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

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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 ...
-3
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2answers
81 views

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

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 ...
22
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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
26 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
27 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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0answers
15 views

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

If we have a $n$-tuple $\mathscr x$ $$\mathscr x := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
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1answer
25 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
54 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
12 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
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?
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1answer
35 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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3answers
177 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 ...
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1answer
17 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 ...
3
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1answer
28 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
90 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

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
42 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
47 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.
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1answer
49 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
41 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 ...
315
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19answers
57k 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 ...
6
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2answers
691 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 ...
0
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0answers
17 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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1answer
10 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')= ...
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0answers
34 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 ...
0
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1answer
129 views

Well-posed problem

In the definition of a well-posed problem it states that a problem is well posed if: 1.A solution exists. 2.The solution is unique. 3.The solution's behaviour changes continuously with the initial ...
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2answers
21 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, ...
1
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1answer
19 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 ...
3
votes
1answer
43 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 ...
1
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1answer
12 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 - ...
6
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2answers
47 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 ...
4
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3answers
64 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
34 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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0answers
24 views

Are there symbols for the map function or the filter function?

I'm going to define two notations, because I'd like to use them, but I'm wondering whether someone has done this before in serious mathematics. The idea comes from functional programming. Definition: ...
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3answers
195 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
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1answer
26 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
24 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 ...
0
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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
706 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...
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3answers
38 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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0answers
10 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
297 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
26 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 ...
0
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1answer
21 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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2answers
224 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 ...