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

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Different two definitions for separable extension

Let $E/F$ be an algebraic field extension and $\bar F$ be an algebraic closure of $F$. Define $[E:F]_{\text{sep}}$ as the cardinality of $$\{\sigma\in \operatorname{Mono}(E,\bar F): \sigma \text{ ...
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1answer
40 views

When is it allowed to “take apart” a limit (multiplication of limits)?

When is it allowed to "take apart" a limit? Here's an example to show what I mean: $\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e^{\frac 1 n}-1}=e-1$ since we can "take apart" ...
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1answer
60 views

Is this definition valid?

I am working on this problem: "Suppose $f:A\times A\rightarrow A$. A set $C \subseteq A$ is closed under $f$ if $\forall (x,y) \in C \times C(f(x,y) \in C)$. Now suppose $B \subseteq C $. The closure ...
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1answer
19 views

The domain of a function as a function: the “domain-function”

The domain of a function $f:X\to Y$ is normally defined as $\operatorname{dom}f\equiv X$, but I would like the domain-function $\operatorname{dom}$ to be a funtion itself, i.e. I would like to define ...
2
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1answer
94 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
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2answers
239 views

Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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0answers
11 views

Correct understanding of col and diag operators

In a scientific paper I am currently working with, a definition of $Col$ and $Diag$ operator is introduced: We use the operator $Col_{k\in K}(x_k)$ which stacks up its vector (or matrix) ...
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3answers
33 views

Continuity definition of a functional

I'm having a hard time understanding the formal definition of continuity of a functional. I'm not sure if such questions are appreciated on this site; so let me know. Definition: The functional ...
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10 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from ...
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2answers
91 views

What is the definition of a finite set $S$? [closed]

This question is intended mainly for beginners. We can say "$S$ is not infinite" or "counting elements of $S$ is a procedure that (theoretically) terminates", a little of maths appears in “$S$ is ...
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1answer
38 views

Definition of an absolutely continuous random variable

Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like: $$\mathbf{P} (A) = \int_A f d \mu$$ for some Borel set $A$. But what is $\mu$? Is ...
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35 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
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0answers
28 views

Notation: column/row projection function for matrix-like objects

If we have a $n$-tuple $\mathscr x$ $$\mathbf{x} := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
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0answers
21 views

Definition of a minimal set of automorphisms generating an orbit?

By definition, the orbit of a vertex v in a graph G is the set of all vertices f (v) such that f is an automorphism of G. I wonder whether there is a definition for a minimal set S of automorphisms ...
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20 views

Are little-o and “error term” the same thing?

I'm reading The Elements of Infinitesimal Calculus by David A. Santos, and I stumbled upon this: 45 Definition Let $m$ and $n$ be non-zero natural numbers with $m < n$. We say that $x^n$ ...
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1answer
15 views

Precision needed in definition of unboundedness

This is a quick question about what it means for a function to be unbounded. Does it mean that the function tends to + or - infinity, or does it just mean that it has no limit?
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1answer
63 views

Define F : Z → Z by the rule F(n) = 2 -3n, for all integers n

I am not sure how to go about solving this problem. Can somebody tell me how to define $F : Z \to Z$ by the rule $F(n) = 2 -3n$, for all integers $n$ ? I am not sure where to even start or what ...
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1answer
26 views

Are there terminologies distinguishing these two ranks?

Definition 1. Let $R$ be a ring with invariant basis number and $M$ be a free $R$-module. Then, the rank of $M$ is the cardinality of an $R$-basis of $M$. ${}$ Definition 2. ...
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1answer
27 views

Understanding quotient map

If I am understanding correctly, a quotient map can be defined in this way (actually I quoted the following from Munkres): Let $X$ and $Y$ be topological spaces; let $p:X \rightarrow Y$ be a ...
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1answer
46 views

Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition

Based on observation after reading few books and papers, I think that Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes ...
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1answer
22 views

On the definition of uniform continuity over an interval.

I was reading some slides and I stumbled upon this definition of uniform continuity in an interval I am unsure on how to trace this back to the definition of uniform continuity that I know: A ...
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2answers
44 views

What's the difference between finite and finitely generated algebras

I didn't understand the difference between the two definitions: I thought the definition of $B[a_1,\ldots,a_n]$ is exactly the one in the item (b), i.e., $B[a_1,\ldots,a_n]=Ba_1+\ldots+Ba_n$. I ...
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1answer
70 views

What is the difference between CW-complex and Cellular complex?

Is every CW-complex is a Cellular space? Is its converse true? If it is true then what is the difference between them? We include the definition of CW-complex in algebraic topology given by ...
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1answer
43 views

Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
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2answers
711 views

Are ideals also rings?

I am learning about rings and ideals. But I am confused about something. My book (Gallian) says that an ideal of a ring by definition is a subring. But I have talked to other people who insist that an ...
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1answer
14 views

What is an example of a non-negative Hermitian form which is still not an inner product?

I was reading the definitions: Let $X$ be a vector space and $f: X \times X \longrightarrow \mathbb K$, where $\mathbb K = \mathbb R$ or $\mathbb C$. $f$ is said to be a Hermitian form on $X$ if: ...
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0answers
22 views

Definition of convergence rate of random variables

What is the definition of rate of convergence of a sequence of random variables? i.e. what does it mean that the convergence rate of the sequence of random variable $X_1,X_2,\ldots X_n,\ldots$ to $X$ ...
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0answers
7 views

What does it mean 'not containing $l^{\infty}(2)$ isometrically'

What does it mean 'not containing $l^{\infty}(2)$ isometrically'? The following is the context: Suppose $X,Y$ are sets and $E,F$ are normed spaces not containing $l^{\infty}(2)$ isometrically. Can ...
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3answers
92 views

What set does $\mathbb W$ denote?

What set does $\mathbb W$ denote? I know this may horribly lack context, but I've seen multiple times on M.SE $\mathbb W$ used in some fairly elementary context I think.
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1answer
19 views

What is the connection between $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ and $V(\vec x) = \frac {1}{2} x^TPx $?

For example, $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ has an equivalent representation $V(\vec x) = \frac {1}{2} x^TPx $ where $P$ is some matrix Can someone make this connection clearer for me ...
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46 views

What is 'free algebra'?

I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring. But I still quite don't get it. Is a free algebra just a free group with additional operation ...
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1answer
37 views

Interpret a relation definition, it's meaning and and how it should be applied

I'm unsure of the definition below : $\mathbb{Z} = (\dots,-2,-1,0,1,2,\dots)$ Take a relation $\rightarrow$ on $\mathbb{Z}^2$ define as $(x,y)$ $\rightarrow$ $(x',y')$ only when: $(x',y')= ...
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2answers
23 views

Is transitive closure defined uniquely?

I'm encountering questions where I'm required to find a transitive closure (and the questions seem to suggest that there is only one), but I probably don't understand something in the definition, ...
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1answer
23 views

Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...
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1answer
20 views

Are these two definitions of separable extension equivalent?

Definition1 - wikipedia Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable. Definition 2 - ...
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61 views

Questions about the definition of convergence

I am having difficulty understanding the definition of convergence. I've been rereading and looking at examples during this past week and I haven't made any progress. Definition: We say that ...
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1answer
49 views

What does it mean for pullbacks to preserve monomorphisms?

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving ...
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3answers
69 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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3answers
44 views

Difference between generators and basis

What is the difference between the terms "generator set" and "basis"? Don't they both just mean a set of objects that you can use to obtain all of the objects in a larger set under some operations? ...
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1answer
27 views

Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
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1answer
35 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
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1answer
14 views

Definition of Cofinal Segment

Recently I encountered the term 'cofinal segment' in the paper 'The Point of Continuity Property, Neighbourhood Assignments and Filter Convergences' by Ahmed Bouziad, example $2.3$. Question: What ...
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3answers
68 views

What does inversion mean?

I am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm ...
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1answer
114 views

If $C=M \times [0,1]$, what is an integral over $\partial C$?

Let $M$ be a compact Riemannian manifold. Define $C=M \times [0,1]$ so that $\partial C = M \times \{0\} \cup M \times \{1\}$. If $f:\partial C \to \mathbb{R}$, is then the integral over $\partial C$ ...
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2answers
321 views

When is it allowed to take a constant out of a series?

When is it allowed to take a constant out of a series? Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$? It's pretty obvious when we know beforehand the series converges ...
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0answers
49 views

Why is delta used to describe difference between two entities

I hear a lot about extracting "delta" between two properties in my current job. I come from a User Interface programming background and I do not really have much math background. I looked up delta ...
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1answer
23 views

How to interpret the definition of inductive set?

I can't understand the sentence below: "A subset Y ⊂ X will be called inductive if, for every x ∈ X such that y ∈ Y for all y ∈ X such that y < x, we have x ∈ Y." please tell me what's the meaning, ...
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36 views

Does this violate the notion of positive definiteness?

From a previous course in linear algebra, I was taught that a function is positive definite if it satisfies $$\left< \vec x \mid v(\vec x) \mid \vec x\right> > 0 $$ In simpler notation, ...
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1answer
141 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
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253 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...