# Tagged Questions

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.

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) ...
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 ...
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,"...
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. ...
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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, ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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? ...
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 ...
48 views

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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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?
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 - ...
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 ...
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 ...
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 ...
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?...