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

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3
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2answers
162 views

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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0answers
28 views

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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0answers
177 views

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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1answer
215 views

Difference between $\Delta f$ and $\Delta f(x)$

What is the difference between $\Delta f/\Delta x$ and $\Delta f(x)/\Delta x$? Are they the same? I've been watching a lecture and the professor seems to describe the slope $$m = \lim_{x\to0} \Delta ...
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1answer
45 views

Lower Central Series and Generators

Let $G$ be a group generated by $S=\{x_1,\cdots,x_n\}$ and let its lower central series be defined as $\Gamma_1=G$, $\Gamma_m=[\Gamma_{m-1},G]$ for $m\geq 2$. By definition $\Gamma_m$ is generated by ...
2
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1answer
220 views

Coercivity definitions

Hi I was given the following definition of coercivity: Let $V$ be a Banach space. The first definition: $A:V \rightarrow V^{*}$ is coercive iff $\exists \zeta: \mathbb{R}^{+} \rightarrow ...
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1answer
97 views

Definition of $C^1, $the vector space of continuously differentiable functions

I asked a question on clarification of the symbol $C^k$. It was confirmed to me that $C^k$ is actually a space of functions. Now my next question in the definition is on $C^0$ and $C^1$. $C^0$ is ...
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1answer
111 views

Definition of Butterfly Effect

The Wikipedia definition of the Butterfly Effect seems to imply that linear functions can exhibit the Butterfly Effect. In particular if the state space is $\mathbb{R}$ with the usual metric then if ...
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1answer
28 views

What is the name of this function? index?

Burton- Number theory p.163 Let $n\in\mathbb{Z}^+$. Let $r$ be a primitive root mod $n$, so that $<r>=\mathbb{Z}_n^*$ Let $a\in \mathbb{Z}_n^*$ Let $k$ be the smallest ...
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1answer
43 views

Definition: Equipollent

Does the term Equipollent simply mean bijective? I have seen that by definition a mapping is equipollent iff it is bijective. What is the point of such a statement? Context: It will be used in ...
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1answer
53 views

Definition of the set of independent r.v. with second moment contstraint

I am trying to nice write the definition of the following set. Def: The set of all distributing of the pair $(X_1,X_2)$ such that $X_1$ and $X_2$ are independent Have second moment constraint ...
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2answers
172 views

Conceptual definition: Injection, surjection and Bijection.

I was wondering if this conceptualisation is correct: Injection means that we don't have two arrows come from an element of the domain towards the range. Hence we don't have one archer standing in ...
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2answers
152 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
2
votes
1answer
71 views

Is the symmetric definition of the derivative equivalent?

Is the symmetric definition of the derivative (below) equivalent to the usual one? \begin{equation} \lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h} \end{equation} I've seen it used before in my computational ...
5
votes
1answer
201 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
2
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1answer
192 views

How the branch cut make a multi-valued function several branches of single valued function?

In the wikipedia article, it describe the branch points and branch cuts: A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued ...
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1answer
35 views

proving that a subset of a set has a functional mapping that is a subset of another

i wanted to prove: Let $f:X\to Y$ be a mapping from $X$ into $Y$. Show that if $A$ and $B$ are subsets of $X$, then: $$(A \subset B) \implies \left(f(A) \subset f(B)\right)$$ but i thought ...
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1answer
85 views

what is a fiber for function mapping?

I have read that a fiber is the pre image of a mapping.. Does this mean that I can think of a fiber as a line that connects x to y where the line is a function? So for example with $f(x)=2x$ there ...
3
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1answer
116 views

Is this “the winding number”?

Note that $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}:z\mapsto e^z$ is a covering map. Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be a closed curve. Let $\gamma$ be any ...
0
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2answers
47 views

True false about direct sum and their bases of vector spaces

I am not entirely sure about the following true/false questions For all the following : $V$ a vector space and $W_1$ and $W_2$ two subspaces such that $V = W_1 ⊕ W_2$ 1) for all subspaces U of V : ...
8
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1answer
365 views

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 ...
2
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1answer
132 views

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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0answers
62 views

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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1answer
41 views

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 $\mu(\lambda)$ ...
3
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2answers
108 views

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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1answer
114 views

How is the area of a set of points in $\Bbb R^2$ defined?

Let $S$ be a subset of $\Bbb R^2$. If no vertical slice of $S$ contains gaps, we could define the area of $S$ through the following. $$A(S) = \int_{-\infty}^\infty\left(\sup\{y\in\Bbb R\mid (x,y)\in ...
3
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0answers
23 views

Does this matrix have a name: $x=(x_0, \ldots, x_n)$, $S(x)_{i,j} = x_{i+j \mod(n)}$. Can you help me generalise the idea?

Apologies if this question is vague/unclear (especially the title). Given a vector $x=(x_0, \ldots, x_n)$ define the matrix $S(x)_{i,j} = x_{i+j \mod(n)}$. $$ S(x) = \left( \begin{array}{rrrr} ...
2
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1answer
27 views

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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0answers
31 views

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 $n=6$ ...
5
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1answer
80 views

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 ...
2
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1answer
182 views

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 ...
3
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1answer
68 views

Caratheodory: Construction

Given a semiring. Consider a premeasure. Regard the following constructions: $$\inf_{A\subseteq S_1\sqcup\ldots\sqcup ...
3
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2answers
700 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
3
votes
1answer
46 views

Why Not Define Connectedness to Mean Path Connected?

All spaces I have seen which are connected are also path connected (apart from examples to show that the two are not equivalent). Is there a reason for using the weaker definition of connectedness ...
2
votes
1answer
53 views

What is a linear equation?

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?
2
votes
3answers
359 views

Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)?

In the expression $\frac{1}{b}\pmod m$, where $(b,m)=1$, is $\frac{1}{b}$: a) a rational number (and so rational numbers are defined in modulo arithmetic using multiplicative inverses)? b) just ...
2
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0answers
60 views

What is the formal definition of a structure?

I can't seem to find one anywhere, and I've looked in several books and pages. I don't even know how to tag this.
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1answer
36 views

Cauchy - Sequences. Different definitons

Is $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall j \ge N |a_j - a_N| < \epsilon $$ equivalent to $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall m,n \ge N |a_m - a_n| ...
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2answers
49 views

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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0answers
29 views

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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0answers
34 views

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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7answers
2k views

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 ...
2
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0answers
38 views

Synonyms for “Theorem”

Some mathematical results, despite being formally proven, are not actually called "theorem". Examples include: Bertrand's postulate Pigeonhole principle Law of large numbers Do these names imply ...
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1answer
161 views

Why is an open interval needed in this definition? (definition of a limit of a function)

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 ...
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1answer
30 views

On the definition of transition maps

When defining a manifold the domain and codomain of the transition maps is usually denote like this: $$\varphi_\eta \circ \varphi_\lambda^{-1}: \varphi_\lambda(U_\lambda \cap U_\eta) \to ...
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0answers
189 views

Anti-Diagonal Matrix

Hi just a quick question. Can a anti-diagonal have zero in its diagonal? I have this definition in my paper. "An anti-diagonal matrix is a square matrix where all the values are zeroes except those ...
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1answer
122 views

What is Bourbaki's definition of subfield? or categorical definition of subfield?

Let $F$ be a field. Let $K$ be a subset of $F$ which is closed under two binary operations $+,\cdot$ Assume $(K,+,\cdot)$ is a field. Is $K$ called a subfield of $F$ in Bourbaki's definition? Or, ...
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1answer
271 views

What is a sparse subset?

In a work about fully homomorphic encryption I found usage of the expression: "sparse subset", as in: Our hint will consist of a set of vectors that has a (secret) sparse subset of vectors whose ...
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1answer
68 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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8answers
1k views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?