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

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Question about function composition

All textbooks and websites I've consulted define function composition thus: Let $f: A \rightarrow B$ and $g: B\rightarrow C$ be functions. The composite of $f$ and $g$ is the function $f \circ g: ...
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2answers
214 views

What is manifold in Geometry?

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand. A ...
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23 views

Definition of open relative, can't understand explanation

I can't figure out this paragraph: $E$ is open relative to $Y$ if to each $p\in E$ there is associated an $r\gt 0$ such that $q\in E$ whenever $d(p,q)\lt r$ and $q\in Y$ Does this look like this:...
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53 views

“Length” of an element in a free group

Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the ...
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2answers
118 views

$(+\infty,+\infty,\cdots,+\infty)$ exists in $\mathbb{R^{n}}$?

$(\mathbb{R}^{n},d)$ is a metric space and $d$ is the standard metric on $\mathbb{R^{n}}.$ Let $(\mathbb{R^{n},\tau_{d}})$ is the topology space induced by metric space $(\mathbb{R}^{n},d)$ .We can ...
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83 views

How to formalize that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0 \implies$ $g$ “grows faster” than $f$?

I understand that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0$ implies that, for sufficiently large values of $x$, $f(x)<g(x)$, as a direct consequence of the definition of limit to $\infty$....
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2answers
241 views

Formal definition of forgetful functor

Given the definition of a category $\mathbb{C}$ (that I rewrite just to agree on the notation), that consists of a collection of objects $\mathsf{Obj} ( \mathbb{C} )$; a collection of $\mathsf{Arr} (...
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151 views

A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”

I'm reading Gian-Carlo Rota's book "Indiscrete Thoughts". In page 220 I came across a strange quotation with very few explanations: We thought that the generalizations of the notion of space had ...
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66 views

About the function $f(x)=\sin x\ln x^2$ and derivative definition

$f(x)=\begin {cases}\sin x\ln x^2 & x\neq 0\\ 0 & x=0\end{cases}$ When I try to find the derivative on $x=0$ with the defintion I get: $\displaystyle\lim_{h\to 0}\frac {f(h+0)-f(0)}{h-0}=\...
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21 views

Example of H-set

A subset $Z$ of a topological space $X$ is said to be an $\textbf{H-set}$ if there exists a transfinite decreasing sequence $\{ F_{\sigma}:\alpha < \kappa \}$ of closed subsets of $X$ such that $Z=\...
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81 views

Does this definition of the “roots” of an element of an arbitary $R$-algebra make sense? If so, where can I learn more?

(All my rings and $R$-algbras are commutative and unital.) Question. I think it makes sense to speak of the "roots" of an element of an arbitary $R$-algebra; a definition is given below. Does it ...
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0answers
30 views

Definition of Dihedral group via semidirect product

Let $G$ be an abelian group and let $\varphi:Z_2\rightarrow Aut(G)$ be the homomorphism where $\varphi(\overline{1})$ is the inversion automorphism. Define $Dih(G)=G\rtimes_{\varphi} Z_2$. Now set $...
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1answer
219 views

Categorical definition of subdomain

Let $R$ be an integral domain and $S$ a subrng of $R$. Definition 1. $S$ is a subdomain of $R$ iff $S$ is an integral domain Definition 2. $S$ is a subdomain of $R$ iff $S$ is an ...
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2k views

How to find the degree of a field extension

I don't quite understand how to find the degree of a field extension. First, what does the notation [R:K] mean exactly? If I had, for example, to find the degree of $\mathbb Q (\sqrt7)$ over $\mathbb ...
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1answer
62 views

How to find out who firstly introduced a mathematical concept?

I am wondering if there is any way that one can find out the introducer of a given mathematical concept. For example, if I want to write that "Reduced free groups were firstly introduced in Habegger, ...
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44 views

Difference between talking about collection $\{G_\alpha\}$ of open sets and finite collection of $G_1,\dots,G_n$ of open sets

Question: What is the difference between talking about "Any collection $\{G_\alpha\}$ of open sets" and "Any finite collection of $G_1,\dots,G_n$ of open sets"? I imagine they are highlighting the ...
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73 views

Equivalence of two definitions of the derivative of a real function

The derivative of $ x $ in an interval $ [a,b] $ on which a function $ f $ is defined is defined as.. $$f'(x)=\lim_{t \to x}\frac{f(t)-f(x)}{t-x}$$ Why is this equal to $$ f'(t)=\lim_{x \to t}\frac{...
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1answer
58 views

$f$ is continuous on $[a,b]$, differentiable on $(a,b)$ , why does that imply that $g(x)=\frac {f(x)} x$?

Let $f$ be continuous on $[a,b]$, differentiable on $(a,b)$, $0<a<b$ and $\frac {f(a)} a= \frac {f(b)}b$. Why does that imply we can define a function $g(x)=\frac {f(x)} x$ and what are the ...
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39 views

What is a regressive set?

Several authors (e.g. Jockusch, Appel, McLaughlin) use a notion of a regressive set, however none of the authors gives a complete definition, they refer to the paper J. C. E. Dekker, Infinite series ...
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23 views

matrices what is the meaning of defined?

Please explain what is meant by "defined". I have a MCQ type question that goes like this: If $A$ is an $m \times n$ matrix, $B$ is an $n \times p$ matrix and $C$ is a $p \times n$ matrix, then a) $...
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291 views

Is $540^\circ$ a straight angle?

The usual definition of a straight angle is a $180^\circ$ angle. however, because a $540^\circ$ angle is also the same shape, is it a straight angle as well?
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1answer
31 views

Can there be a vacuous tautological consequence $F\vDash F$?

Can there be a vacuous tautological consequence $F\vDash F$? Since $α⊨φ \iff ⊨α→φ$ then is: $(k∧¬k)⊨(p∧¬p)$ for example considered a tautological consequence?
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65 views

What does $M_n(F)$ means

What does $M_n(F)$ means? I am a new comer to matrix algebra. Please help, thank you. Thank you so much. $F$ stands for vector space guess.
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2answers
32 views

For a composition to be defined: $Domf\circ g\subseteq Dom f, Im f\circ g \subseteq Im g $?

For a composition to be defined, is the following two a must? $$f:A\to B, g: C\to D\\ f\circ g : C\to B \\ Domf\circ g\subseteq Dom f\\ Im f\circ g \subseteq Im g $$ Are there other conditionals for ...
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23 views

Question about definition of independent discrete random variables.

In my lecture notes, I am given the definitions for: -the independence of two discrete random variables -the independence of a set of discrete random variables -the pairwise independence of a set ...
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1answer
75 views

Adjoint Functors: Naturality?

Given a pair of functors: $$F:\mathcal{B}\to\mathcal{A}\quad G:\mathcal{A}\to\mathcal{B}$$ Consider an identification: $$\alpha:F(B)\to A\leftrightarrow\beta:B\to G(A)$$ Then they're adjoints if the ...
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1answer
142 views

Choosing the definition of $\frac{\partial^2}{\partial x\partial y}$

Today, I answered this question and discovered that the definition of $\dfrac{\partial^2}{\partial x\partial y}$ is a matter of convention. For example this .edu link and this other .edu link use the ...
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0answers
36 views

Correct way of defining a mathematical object (linguistic)

I am writing my thesis and my advisor made a correction to the sentence below: Transitional Rule commonly denoted by $\phi$ is defined by the map $\Sigma^n \rightarrow \Sigma$. He has changed ...
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4answers
1k views

Euclid: What is the difference between a 'surface' and a plane 'surface'?

I've begun to study Euclid's Elements and i've a few questions regarding the difference between a surface and plane surface. A surface is said to be "that which has length and breadth only", it then ...
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6answers
279 views

How do we define arc length?

In trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...
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170 views

What is a Single Objective Optimization problem?

I can't find any definition of this problem on the Internet. Could you help me by providing some definition?
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1answer
44 views

Russell's Paradox for the zero set and a set with the zero set.

So I have a question: Let: Allow set B = {x: x $\notin$ x}. Then, B $\in$ B $\iff$ B $\notin$ B ? Does this apply for the zero set? Because I'm a bit confused. The definition is a zero set is always ...
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1answer
40 views

Question about “integrable” random variable

I was reading the definition of Markov's Inequality on Wikipedia and it says If $X$ is any nonnegative integrable random variable and $a > 0$, then $\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}...
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1answer
108 views

Why is $\mathrm{arctan}(0)$ not infinity?

$\arctan x$ is defined as: $$\arctan x = \frac{1}{\tan(x)} = \frac{1}{\frac{\sin(x)}{\cos(x)}}$$ if I now have $x = 0$ I should get: $$\frac{1}{\frac{\sin(0)}{\cos(0)}} = \frac{1}{\frac{0}{1}} = \...
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1answer
209 views

Different definitions of subnet

I encountered two different definitions of subnet. The first is Let $(I, \preceq_I ), (J,\preceq_J )$ be two directed sets and $X$ be the underlying set.$\{ \eta_j \}_{j \in J}$ is a subnet of $\{ ...
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1answer
42 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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1answer
52 views

Closed vector space and a subspace of a vector space [duplicate]

What is a closed operation in a vector space? I don't see any difference between a closed operation in some vector Space R$^n$ and the open operation. What I mean by the closed operation is addition ...
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1answer
80 views

Interpretation of the radius of convergence

What interpretation should one give to the radius of convergence of a series $\sum a_nz^n$ ? I do know how it is mathematically defined and what it implies for convergence/divergence, but I'm having ...
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3answers
62 views

E is closed if every limit point of E is a point of E?

E is closed if every limit point of E is a point of E? Should that be "E is closed if every point of E, is a limit point"? I don't understand. Limit points are essentially points that hug other ...
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1answer
79 views

Understand the definition of convex metric spaces

I am trying to understand the following definition: We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$ Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$ Clearly ...
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131 views

Ordinary differential equations of order zero?

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$? If not, can we define differential ...
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1answer
36 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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99 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. Shafarevich definition (pg 128) - A variety is non-singular in codimension one if the singular locus has codimension $> 1$. ...
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140 views

Please help with understanding a logic definition: Subformula

Alright, so I am reading "Computability and Logic" by Boolos and Jeffrey, specifically I'm on chapter 9 "A Precis of First-Order Logic: Syntax. There has been more than a handful of definitions in ...
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2answers
3k views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these $...
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35 views

Sense of the graph of a function

What makes it necessary to define the graph of a function $f:A\rightarrow B$ as $$\{(x,f(x))\mid x\in A\}$$ which makes it a subset of $A\times B$, when this is equal to the function itself, which is ...
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192 views

Spectral gap vs. algebraic connectivity

Can someone please clarify how the spectral gap of a graph relates to its algebraic connectivity (aka Fiedler value) and whether these use the adjacency matrix or laplacian matrix?
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90 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function $p:\...
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106 views

What is the oscillation of a function?

Define the oscillation of a function at a point $x$ to be (for an open interval $I$): $$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$ I am a bit confused about the definition above. How am I ...
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78 views

Theory of definitions

I am reading "Introduction to Logic" by P Suppes at the moment. In the Chapter 8 - Theory of definitions of it, I 've some confusion, actually about the Conditional Definition. The brief explanation ...