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

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307
votes
18answers
55k 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 ...
15
votes
4answers
734 views

What's wrong with this “backwards” definition of limit?

Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?: $\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if ...
2
votes
1answer
274 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
2
votes
2answers
174 views

Motivation behind definitions of the Integral without reference to Derivatives

If a (definite) Integral can simply be calculated as the difference of two Antiderivatives and Antiderivatives are simply the "reverse process" of Differentiation. Then it seems to me that the ...
1
vote
1answer
139 views

Meaning of Point Evaluation

I read in some general measure theory books and there is always like "define measure $x$ to be the point evaluation at $y$..." but when I look around online and some other books there is no mention on ...
2
votes
1answer
192 views

Discussion: Differing definitions for the rank of a set

I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.). $\text{rank}(A)=\min\{\alpha\mid ...
1
vote
1answer
93 views

What's the meaning of $C$-embedded?

What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.
1
vote
2answers
269 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
2
votes
2answers
118 views

What is a non-degenerate module?

I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$) I came across a module being called ...
7
votes
1answer
149 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
4
votes
3answers
294 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
1
vote
3answers
218 views

Time independence in SISO systems

I started learning dynamical system a couple of weeks ago and the lecture tried to define what is a SISO system in the last lecture. The lecture wasn't very clear and did not give a formal definition ...
8
votes
6answers
461 views

What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?

I thought I'd bring this question to math.SE, as it could spark some interesting discussion. When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...
4
votes
2answers
198 views

Which algebraic structure captures the ordinal arithmetic?

Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these ...
1
vote
1answer
104 views

What is the meaning of the term “inductively P map”?

In this page is the definition of an inductively open map. But in this pdf is the definition of a inductively P map, where P is a property of maps. But there is a difference in the definitions. In ...
6
votes
2answers
792 views

On the definition of the Hausdorff distance

$\newcommand{\dist}{\mathrm{dist}\,}$ Let $M$ be a metric space and $\emptyset\neq A,B\subset M$ bounded closed subsets. The Hausdorff distance is defined as $$h(A,B)=\max\{\dist(A,B),\dist(B,A)\},$$ ...
2
votes
1answer
492 views

What is unitary space

In http://www.encyclopediaofmath.org/index.php/Unitary_space, unitary space seems to be Hilbert space. But in http://www.answers.com/topic/unitary-space, "finite dimensional" is required. My question ...
1
vote
1answer
264 views

What is a differentiable functional?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
3
votes
2answers
351 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
2
votes
1answer
82 views

What is the name of the group linear functions on a finite field?

More precisely what is the name for the group $$\{ X\mapsto \alpha^2X+\beta : \alpha,\beta \in GF(q), \alpha \neq 0\}$$ I've always called it the special affine group, but I see that can mean ...
2
votes
1answer
169 views

Question about inverse limit

I'm puzzled by the definition of inverse limit in this Wolfram article. I thought if an object was defined by a universal property it meant that the object is unique up to unique isomorphism. This ...
7
votes
2answers
661 views

Domain of an operator in functional analysis

I used to think that if we say $f$ is a function from a set $X$ to $Y$ then this implied that $f$ was defined on all of $X$. Because the definition of function is that it's a set $\{(x,y) \mid \text{ ...
5
votes
3answers
456 views

Spectrum of a field

Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals. Thanks a ...
4
votes
1answer
231 views

Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
29
votes
3answers
5k views

difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
6
votes
5answers
705 views

Why is associativity required for groups?

Why is associativity required for groups? I'm doing a linear algebra paper and we're focusing on groups at the moment, specifically proving whether something is or is not a group. There are four ...
2
votes
2answers
913 views

Characteristic time?

Could somebody tell me the definition of a "characteristic time"? For example, what is the characteristic time for a function $f(t)=\operatorname{tanh}(t)$ to reach 1? I tried looking up a definition, ...
2
votes
1answer
278 views

What does “quotient-ring” mean?

I am reading a paper about rings (http://malaschonok.narod.ru/publ/ma01.ps, page 3). In this paper the term "quotient-ring" appeared. What is a quotient-ring? (Note: The text in the original ...
3
votes
0answers
64 views

Isotropic subspaces [duplicate]

Possible Duplicate: Etymology of the word “isotropic” Let $V$ be a vector space and we have a symmetric, non-degenerate bilinear form with signature $(n,n)$ on it. A subspace ...
8
votes
1answer
2k views

what exactly is mathematical rigor?

I am a programmer moving up to computer science and even in my past life as a mechanical engineer I used sufficient maths to get by but I have always wondered what exactly is this thing called rigor. ...
3
votes
2answers
224 views

Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate]

Possible Duplicate: Alternative ways to say “if and only if”? So when I come across mathematical definitions like "A function is continuous if...."A space is compact ...
1
vote
2answers
96 views

Quibble with terminology

Proposition 5.15 on page 63 in Atiyah-Macdonald goes as follows: Let $A \subset B$ be integral domains, $A$ integrally closed, and let $x \in B$ be integral over an ideal $ \mathfrak a$ of $A$. Then ...
1
vote
1answer
108 views

Typo in lecture notes?

The following is an example in my lecture notes: "Let $X$ be a locally compact topological space (that is, a topological space in which every point has a compact neighborhood). Then $C_0(X)=\{f \in ...
0
votes
1answer
214 views

a question about definition of regular surface

While I am reading Do Carmo's differential geometry,I have several questions about the definition of regular surface. From condition 2,the author said : "...... $x^{-1}:V \cap S \rightarrow U$ ...
2
votes
2answers
142 views

Am I allowed to realize one object twice within one set-theory?

Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing. As I understand it, stating the axiom allows me to make a definition like $$(a,b):=\{\{a\},\{a,b\}\}$$ and ...
3
votes
4answers
329 views

Is this mathematical definition iterative? If not, what does an iterative function look like?

I was debating with someone about iterative vs recursive in programming. I was defending the iterative side. He then said me that the true definition of Fibonacci number is this: $$f(n) = f(n-1) + ...
0
votes
1answer
136 views

Meaning of the term single letter formula

It is common in information theory to look for single letter formulas or to dismiss a result as suboptimal if no single letter formulas are available. Could someone clarify the meaning of what is a ...
0
votes
1answer
267 views

Definition of effective enumerability and empty set

Let $S$ be a set. We say that $S$ is effectively enumerable iff (by definition) there exists a function $f \colon N \to N$ which has $S$ as codomain. My question is: is the empty set an effectively ...
2
votes
1answer
253 views

Free boolean algebra

Consider the following definition: Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X ...
3
votes
0answers
50 views

Defining a Certain Class of Plane Graphs

I'm having problems finding the right words to formulate the following class of graphs in a definition. I'm defining a class of plane graphs with the following properties: Removing any vertex of ...
8
votes
5answers
471 views

Proving that $a + b = b + a$ for all $a,b \in\mathbb{R}$

Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting ...
0
votes
0answers
71 views

Concerning the point stabilizing group and coset stabilizing group.

I would like to know more about the point stabilizer group and the coset stabilizer group, like the definitions, why they are used in group theory, who developed them and there importance.
1
vote
1answer
91 views

How to formally describe this Uppaal automata?

I have the following simple automata: What I'm looking for is a formal description of this based on the definition here $A=(\Sigma,\Gamma,S,s_0,\delta,\omega, F)$ How to declare all the ...
4
votes
3answers
603 views

well-defined functions

I am asked to argue whether or not the following two functions are well-defined (textbook definition: a) define $y$ for all $x$ in domain, and b) any is mapped to exactly one y). Both of the below ...
0
votes
1answer
423 views

What is the winding number?

I tried to study the concept of winding number in a general way (the algebraic topology way) but i only find for example the definition from differential geometry and then i find this Winding number ...
2
votes
2answers
315 views

What are the carried numbers called in an Addition problem

What is the 1 that is carried called? These are all Latin, would this make sense? The Latin word for "carry" is "porto", would it be called Porto? Just guessing here Example: ...
8
votes
2answers
307 views

Usage of the word “formal(ly)”

This is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems to ...
10
votes
3answers
502 views

What is the purpose of the $\mp$ symbol in mathematical usage?

Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please ...
2
votes
1answer
102 views

Is there a name for a collection of open sets where arbitrary intersections are open?

Let $\mathcal{U} = \{U_i\}_{i\in I} $ be a collection of open sets with the property that the set $\bigcap_{i\in J} U_i $ is open for all subsets $J$ of $I$. Is there a name for such collections of ...
19
votes
3answers
1k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ ...