1
vote
1answer
31 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
6
votes
1answer
140 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
5
votes
1answer
81 views

What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules ...
0
votes
2answers
43 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
6
votes
3answers
288 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
4
votes
1answer
96 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
3
votes
4answers
97 views

The differentiation of $ \sin, \cos$ through a Taylor Series

This question has been asked quite a lot on math SE, however, please before you mark this as a duplicate carry on reading, I will try to highlight my doubts and concerns as clear as possible. First ...
2
votes
4answers
275 views

What is the Riemann Sphere?

Reading from wikipedia I understood that Riemann Sphere is used to represent extended complex plane. But it would be great if someone could explain it in a less technical manner.
0
votes
1answer
46 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
1
vote
0answers
65 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
1
vote
1answer
52 views

How does $\exp(0)=1$ follow from the definition $\exp(z):= \sum_{n=0}^\infty \frac{1}{n!} z^n$

We introduced the Exponential function as follows: $$ \exp: \begin{cases} \mathbb{C} & \longrightarrow \mathbb{C} \\z & \longmapsto \displaystyle \sum_{n=0}^{\infty} \frac{1}{n!}z^n ...
0
votes
2answers
95 views

definition of separation axioms in topology

I am learning the Separation Axioms and came across the definition of regular space. In the definition they say, "Suppose the one point sets are closed in $X$" My question is: how can one point sets ...
4
votes
5answers
168 views

Trigonometry confusion

I was doing a bit of trigonometry, as I have been for a couple of years and it suddenly dawned on me that I don't really understand the trigonometric functions, at all. You first learn the basic trig ...
0
votes
2answers
87 views

Lebesgue Measure Definition

Given a subset $A \subset \mathbb{R}$ with the length of an open interval $\mu_L(I_k) = b_k -a_k : I \doteq [a_k,b_k]$ The lebesgue measure is defined as $$ \lambda^{\ast} (A) \doteq \inf \Big\{ ...
1
vote
2answers
124 views

Problem with definition of regular surface in classical differential geometry

I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this: I have few doubts about this definition: 1) Why we need to find ...
5
votes
1answer
223 views

The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
2
votes
0answers
73 views

What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function ...
3
votes
1answer
83 views

What's the intuition behind this definition of ordered pair in the $\lambda$-calculus?

On this page, we have the following definitions. pair = λabf.fab first = λp.p(λab.a) second = λp.p(λab.b) So I tried computing "first (pair a b)," and sure ...
0
votes
3answers
263 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
0
votes
2answers
54 views

Constraint satisfaction problem - Arc consistency

The Constraint satisfaction problem (CSP) is roughly speaking a formalism that defines a finite set of relations over a domain. The relations are simply defined by enlisting elements in certain ...
10
votes
3answers
305 views

How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
11
votes
3answers
380 views

It is possible to define our intuitive notion for probability in subsets of $[0,1]$

I've always heard and read the sentence: If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$. What is the meaning for that? Is this the "real" ...
0
votes
1answer
109 views

How information works?

I am really confused after reading wikipedia... What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information. For ...
3
votes
1answer
171 views

The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$ I have three somewhat broad questions about this: Why is it related to ...
6
votes
2answers
270 views

What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$. It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
6
votes
1answer
207 views

Axioms vs. Universal Constructions/Properties

What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...
4
votes
4answers
131 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
15
votes
2answers
1k views

Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
2
votes
1answer
263 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
7
votes
2answers
210 views

Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?

What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
1
vote
2answers
194 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
7
votes
1answer
139 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
3
votes
2answers
324 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
2
votes
1answer
240 views

Free boolean algebra

Consider the following definition: Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X ...
51
votes
9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
16
votes
5answers
1k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
1
vote
3answers
103 views

Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
18
votes
7answers
2k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
8
votes
6answers
4k views

What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...