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.

learn more… | top users | synonyms

1
vote
1answer
135 views

What is the motivation behind the arbitrary union topological axiom?

1. Why is the arbitrary union axiom in the definition of topology necessary? 2. Why is it useful? Why might we expect ("intuitively") that it should be useful? 3. What is the (historical) ...
0
votes
1answer
65 views

Is there a generalisation of norm catering for $\|a\mathbf{v}\|=\|\mathbf{v}\|$?

I'm working with a function $p$ which gives a kind of "size" of the vectors in my vector space, and it has all the properties of a norm except that $$p(a\mathbf{v})=p(\mathbf{v}).$$ Ordinarily a norm ...
53
votes
10answers
5k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
0
votes
1answer
45 views

Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
10
votes
1answer
220 views

Are there parts of Integral Calculus that just *have* to be memorized?

Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view. When studying Differential Calculus, I found that ...
0
votes
2answers
86 views

Why are there different metrics? [closed]

This is a general question and I am asking out of curiosity. There are many metrics such as Euclidean norm, sup norm etc. Can you give examples/reasons why we need all these different metric notions?
2
votes
1answer
56 views

A nontechnical way to comprehend $\aleph_2$

This is possibly a dumb question, but I do not know where to look for an answer. Without getting technical, one can show why $card(\mathbb{N}) = card(\mathbb{Q}).$ (Typically by showing how the two ...
2
votes
2answers
51 views

Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
1
vote
1answer
47 views

What is the required group theory knowledge needed to understand Verhoeff's algorithm?

The Wikipedia page tells me I need to understand permutation groups and dihedral groups. Can someone clearly outline what exactly the perquisites of understanding this is and how much time I'll take ...
0
votes
1answer
32 views

Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons. I was thinking about this recently and realised Don't all vector spaces have norms on them? If they all ...
1
vote
1answer
67 views

Books about the foundations of (calculus) functions?

I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe ...
1
vote
1answer
125 views

Why am I so bad at math? [closed]

Ever since I was young, I've always struggled at math. Bear in mind I could count before I entered kindgergarden, but even when I had to learn simple things like long division, I was always behind ...
0
votes
1answer
77 views

Is it meaningful to search for “elegant” representations of mathematical objects?

For centuries we struggled with the concept of spatial rotations. We used to represent them in many different ways: mostly, Euler Angles and matrices. Those all had drawbacks and failed in specific ...
0
votes
0answers
33 views

Serendipitous discoveries in mathematics [duplicate]

I have recently been reading about serendipitous discoveries in science and I found them quite inspiring. Most of those discoveries are in Chemistry. I'm looking for examples of these kinds of ...
0
votes
1answer
32 views

Mathematical Documentaries [duplicate]

Do you know of any documentaries Like Fermat's last theorem or N is number? Story of a mathematician or a theorem.Dont know if this is a good site for the question.
9
votes
3answers
124 views

The case for L'Hôpital's rule?

While this may seem very subjective — and, admittedly, my own dislike of L'Hôpital's rule is not entirely devoid of subjectivity — I am looking here for argumented, factual answers. From what I ...
3
votes
0answers
39 views

Proof by “continuous induction”

There’s a method of proving inequalities over some interval of real numbers using differentiation. For instance to prove that $x-\log(1+x) \geqslant 0$ whenever $x \geqslant 0$ we can differentiate ...
1
vote
1answer
53 views

Spivak's Calculus for a soon to be physics undergraduate.

I'll soon start my undergrad. studies in physics and because of that I picked up Spivak's Calculus a while ago to get a solid foundation in single-variable calculus before I start my studies. However, ...
2
votes
2answers
76 views

Could I learn linear algebra before or along with calculus? (Same with differential equations)

I want to plan the next few subjects I learn (by self-study) in mathematics. I have made it through the equivalent of maybe half a Calculus I class so far, but I would like to start a bit on linear ...
7
votes
2answers
156 views

In category theory: What is a fibration and why should I care about them?

I stumbled upon the "fibration of points" in the definition of a protomodular category and apparently this is indeed an instance of a fibration. What are fibrations intuition-wise and how are they ...
0
votes
0answers
26 views

What are Bernoulli numbers? Is there any identity to find these numbers?

What are Bernoulli numbers? Is there any particular pattern to find these numbers like the Fibonacci numbers? How are they applicable in Taylor expansions of hyperbolic tangents?
0
votes
1answer
33 views

Possible Masters degrees for a maths Graduate

So I will be finishing my degree in Maths at the end of next year and was considering doing a masters in Maths combined with another discipline such as biology , Earth sciences , oceanography etc I ...
2
votes
2answers
80 views

Studying mathematics as an Undergraduate [closed]

This is not a question related to a certain topic in mathematics, but more a question about studying mathematics generally. (please forgive me if my vocabulary isn't appropiate, I'm not a native ...
2
votes
0answers
63 views

Intuitive motivation for the $\diamondsuit$-principle?

I'm having difficulty internalizing the $\diamondsuit$-principle. I understand what it says, and can see why some of the constructions that I've seen that use it work. But I don't think I understand ...
3
votes
2answers
110 views

Maclane/Birkhoff's “Algebra” as a first book on the subject?

Would the more knowledgeable and well-versed members of this community be so helpful as to give their opinion on using Birkhoff & MacLane's famous "Algebra" for a first course in Abstract Algebra? ...
3
votes
2answers
133 views

Spivak's Calculus?

I have seen many users here asking questions about problems in what they call "Spivak's Calculus Book". I have never seen the book, and information online is scarce. From what I've gathered, it is ...
1
vote
3answers
48 views

Decision theory references for advanced undergrad/early grad students?

I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
1
vote
1answer
54 views

Correct notation for presenting solutions to equations

Let's say I have a cubic equation $(x-a)(x+b)(x-c) = 0$, and I want to represent the solutions to this equation, what is the formal/conventional way that one would arrive and state the solution to the ...
3
votes
1answer
63 views

Calculus by Spivak - looking for a piece of advice from someone who has been through it.

I am currently self-studying Calculus by M. Spivak and before I started I was well aware of the fact that it is not an easy read. Right now I am going through the problems at the end of the very first ...
1
vote
1answer
17 views

Translating between Set Theoretic and Interval notation

Given an open interval (or closed it makes little difference to the question) on the Real Field $(a, b)$, where $a,b$ are real numbers, and an arbitrary predicate $P(x)$, which is true for all ...
0
votes
1answer
23 views

meaning of the notation a belongs to $ C^n[a,b]$

This notation stands for what : $X$ belongs to $C^n[a,b]$ ? I think $x$ takes values continuously between $a$ and $b$ in $n$ tuples.
5
votes
2answers
199 views

How many sorts are there in Terry Tao's set theory?

In his 2010 post, A computational perspective on set theory, Terry Tao writes: The standard modern foundation of mathematics is constructed using set theory. With these foundations, the ...
0
votes
0answers
14 views

Specific examples of Side Information?

I'm starting to apply information theory to gambling. There is something called Side information (see details in [1]), which I understand is additional information about the outs of the game. It could ...
163
votes
20answers
32k views

What are some examples of when Mathematics 'accidentally' discovered something about the world?

I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood ...
2
votes
0answers
38 views

Congruent Numbers and Integral Points on Elliptic Curves

As you probably know, congruent numbers $N$ and elliptic curves of the form $$E_N:y^2=x^3-N^2x$$ are intimately connected. While playing around with curves of this form, I found that $E_N$ will have ...
2
votes
3answers
47 views

Is there any example of a set, together with the “sum” operation which is non conmutative?

I was wondering if there is any mathematical structure (Im not even sure this is the correct way to name what I have in mind) but basically, any set, together with the operation sum, (I don´t say just ...
2
votes
1answer
29 views

Sum of powers of eigenvalues

The sum of the eigenvalues $\lambda_k$ of an $n\times n$ matrix is equal to the trace of the matrix, i.e. $$\sum_{k=0}^{n-1}\lambda_k=\text{tr}(A).$$ Is there a "closed form" sum of positive integer ...
20
votes
6answers
749 views

Non-trivial “I know what number you're thinking of”

Consider the following 'trick' (WARNING: very lame) Think of a number. Multiply this number by two. Add four. Divide the number by two. Subtract the number you were originally thinking of. I guess ...
10
votes
4answers
380 views

Examples for Hilbert's Quote

Hilbert once said, “The art of doing mathematics consists in finding that special case which contains all the germs of generality.” What would be (relatively) simple examples?
7
votes
4answers
700 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
0
votes
1answer
75 views

How do I get good at calculus in specific, Mathematics in general? [closed]

I understand that this question might look like a duplicate to some others asked before, but I assure you, read on, you'll find my case different (hopefully). I am a 12 grade student from the ISC ...
0
votes
0answers
39 views

Alternative view of matrix inversion (explanation required)

We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we ...
3
votes
3answers
416 views

Do Gödel numbers have a practical use?

Is there any example of Gödel numbers being actually used in practice? If so for what purpose?
0
votes
0answers
17 views

Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
3
votes
2answers
41 views

Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
1
vote
1answer
42 views

Good introduction to cohomology of spaces?

I'm trying to read chapter $3$ of Hatcher but I find it a bit difficult to read. I really only made it through the first two chapters because I had in-class lectures to go along with the reading. Does ...
16
votes
4answers
608 views

Gaining Mathematical Maturity [closed]

I was redirected here by a kind fellow from math.overflow. This is not a typical math question, so I apologize if that is discourteous. I am currently a sophomore in my undergraduate mathematics ...
0
votes
0answers
15 views

Unsolved problems involving or concerning the unitary DFT

Does anyone know of any unsolved problems involving or concerning the unitary discrete Fourier transform matrix $F_n=n^{-1/2}(f_{j k})$ where $f_{jk }=e^{2\pi j k i}$ and $i=\sqrt{-1}$, or its inverse?...
12
votes
1answer
163 views

Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and https://en.wikipedia.org/wiki/...
1
vote
0answers
63 views

Rigorous Approach to Precalculus

I've made the mistake of looking at more advanced texts that deal with precalculus-level mathematics in a more formal, rigorous way than usual. Perhaps this isn't a mistake, but now that I've glimpsed ...