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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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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181 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 : "...... $x^{-1}:V \cap S \rightarrow U$ ...
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2answers
138 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 ...
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4answers
313 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
129 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 ...
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232 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 ...
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1answer
234 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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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 ...
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5answers
458 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
83 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 ...
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3answers
563 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 ...
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1answer
370 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 ...
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2answers
271 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: ...
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2answers
241 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 ...
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3answers
461 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 ...
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1answer
99 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 ...
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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: $$ ...
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1answer
307 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
95 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: ...
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1answer
131 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 ...
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1answer
36 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 ...
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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 ...
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1answer
119 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 $x$?
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1answer
174 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 ...
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402 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 ...
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1answer
181 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 ...
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1answer
499 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, ...
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1answer
272 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 ...
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2answers
937 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 $ ...
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2answers
293 views

What does it mean for a stochastic process to be independent of a sigma algebra?

Does anybody know what it means for a stochastic process $ X = (X_t)_{t \geq 0} $ on a filtered probability space $ (\Omega, \mathcal{F}, \mathbb{F}, P) $ to be independent of a sigma-algebra $ ...
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1answer
174 views

What is the nature of the definition symbol?

A question about definitions and also notation, illustrated with a trivial example: Let $a,b,c\in\mathbb{N}$ and $$a:=2,$$ and $$b:=1$$ I postulate the the following formula holds ...
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1answer
188 views

Locally Euclidean

A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a ...
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1answer
343 views

How to motivate the axioms for the inner product

Typically, one doesn't just write down lists of axioms and then sees if there are enough interesting examples that satisfy them; they evolve over time, usually from a couple of very ...
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3answers
796 views

What are the “whole numbers”?

Just recently, I attempted to answer a question involving "whole numbers", but discovered that my long-held assumption (that they're the same as the integers), is not universal. [In fact, it seems I ...
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0answers
82 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
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1answer
306 views

MATLAB's implementation of cross correlation

Wikipedia gives the cross-correlation as $$ \begin{align*} (f \star g)[n] = \sum^{\infty}_{m = -\infty} f^{*}[m] g[n+m] \end{align*} $$ MATLAB's documentation gives ...
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1answer
140 views

What is the X, Y, Z “resolution” of a three-dimensional grid of points?

I came accross a software which requires the X, Y and Z resolution of a three-dimensional grid of points as Integer. What is a "3D grid resolution" and how do I find it? From what I understand, the ...
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1answer
466 views

Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?

I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$: $$ \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e} $$ Now, I know ...
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1answer
76 views

In category theory, is there any such thing as “compatibility” for arrow composition?

Is this a properly defined category? Objects $\{P, R, S\}$ Arrows $f_{1} : P \rightarrow R$ $f_{2} : P \rightarrow R$ $g : R \rightarrow S$ $h_{1} : P \rightarrow S$ $h_{2} : P \rightarrow S$ ...
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3answers
109 views

Proving the relationship $1 + r \leq \left(1 +\frac{r}{m}\right)^m$ for any $m \geq 1.$

I am currently trying to prove the following relationship $$1 + r \leq \left(1 +\frac{r}{m}\right)^m\quad \text{for any }m \geq 1.$$ Would you be so kind and provide some hints/solutions to the ...
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4answers
111 views

What are possible variations of the definition of $\sigma$-additivity?

From what I've read in Wikipedia, $\sigma$-additivity is defined in the following way: Let $\mathcal{A}$ be a $\sigma$-algebra of some underlying set $X$ and let $f$ be a mapping ...
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1answer
601 views

Free group and universal property

I'm trying to understand universal properties. An example is the definition of a free group (as I understand it so far): Revised definition: A free group $F_S$ over a set $S$ is a pair $(g,F_S)$ ...
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613 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
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1answer
131 views

The precise definition of a “sheaf of rings”

Let $X$ be a topological space. Let $Op(X)$ be the category of open sets. A sheaf of abelian groups is a (contravariant) functor $F:\mathcal{C} \to Ab$ satisfying the sheaf condition: the following ...
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Definition of fragility

What does it mean for a solution to a system of differential equations to be fragile? A context for the term can be found here: This is taken from here in Mathematical Methods for Mechanics: A ...
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2answers
1k views

real numbers and number line

While reading some articles, I got a bit confused by the definitions of numbers. Specifically, Can the number line contain decimal values? I read that Real numbers = All numbers on the number ...
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5answers
16k views

What is the difference between Singular Value and Eigenvalue?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for ...