Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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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
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1answer
411 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
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1answer
232 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
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2answers
334 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
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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
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1answer
163 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
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2answers
561 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{ ...
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3answers
402 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 ...
3
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1answer
224 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 ...
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3answers
4k 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
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5answers
647 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
727 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
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1answer
276 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 ...
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57 views

Help in Analysis

I'm studying the article "On W1;p Estimates for Elliptic Equations in Divergence Form" of L. A. CAFFARELLI and I. PERAL. There, you can find We use the classical Hardy-Littlewood maximal operator, ...
3
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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
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1answer
1k 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
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2answers
221 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 ...
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2answers
91 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 ...
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1answer
107 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 ...
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1answer
189 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
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2answers
140 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
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4answers
322 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) + ...
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1answer
132 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
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1answer
239 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
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1answer
244 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 ...
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0answers
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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 ...
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5answers
463 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 ...
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0answers
69 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.
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1answer
89 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
576 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
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1answer
381 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
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2answers
286 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
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2answers
265 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
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3answers
477 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
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1answer
100 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 ...
18
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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: $$ ...
3
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1answer
309 views

“sheaf” au sens de Serre

I learned the definition of sheaves from Algebraic Geometry by Hartshorne, while reading Serre's GAGA, I was wondering if there was another definition of sheaves. [Here is the link of the English ...
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0answers
96 views

How are the various numbers in the standard 2.2 gamma correction for RGB derived?

Here is the standard fwd Gamma 2.22 (1 / 0.45) correction formula: ...
2
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1answer
134 views

Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I'm reading requires that the actions of a group are "deterministic"; but the formal definition given makes no mention of this property: A set G is a group if ...
0
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1answer
38 views

Dimension of the dual image space

Is it ok to assume that $\operatorname{dim}(\operatorname{Im}(T^*))=\operatorname{dim}[(\operatorname{Im}(T))^*]$, where $T$ is a linear map acting on a finite dimensional space. i.e. just taking the ...
0
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1answer
161 views

Definition of conjugate transpose in this case

Would somebody mind clarifying the following for me please? Suppose $\psi(f,g):=\int_a^b f(t)\overline{g(t)}dt$ where $f,g$ are complex functions of $t$, what does it mean to say that it is ...
0
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1answer
126 views

Defintion of totally geodesic flat submanifold

I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian ...
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5answers
2k views

What does it mean when a function is finite?

When someone says a real valued function $f(x)$ on $\mathbb{R}$ is finite, does it mean that $|f(x)| \leq M$ for all $x \in \mathbb{R}$ with some $M$ independent of $x$?
2
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1answer
182 views

define simultaneous substitution recursively

Can you help me with my approach to the following task: Define simultaneous substitution $\phi[\psi_1,...,\psi_k/p_1,...,p_k]$ recursively. Usually we have recursive definitions about formulas, but ...
6
votes
1answer
414 views

What do these arrows mean? (Froda's Thm)

I was reading "A Course in Probability Theory" by Kai Lai Chung, and in the book he was discussing discontinuity of monotonic functions, and after doing some searching online to learn more about the ...
4
votes
1answer
192 views

intrinsic and geometric definition of blow-up

Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ...
3
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1answer
532 views

Definition of Separability Degree

For an assignment, I am trying to determine the separability degree of some algebraic field extension $L/K$. The definition of the separability degree of polynomial is not difficult to find at all, ...
6
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1answer
285 views

Are these definitions of submanifold equivalent?

Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
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3answers
2k views

Question about basis and finite dimensional vector space

I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5) I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
3
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2answers
1k views

Local definition of Hölder continuity

What does it mean for a continuous function $ f $ on $ \mathbb{R} $ to be Hölder continuous with exponent $ \alpha $ at a point $ x_0 $ ? I only now the global definition: A function $ f $ on $ ...