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

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3
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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 ...
0
votes
1answer
84 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 $\infty$....
4
votes
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} (...
3
votes
1answer
153 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 ...
0
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1answer
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}=\...
0
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1answer
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=\...
1
vote
2answers
82 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 ...
3
votes
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 $...
1
vote
1answer
228 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 ...
5
votes
1answer
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 ...
3
votes
1answer
63 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, ...
1
vote
1answer
45 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 ...
2
votes
1answer
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{...
0
votes
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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) $...
7
votes
6answers
293 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?
1
vote
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?
0
votes
1answer
69 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.
0
votes
2answers
33 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 ...
0
votes
1answer
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 ...
2
votes
1answer
76 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 ...
5
votes
1answer
144 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 ...
2
votes
0answers
38 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 ...
1
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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 ...
2
votes
6answers
284 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 ...
0
votes
1answer
176 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?
1
vote
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 ...
1
vote
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)}...
5
votes
1answer
109 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}} = \...
5
votes
1answer
218 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 $\{ ...
0
votes
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 ...
1
vote
1answer
53 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 ...
0
votes
1answer
81 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 ...
0
votes
3answers
64 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 ...
-2
votes
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 ...
0
votes
1answer
143 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 ...
1
vote
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 ...
2
votes
1answer
100 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 $> 1$. ...
0
votes
1answer
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 ...
1
vote
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 $...
0
votes
1answer
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 ...
2
votes
1answer
204 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?
0
votes
1answer
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:\...
1
vote
1answer
107 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 ...
1
vote
1answer
79 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 ...
1
vote
1answer
351 views

Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending? [duplicate]

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
0
votes
1answer
47 views

Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
3
votes
2answers
25 views

What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
0
votes
1answer
74 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...