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

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4
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1answer
93 views

Left Hand Derivative Definition

What is the actual definition of Left Hand Derivative ? I bumbed into this site and the second white box on their site gives the definition . Is that wrong ? What is the correct one then ?
0
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1answer
76 views

Precise definition of distinct vectors

What is the precise definition of 'distinct vectors'? In particular, are the vectors (2, 1) and (4, 2) distinct, seeing as they are multiples of each other?
1
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2answers
54 views

Definition of algebraic structure

Is there a definition of algebraic structure? Wikipedia says: a set (called carrier set or underlying set) with one or more finitary operations defined on it. In particular, what is the ...
0
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0answers
19 views

Complex conjugate of operator on differential forms

In a Lecture note on Kahler manifolds, the author writes the following two identities are equivalent as they are conjugates of each other : $[\Lambda, \bar{\partial}]=-i\partial^{*} $ and ...
0
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0answers
51 views

How to say this in math lang?

Well, I have a paper to write in which I have to formalize the following definitions: A transaction is composed of actions (and cascading actions), which affect tables, which are made of records. ...
0
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1answer
38 views

Is it legal to define a function to be undefined for some x?

Is the following definition legal: f(x) = 1 for 0<x<1, and undefined otherwise. Such functions clearly exist, but the question is if it is legal to just ...
0
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0answers
24 views

Is $\int^{\infty}_{-\infty}\delta(x-x_0)f(x) \, dx = f(x_0)$ sufficient to define delta distributions?

Most of the sources start introductory section of the delta distributions by defining \begin{eqnarray} \delta(x-x_0)&=&\begin{cases} \infty, & \text{if $x=x_0$}.\\ 0, & ...
1
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1answer
44 views

Definition of Euclidean domain

Today we learned about Euclidean domains in class but I don't understand why we need one of the conditions stated in the definition . We called an integral domain R a Euclidean domain if there exists ...
1
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1answer
67 views

A strange definition for a strange set.

Given a ring $A$ we know that its center $C=\{x : yx= xy \;, \forall y \in A\}$ is a well defined subset of $A$. Now I want define a set that, intuitively, is '' The set of all elements that commute ...
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2answers
66 views

Is definable $\emptyset$-definable?

A set $A$ is $X$-definable in an $L$-structure $\mathcal{M}$ with a domain $M$ iff there are an $L$ formula $\phi(a,\bar x)$ and $\bar x \in X^n$ such that $A=\{a \in M^m | \mathcal{M} \models ...
1
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1answer
87 views

Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?
0
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1answer
42 views

Prove a limit using the formal definition of the limit

So I have a sequence {a_n} = π/2^n where n=1,2,3,4.... And I need to prove that its limit is 0. Here is what have done, can someone check and tell me if this is correct.? Definition: A sequence ...
1
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1answer
59 views

What is $g^1_3$?

I'm trying to find the definition of $g^1_3$ in algebraic geometry Hartshorne's book, anyone who is used with this book could help me to find this definition? Thanks Remark: this extract is from ...
8
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2answers
145 views

Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
3
votes
1answer
71 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
5
votes
3answers
101 views

why does we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to show metric space?

In metric space some axioms must be satisfied . I wonder why we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to be metric space. If this axiom is not satisfied, does any problems occur? ...
1
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1answer
21 views

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: ...
0
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2answers
90 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 ...
1
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1answer
14 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 ...
1
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1answer
36 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 ...
3
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2answers
114 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
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1answer
56 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 ...
4
votes
2answers
190 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
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1answer
137 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
53 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 ...
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 ...
1
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1answer
65 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
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0answers
22 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
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1answer
18 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 ...
0
votes
1answer
107 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
45 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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1answer
34 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 ...
2
votes
1answer
47 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 ...
0
votes
1answer
56 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
25 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
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1answer
18 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
244 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
25 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
31 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
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2answers
22 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
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1answer
17 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
57 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
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1answer
123 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
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0answers
29 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
165 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
153 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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0answers
10 views

Oscillation Index of Thomae function

In this paper, for a real-valued function $f$ and a fixed $\epsilon>0$, the authors defined $$D(f,\epsilon,P)=\{ x \in P: \text{for all neighbourhood } N_x, \text{there exists } x_1, x_2 \in P ...
0
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1answer
25 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
36 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
28 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 ...