For requesting, clarifying, and comparing definitions of mathematical terms.

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3answers
512 views

Quotient spaces in linear algebra

There's a statement in some notes I'm reading that goes like this: "...$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces). I'm still ...
2
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2answers
225 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 ...
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1answer
425 views

Is this definition of limit at infinity of complex functions correct?

In my book (Churchill), a limit of a function at infinity is defined as: $$ \lim\limits_{z \to \infty}f(z) \equiv \lim\limits_{z \to 0}f(\frac{1}{z}) $$ But why can't you define the point at infinty ...
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1answer
25 views

Why isn't $r^{\frac{n}{2}}$ classified as an involution?

Suppose you have a $(2n)$-gon for some $n \in \mathbb{N} : n > 1$. Then the rotation $r^{\frac{n}{2}}$ where $n$ is the number of vertices imposed on the $(2n)$-gon is the same as an involution. ...
5
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1answer
61 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
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3answers
70 views

Definition of Linear Differential Equation

I am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I ...
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2answers
97 views

Similar linear operators and change of coordinates

Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of similar: "We say that operators $S,T \in \mathcal{L}(V)$ ...
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2answers
1k views

Why do functions with compact support include those that vanish at infinity?

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set". Functions with compact support in $X$ are defined in ...
2
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1answer
113 views

What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
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1answer
963 views

Why is the Logarithm of a negative number undefined?

The Definition of a Logarithm is: If $$x^y=a$$ Then $$\log_xa=y$$ Given this definition, since $$e^{i\pi}=-1$$ Then shouldn't $\ln(-1)=i\pi$? What is wrong with it?
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0answers
87 views

Definition of a Turing Machine

Could someone explain the following definition of a Turing Machine? A Turing Machine $M$ is defined formally by a tuple $(\Sigma, Q, \delta)$ Where $\Sigma$ is a finite set representing the number ...
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1answer
96 views

What is a “lemma”? [duplicate]

As per title, what is a "lemma"? How is it different from "theorem"? ASAIK, I have to prove a self-proposed theorem in my paper. Do I also have to provide the proof for a self-proposed lemma?
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3answers
59 views

Topology generation

What does it mean for a topology to be generated? For example $X=\mathbb{R}$ be topology generated by $[a,b)$. Isn't the topology a collection of open sets? $[a,b)$ is not open though.
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2answers
50 views

How do we know we can do cancellation in $\mathbb{Z}$?

For example, $2*x = 2*5$ implies $x = 5$ but how come, if $2$ doesn't have an inverse?
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4answers
113 views
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4answers
345 views

Why do we think of a vector as being the same as a differential operator?

I'm reading Frankel's The Geometry of Physics, a pretty cool book about differential geometry (at least from what I understand from the table of contents). In the first chapter, we are introduced to ...
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2answers
158 views

Spacelike curves definitions

Well, I am looking for the definition, if there is any, of points separated by a spacelike curve in a Lorentzian or more generally in Semi-Riemannian space?
2
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1answer
110 views

Differing definitions for 'Algebra of subsets'

For a collection, $A$ of subsets of a set $X$ to be an algebra of subsets it must satisfy the following properties: $A$ is non-empty If $E \in A \implies E^c \in A$ If $E, F \in A \implies E \cup F ...
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0answers
58 views

evaluating a meromorphic section of a line bundle at a point

Let $D$ be a Cartier divisor of a variety $X/K$ with associated line bundle $\mathcal{O}(D)$ and meromorphic section $s_D$. How do you define $s_D(P) \in K$ for $P \in X(K) \setminus ...
2
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1answer
41 views

How to define “closer to proportion”

$\def\prop#1#2#3{#1:#2:#3}$Let's say I have a proportion, $\prop 135$ And then a set of other ones, $$ \begin{array}{l} \prop 235\\ \prop 145 \\ \prop 136 \end{array} $$ Which one is closer to ...
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1answer
69 views

$K$-Category $M(0, 0) = M(A, 0) = M(0, A)$ using definition from Swan's 'Sheaf Theory'

I'm using the following definition: A category $\mathcal C$ is given by the following: A collection of objects $A$ A set $M(A, B)$ for any two objects $A, B \in \mathcal C$. A function $M(B, C) ...
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2answers
128 views

If $a$ is proportional to $b$ does it imply that $b$ is proportional to $a$?

If $a$ is proportional to $b$ does it imply that $b$ is proportional to $a$? This is a simple question which I have found others giving different answers.
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2answers
619 views

What is an indeterminate in a polynomial ring?

I am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where ...
5
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1answer
732 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 ...
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2answers
142 views

How do the terms “countable” and “uncountable” not assume the continuum hypothesis?

Every countable set has cardinality $\aleph_0$. The next larger cardinality is $\aleph_1$. Every uncountable set has cardinality $\geq 2^{\aleph_0}$ Now, an infinite set can only be countable or ...
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7answers
798 views

What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
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1answer
137 views

What does it mean when a set is said to be a “finitely generated vector space”?

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means. Generated by what? Generating what? And for what? I ...
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1answer
78 views

About definition of endomorphism on vectos space

"let $ f $ be a homomorphism between two vector spaces $V$ and $W$, $f$ is endomorphism on $V$ if $im(f) \subseteq V$" is correct? Thanks in advance!
2
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0answers
87 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
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0answers
494 views

What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
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1answer
661 views

Why is the direct substitution property so specific

Mt text book states the Direct Substitution Property as If f is a polynomial or a rational function and a is in the domain, then $$\begin{align*} \lim_{x\to a} f(x)=f(a) \end{align*}$$ Why does ...
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1answer
203 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
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1answer
93 views

Is $\Bbb Q[\sqrt2]$ cyclotomic?

This overview of Galois Theory claims that a field extension of $F$ is cyclotomic if it's obtained by adjoining an $n$th root of any element of $F$. Wikipedia claims you have to adjoin a root of unity ...
6
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2answers
149 views

Differential — Mathematically conform?

In calculus, I know that one defined the differential quotient $$\frac{dy}{dx} := \lim\limits_{h \rightarrow 0}{\frac{y(x+h)-y(x)}{h}}$$ I learned that it is not a quotient, but can be treated as one ...
2
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0answers
94 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
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2answers
43 views

Why is two-element null semigroup excluded from the $0$-simple semigroup definition?

My question is probably a little bit silly, but still.. The definition of $0$-simple semigroup states, that a semigroup $S$ with zero is called $0$-simple, if $\{0\}$ and $S$ are it's only ideals and ...
1
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1answer
84 views

What does it mean to be compatible with the isomorphism structure of a class?

Let $\mathrm{UR}$ denote the class of all unital rings and $\mathrm{Set}$ denote the class of all sets. Actually, perhaps it would be better to view $\mathrm{UR}$ as the groupoid whose objects are ...
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1answer
137 views

Function being continuous at a point

I've been looking at the $\epsilon-\delta$ arguments for determining whether a function is continuous at a point. I'm really stuck on how to choose your $\epsilon$. Specifically lets look at the ...
0
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2answers
198 views

Complex Analysis Question About Analytic Functions

I have some questions about knowing where and where not functions are analytic. Here's a function, f(z)= $\frac{Log(z+4)}{z^2+i}$ -I know that this function is not defined for ...
5
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1answer
294 views

Difference between a type and a set

I've been trying to understand this distinction for a while, buts its still not making sense to me. Originally, I thought the distinction between type and set was as follows. The relationship ...
2
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2answers
80 views

Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $?

Is the equality below true? $$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
2
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2answers
240 views

Can multiplication be defined without addition?

I'm struggling to understand how to define multiplication and addition, now that I've been told that multiplication is not just repeated addition. It seems that the axioms for the two are ...
5
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5answers
272 views

No difference between $0/0$ and $0^0$?

I have seen discussions about both $0/0$ and $0^0$ and they differ a bit in the way that most seem ok with calling $0/0$ "undefined", while the $0^0$ discussion still seems like a dispute. If this is ...
3
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5answers
1k views

What is the difference between a Definition and a Theorem?

This may get into a discussion, but I have a homework problem and it tells me there is a difference between a definition and a theorem. I don't know how to differentiate the two in this question: ...
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0answers
375 views

Arc Length Parametrization

Professor was a little fuzzy on this topic, so I just wanted to make sure I have this definition correct: Given a function $\alpha : T \to C \mid t \in [a,b]$ , where $t$ is the parameter and $C$ is ...
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0answers
43 views

What's a algebraically isolated singularirty?

What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable $1$-forms here, though the definition may be more general.
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1answer
86 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
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1answer
125 views

A function “extends” to the cone on X

I have the following statement: A map $f : X \rightarrow Y$ is nullhomotopic if and only if it extends to the cone on $X.$ My problem is that I have no idea what "extends" means in this statement (I ...
7
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4answers
264 views

The interest rate last year was 2%, this year it is 3% - did interest rates go up 1% or 50%

I've heard some experts say 1% and other experts say 50% to describe this same scenario. Can both be correct? Which one is more mathematically correct? How do you remove ambiguity when trying to ...
2
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3answers
57 views

What does the sentence “The only sub-algebras of $\mathbb{R}^{2}$ are $0,\mathbb{R}^{2},\mathbb{R}(0,1),\mathbb{R}(1,0),\mathbb{R}(1,1)$” mean?

I started studying functional analysis a couple of days ago, I have reached the Stone-Weierstrass theorem which is stated in my lecture notes as Let $X$ be a compact metric space, $A\subseteq ...