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

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1answer
20 views

What's the correct definition of generated ideal in a pseudo-ring?

Given a ring (with $1$) $R$, one defines what, say, a left ideal is. There's also a natural definition of ideal generated by a subset Definition A: $_R(S):=\bigcap\{I\supseteq S:I\text{ is a left ...
6
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1answer
2k views

Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
0
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0answers
14 views

global manifolds

Can you also explain why the global stable manifold is the union of the flow of the local stable manifolds for t < 0? Why do we not include all t?
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2answers
23 views

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
34 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
56 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 ...
14
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3answers
685 views

True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
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2answers
841 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
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3answers
724 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
2
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1answer
15 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 ...
317
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21answers
58k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
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2answers
198 views
+50

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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2answers
203 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
3
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1answer
72 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 ...
2
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1answer
70 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 ...
2
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3answers
190 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
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0answers
17 views

Definition of frontal map

In the lecture about singularities of curves and surfaces the lecturer gave the following defintion: A smooth map $f: U \subseteq \mathbb R^n \to \mathbb R^m$ is called frontal if and only if for ...
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4answers
496 views

What precisely is a vacuous truth?

Is there a proper and precise definition that goes something like this? Definition. A statement $S$ is a vacuous truth if ... ...
8
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2answers
224 views

Can someone explain definition of number 1?

I've found the next piece of text: As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti: $$1 = ...
0
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0answers
22 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
8 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) ...
7
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6answers
349 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
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3answers
29 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 ...
0
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2answers
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
86 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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3answers
2k views

Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is ...
3
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0answers
32 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 ...
0
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1answer
31 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 ...
0
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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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0answers
55 views

Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
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0answers
17 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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0answers
13 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 $x^n$ ...
0
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1answer
14 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?
4
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1answer
48 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 ...
0
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3answers
178 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
0
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1answer
18 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 ...
4
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1answer
32 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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3answers
91 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.
0
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1answer
19 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 ...
0
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2answers
43 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
48 views

What does it mean by a set is bounded. [closed]

Given a subset $S\subset R^m,$ what does it mean by $S$ is bounded? I missed a class so didn't get the definition... Please help.
1
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1answer
53 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 ...
1
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1answer
42 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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4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
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2answers
697 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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0answers
20 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 $X$ ...
0
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1answer
12 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
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 ...
0
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1answer
17 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 ...
1
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1answer
32 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')= ...