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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.

1
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0answers
24 views

Why is Lebesgue measure theory asymmetric?

A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only external ...
2
votes
2answers
39 views

Equality occurring at two different values for a three way inequality

The question (which was posted by another user) says to prove $$\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2+ab}{3}\leq \dfrac{a^2+b^2}{2}$$ for any $a,b\in\mathbb R$. This part I was able to solve but ...
1
vote
0answers
31 views

Amount of knowledge required of set theory and logic to pursue undergraduate mathematics

Currently i am studying undergraduate mathematics. My current understanding of set theory and logic comes from chapter 1 of Munkres' topology and from Rudin. I am of the opinion that this is not ...
0
votes
2answers
26 views

Simple variable understanding question

A function $f \colon X \to Y$ is a rule which assigns to each element $x \in X$ a unique element $y \in Y$. I don't understand what it is meant by each element $x$. If it was “all $x$” or “each $x$” ...
2
votes
0answers
47 views

Intuition/Simple Proofs required about $kernel$, $rank$, $co-rank$

This question might be so elementary , but I love to see some geometric/algebraic approach for these facts concerning transpose matrix. I know some basic thing about DUAL space but those are not ...
-1
votes
1answer
62 views

Best math workbooks?

I'm learning math with Khan Academy, and currently I'm doing polynomials (division, factoring, quadratics etc.), and I want to do more exercises than it's available there. Do you know any good ...
6
votes
0answers
84 views

Is it okay that the objective of a math thesis is to give a new proof of old theorem?

In a math thesis, no matter it is in undergraduate or PhD, is it okay that the objective of a math thesis is to give a new proof of old theorem? Even though the new proof may be more complicated or ...
2
votes
0answers
65 views

Asimov's psychohistory and real math [on hold]

In Asimov's Foundation, math based psychohistory is referenced to model and predict not less than the history of the human kind. Given that is of course just fiction, is there any existing ...
6
votes
3answers
1k views

How to know when to give up on a ${}$ math problem, which may not have a solution? [on hold]

When you're working on problems from textbooks, you know that the problem has a solution, because the author claims that it has. But when you're doing original research, you generally don't know ...
0
votes
0answers
42 views

Are there any advantages of treating four-dimensional Euclidean space as $\mathbb{C}^2$ instead of $\mathbb{R}^4$?

When we treat four-dimensional Euclidean space as $\mathbb{R}^4$ we can easily define a line, plane, and 3-plane by linear equations, whereas if we wish to do the same with $\mathbb{C}^2$ we have a ...
0
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0answers
33 views

Concerning a Gentle Introduction to Group Theory.

Can someone please suggest which of the following texts should iuse for Elementary Group Theory. I have consulted the following texts John F. Humphrey's A Finite Course in Group Theory. Dummit and ...
1
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0answers
46 views

Is exterior calculus efficient for simple vector calculus problems?

Exterior calculus and invariant formulations are important and lead to many breakthroughs and great insights in physics and mathematics. But for daily vector calculus tasks, I still struggle to apply ...
0
votes
1answer
111 views

Learning Differential Geometry through General Relativity

I was wondering whether it is possible to get a fairly good understanding of the subject of differential geometry through learning General Relativity (eg from Woodhouse's book)? Ie the sort of ...
0
votes
0answers
40 views

Lemmas used to prove a theorem which is discovered to be useful afterwards [closed]

I would like to know some history about such things: Could someone give some interesting examples of lemma which is originally proved in order to serve as lemmas of big theorems, but is discovered to ...
1
vote
4answers
87 views

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$ Here [.] denotes the greatest integer function. My Try : I know how to solve this kind of problem :$\lim_{n \to \infty}...
2
votes
0answers
80 views

How can I learn about the Monster group?

There are several questions about the Monster group on this site, but none really answer the question in the title. While reading about groups in a first year algebra course, I was told about the ...
0
votes
5answers
100 views

Natural example to “negative $\times$ negative = positive”

Please help me to find some natural example of the meaning of a negative $\times$ negative = positive, except velocity, distance etc in physics. Simplest example to mind is $$a(t) \times v(t)$$ but I ...
84
votes
11answers
12k views

Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?

In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object,...
8
votes
3answers
163 views

Where are the model theory concepts from?

Look at the following definition. Definition. Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we ...
3
votes
2answers
99 views

Can $\mathbb R$ be written as $(-\infty , \infty)$?

I was thinking about if $\mathbb R$ could be written as $(-\infty , \infty)$. I'm not sure if it's okay, because I've read somewhere (I can't remember where) that $(-\infty , \infty)$ declares ...
10
votes
4answers
570 views

Vladimir Arnold on formal thinking

In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically ...
5
votes
0answers
70 views

How to make a question regarding proof verification more interesting and more attractive? [migrated]

Since I self-study mathematical analysis without formal teacher, I can only appeal to help from out site most of the time. It's obvious that to grasp the underlying concepts in mathematics, we must ...
1
vote
0answers
48 views

How can I get better at replicating/remembering proofs? [closed]

Whenever I read proofs, I make sure to understand every detail, and the steps in between those details. I am very diligent in this. However, I notice that if I think back to a theorem some time later,...
8
votes
0answers
94 views

What is the largest number used in a useful mathematical proof that isn't just an upper or lower bound? [on hold]

There are quite a few famous gigantic numbers used in useful mathematical proofs, like Skewes's number, Graham's number, and the number $2 \uparrow \uparrow 10^{10^6}$ from Coward and Lackenby's 2011 ...
0
votes
0answers
40 views

Roadmap for Mathematicians to join Space Research Institutes [on hold]

I am a student of Pure Mathematics,I have completed my Masters in it. I have a dream of joining NASA as a Mathematician,so I was wondering what is a possible roadmap to join it. I found from their ...
1
vote
0answers
30 views

How rich is the class of “vertex-transitive graphs”?

I wonder how rich the class of vertex-transitive graphs is. Of course, "richness" is only vaguely defined, but let me give an example for what I mean. Example. Take the graphs of maximum degree three....
-1
votes
2answers
98 views

Prime Sequences in Nature [closed]

I've heard that prime numbers are considered to be important in the field of cryptography, however are there instances in nature where prime numbers emerge? I wonder if there any examples in nature ...
2
votes
2answers
51 views

A suitable device or software for taking note in class when there are many mathematical formulas. [closed]

I will do a mathematical degree. and I need take note in class. But I do not want to use the paper to take note. Any suggestions? I need to write a lot of mathematical formations alongside text many ...
0
votes
4answers
30 views

What is the need of homogeneous function? [closed]

I understand what an homogeneous function is, but I just can't seem to understand the purpose of having this concept of function. What does it even tell us? I mean if $f(tx,ty) =t^n f(x,y)$ then what ...
0
votes
1answer
40 views

Classical groups - applications?

Are there any applications of classical groups in subjects like algebraic geometry, algebraic number theory, algebraic topology or arithmetic geometry? If there is, then can anyone please give some ...
3
votes
3answers
87 views

How do we get past how **every** outcome is very unlikely?

Edit: This question is about rejecting the null hypothesis. Last month my evil twin and I were at a game show. The rules are as follows: There is a sealed booth with two magic boxes. Box A has a ...
-2
votes
1answer
79 views

What is the need of topology? [closed]

I am not getting that what is the actual notion of topology and what are the best books of topology for beginner. I always try to visualise the topology but I fail. I am not getting any idea. How ...
6
votes
0answers
57 views

$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$

It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations: $\sum_{n=0}^\infty \frac{1}{p^n}=q;$ $\sum_{n=0}^\infty \...
0
votes
0answers
18 views

Elements over transcendental extension

Let $F$ a field and suppose that $E$ is a extension field of $F$. Now, take $\alpha\in E$ trascendental over $F$. My question is about the form of the elements of $F(\alpha)$. I think that $$F(\alpha)=...
13
votes
1answer
174 views

Concrete Problems that can be solved by appealing to a Moduli Space

I have always enjoyed the idea of creating "parameter spaces" or "moduli spaces," but it is only recently that I have seen very concrete applications of studying the moduli space. Because of how ...
-5
votes
1answer
50 views

Assumptions and axioms [closed]

By Godel, if mathematics is consistent it must be incomplete. Given that mathematics is essentially believed to be a tautology it must be incomplete. Can we assume mathematics itself as an axiom? ...
6
votes
1answer
101 views

Books written similar fashion to the book of Munkres, Analysis on Manifolds, on any subject

Question: I'm looking for writers and their books that are written in similar fashion to the book of Munkres Analsis on Manifolds. In other words, books with the properties (or at least with a ...
0
votes
0answers
37 views

User Profile Idea [migrated]

Generally, I do not post such a question, but I am just throwing out an idea for this site. I know this might not even be the right community to post such a question, but any help would be appreciated....
4
votes
6answers
377 views

Some good book suggestions beyond linear algebra?

High school student here... Recently I was told I could go to the book store and pick out any math books I want. (2 or 3) Does anyone have some good suggestions? I'm comfortable with anything ...
3
votes
0answers
66 views

Is Godel's incompleteness theorem unavoidable?

So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other ...
0
votes
0answers
32 views

standardized form for products of surds

As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be ...
7
votes
2answers
115 views

A list of proofs of Fourier inversion formula

The reason for this question is to make a list of the known proofs (or proof ideas) of Fourier inversion formula for functions $f\in L^1(\mathbb{R})$ (obviously adding appropriate hypothesis to get a ...
0
votes
0answers
21 views

Notation of branch cut of logarithm

Just a small question on the notation of showing how the branch cut is chosen: $\log(z-c)$ is defined as $\arg(z-c)\in[\theta_0,\theta_0+2\pi)$ with straight branch cut. VS $\log(z-c)$ is ...
14
votes
4answers
163 views

What about linearity makes it so useful?

Among all areas of mathematics, linear algebra is incredibly well understood. I have heard it said that the only problems we can really solve in math are linear problems- and that much of the rest of ...
3
votes
2answers
86 views

Can Brownian motion be regarded as chaos?

I want to know if it is included in chaos. Does it have boundedness, deterministic, initial value sensitivity that is characteristic of chaos?
3
votes
6answers
143 views

What is a number in math? [closed]

Before I begin, let me give you so background. I previously asked a question on "How to prove that −x is not equal to x just because they yield the same result when in $x^2$". This got me thinking. ...
3
votes
0answers
57 views

Does the definition of division by zero in Wheel theory actually make sense?

I came across this question : Will Division by Zero be Defined Eventually? and was very surprised that there is a theory, called Wheel theory, which tries to make the division by zero meaningful. ...
1
vote
4answers
313 views

Why complex plane is defined? [duplicate]

I am not getting the motivation behind defining Complex plane. I mean why defining such plane helps to understand $i$? So if I want, can I replace any other number in place of $i$( suppose, I replace ...
1
vote
1answer
61 views

How can I propose and recieve ideas of a hard problem? [closed]

Where can I propose and discuss ideas, get more mathematicians involved in discussions and propose ideas on a mathematical problem? I have the feeling that Math StackExchange is not the right forum ...
0
votes
1answer
61 views

What is the difference working in $L^2$ and $C^2 $ for Fourier series expansion of functions?

Once one of my lecturer said that if we worked in $L_2$, the complex inner product $$(u,v)=\displaystyle\int^L_0 u\ \bar v\ dx $$ works flawless for Fourier series, instead of working in $C^2$. ...