitrary structure as a guideline while trying to play identically.

The following two player game is an (inadequate) attempt to capture an idea I am very interested in exploring. If any of the learned folks here recognize the problem I'm playing with and can point me ...
1
vote
0answers
18 views

Wrapping an ellipse onto a semi-ellipsoid

I've seen many discussions online and in papers on how circles can be wrapped onto spheres. However, if I wanted to, say, wrap an ellipse so that it perfectly covers the top half on an ellipsoid (thus ...
2
votes
1answer
93 views

Are these statements always true?

I haven't found an answer in my books. Although the question seems very simple, I want to ask. Are these statements always true? a) For any infinity non-negative integer sequence, if there is an ...
0
votes
1answer
74 views

How to improve at handling the (mild) complexity of high school trig?

I'm having trouble in Khan Academy trigonometry where there are two steps, that aren't a sequence of algebraic manipulations. I get immersed in the second step, and give that as the answer -- ...
20
votes
2answers
3k views

Why, historically, did Gödel think CH was false?

Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common ...
0
votes
1answer
45 views

What makes a good mathematical theory?

I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when ...
2
votes
1answer
53 views

Intuition behind Covering Axioms

Many concepts in General Topology are the direct abstraction of very profound and natural concepts (think of structures as topology or uniformity themselves, separation axioms, quotient and ...
1
vote
1answer
89 views

Are there any power identities which don't belong to this list?

The problem of finding expansions of monomials, binomials etc. is classical and there is a lot of beautiful solutions have been found already, the most prominent examples are Binomial Theorem, ...