# Tagged Questions

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

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### Definition of division algebra

The definition on Wikipedia of a division algebra $D$ is given as: Given $a,b \in D$, $b \neq 0$ there exists a unique $c\in D$: $a = bc$ and a unique $d \in D$: $a = db$. My question(s) are: What ...
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### Definition of inverse function

I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative ...
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### Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
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### Tangent Space: Identifications

Given a manifold $M$. Denote a chart by $\kappa$. Introduce the directional derivative: $$\partial:\mathbb{R}^n_a\to T_a\mathbb{R}^n:v\mapsto\partial_v\rvert_a$$ That is an isomorphism with inverse ...
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### Confusion in the definition of set

Which of the following is the correct definition for set? Set is a well defined collection of objects. Set is a collection of well defined objects.
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### Is this how the limit of a sequence of sets is commonly defined?

I was looking at the wikipedia page for the Cantor set which defined the set using a limit. I had not previously seen a limit expression involving a sequence of sets rather than real numbers, so I got ...
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### Most general definition of homomorphism and isomorphism

What is the most general definition of homomorphism and isomorphism? It is clear what they mean when there is an algebraic structure to be preserved, but what about when there is no such structure? ...
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### What is the domain of continued fraction?

I'm trying to formally define (generalized) continued fraction. Consider $[i;\sqrt2,i,i]$. This is not well defined since $i+\frac{1}{i}=0$. What would be a domain of continued fraction? (As a ...
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### From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
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### Is the symmetric definition of the derivative equivalent?

Is the symmetric definition of the derivative (below) equivalent to the usual one? $$\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$$ I've seen it used before in my computational ...
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### Meaning of the word “conjugate” across mathematics?

Clearly, the word conjugate or conjugation is used for a myriad of different concepts across mathematics and even in science (see the Wikipedia page). Its meaning can range from the fraction used to ...
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### Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.

These rules are part of an attempt to define an additive identity in terms of division in basic standard arithmetic. The difficulties with defining division by $0$ are well known. In order to ...
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### Definiton of No Tear and No Paste

Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it. First of all, ...
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### Two different definitions of “scale invariance”

I found the following definition of "scale invariance" in the book Critical Phenomena in Natural Sciences by Didier Sornette. A function $f$ is scale invariant if there is a number $\mu(\lambda)$ ...
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### what is the basic difference between a mapping and a function? [duplicate]

what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?
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### What is the definition of “sheet”?

Here is a definition for evenly covered set. Definition Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function. Let $U$ be an open subset of $X$. Then $U$ ...
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### Composite residuosity statement.

Consider the following definition. A number $z$ is said to be $n$-th residue modulo $n^2$ , if there exists a number $y \in \mathbb{Z}_{n^2}^*$ such that $$z\equiv y^n \mod n^2$$ Let us take $n=6$ ...
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### Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small $\mathbf{Ab}$-...
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### Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
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### What is effectively continuous?

In Soare's book Recursively Enumerable Sets and Degrees I saw a sentence: $\Phi_e$ is an effectively continuous functional from the Cantor space $2^\omega$ to itself. What does it mean for a ...
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### Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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### trail vs path in Graph Theory v/s Graphical Models

In my course on probabilistic graphical models, I learnt (quoting from page 36 of the book Probabilistic Graphical Models: Principles and Techniques by the same author) Path: We say that X1 , . . . ,...
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### Seeking elegant proof why 0 divided by 0 does not equal 1

Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ...
Here's a part of the definition Ross' Elementary Analysis states for limits of a function: 20.3 Definition (a) For $a\in\mathbb R$ and a function $f$ we write $\lim_{x\to a} f(x)=L$ provided \$\...