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

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7
votes
4answers
346 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 ...
0
votes
2answers
175 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
votes
1answer
122 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 ...
1
vote
0answers
60 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
votes
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 ...
0
votes
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) ...
0
votes
2answers
131 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.
0
votes
2answers
665 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
votes
1answer
968 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
2answers
145 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 ...
8
votes
7answers
842 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 ...
1
vote
1answer
140 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 ...
0
votes
1answer
82 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
votes
0answers
89 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
0answers
522 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, $f(z)=\sqrt{1-z}$.
0
votes
1answer
696 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 ...
0
votes
1answer
216 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 ...
1
vote
1answer
95 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
votes
2answers
153 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
votes
0answers
101 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: ...
1
vote
2answers
47 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
vote
1answer
88 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 ...
1
vote
1answer
139 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
votes
2answers
205 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 ...
6
votes
1answer
335 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
votes
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
votes
2answers
256 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
votes
5answers
275 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
votes
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: ...
1
vote
0answers
380 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 ...
1
vote
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.
1
vote
1answer
88 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 ...
1
vote
1answer
130 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
votes
4answers
265 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
votes
3answers
58 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 ...
0
votes
1answer
33 views

Question about theorem involving GCDs?

From http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/11.html, it says Theorem 1.1.4. Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either ...
1
vote
1answer
43 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
0
votes
2answers
143 views

Donald Knuth's notations on multiple sum

I am reading Donald Knuth's Concrete Mathematics (2nd Edition) and I am on chapter 2 (Sums). I have problems in understanding his some notations on multiple sums. I quote his explanations I can't ...
4
votes
1answer
112 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
2
votes
2answers
73 views

definition of rectangle

I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle. In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. ...
2
votes
1answer
65 views

What is the function that is not a binary function called?

A binary operation is a calculation involving two elements of the set and returning another element of the set. Suppose it doesn't return an element of the set. What is the function called? For ...
4
votes
1answer
161 views

Multiple-valued analytic functions

Although our definition requires all analytic functions to be single-valued, it is possible to consider such multiple-valued functions as $\sqrt{z}$, $\log z$, or $\arccos z$, provided that they ...
4
votes
2answers
857 views

What does “characteristic” mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely". ...
3
votes
1answer
93 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
2answers
671 views

Understanding the definition of domain in Complex Analysis

I have a definition in my book which states, "a nonempty open set that is connected is called a domain." I understand what an open set is (a set containing none of its boundary points and I know what ...
4
votes
1answer
53 views

Is there a specification language that covers all of these classes of structures?

Classes of mathematical structures abound in modern math. Examples include: The class of all groups. The class of all partially-ordered sets. The class of vector spaces. The class of ordered fields ...
1
vote
1answer
198 views

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? I'm looking at a proof where they only show that $f$ is continuous and 1-1. Then I looked ...
-1
votes
2answers
174 views

Example of a manifold?

Why is this picture an example of a $1$-dimensional manifold? My thought process is: the circle must have a point removed from it because otherwise it would be self-intersecting, and self-intersection ...
0
votes
1answer
52 views

Does this definition say what I want it to?

I don't have a real background in math but I should still be able to define stuff in my MSc thesis, although the thesis does not involve a lot of math. I want to define an object $o$'s ...
3
votes
1answer
93 views

How to define $a^x$?

It's so common that we use the function $f(x)=a^x$. But actually how do we define it? In simple language we can say $a^n$ is the number $a$ multiplied with $a$ $n$ times for any $n$ in $\mathbb{N}$ ...