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

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0
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1answer
434 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
322 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
342 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
516 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
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 ...
20
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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
votes
1answer
324 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
98 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
votes
1answer
138 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
votes
1answer
45 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
165 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
140 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
3k 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
votes
1answer
202 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
465 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
210 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
629 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
327 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 ...
10
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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
votes
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 $ ...
2
votes
2answers
333 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 $ ...
1
vote
1answer
181 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 ...
0
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1answer
221 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 ...
9
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1answer
408 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 ...
3
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3answers
1k 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 ...
3
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0answers
88 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 ...
0
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1answer
318 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 ...
0
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1answer
150 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 ...
7
votes
1answer
515 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 ...
1
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1answer
81 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 $f_{1} : P \rightarrow R$ $f_{2} : P \rightarrow R$ $g : R \rightarrow S$ $h_{1} : P \rightarrow S$ $h_{2} : P \rightarrow S$ ...
2
votes
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 ...
2
votes
4answers
112 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 ...
4
votes
1answer
778 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 $(g,F_S)$ ...
20
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3answers
652 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 ...
2
votes
1answer
133 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 ...
1
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0answers
69 views

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 ...
0
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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 ...
23
votes
5answers
22k 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 ...
5
votes
1answer
191 views

What set operation is this?

Given two sets $ A = \{\{1\} , \{2 , 6\} \}$ and $ B = \{\{2\} , \{3\} , \{4 , 5\} \}$, what set operation can produce $$ C = \{ \{ 1 , 2 \} , \{ 1 , 3 \} , \{ 1 , 4 , 5 \} , \{ 2 , 6 , 2 \} , \{ 2 , ...
0
votes
1answer
80 views

Which distribution can be abbreviated as “LD”?

Which distribution can be abbreviated as LD and which PDF is expressed as a formula with sum of erfc() functions? $$p(o)=\frac{1}{4\ell} ...
5
votes
2answers
397 views

the definition of the area of a surface

When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining ...
0
votes
1answer
100 views

Definition of the 1-dimensional $\mathbb{C}GL(V)$ module “$\det ^n$”

I'm reading through my notes on representation theory of $S_n$ and $GL(V)$, and have come unstuck on a definition which I can't understand - furthermore I can't seem to find any information on it ...
1
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1answer
129 views

Definition of “succession of central extensions of abelian groups”

What is the meaning of the phrase: "A group $G$ can be realized as a succession of central extensions of abelian groups"?
3
votes
2answers
1k views

A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
0
votes
1answer
52 views

Math vocab: operator on $S$ and into $S =$?

Is there a special name for a binary operation on the set $S$ that is also into $S$, that is unambiguous with other uses. I.e. if it's "operator on $S$", I've heard that in other places meaning the ...
3
votes
1answer
130 views

What do mathematicians call the Two's Complement on 8-bits group?

It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$ What is an elegant ...
5
votes
2answers
326 views

Is 2+2=4 an identity?

I know this seems like a silly question, but someone was trying to debate with me about how 2+2=4 should be called an identity and not an equation. I mentioned how it has no variables and isn't true ...
6
votes
5answers
615 views

What's the difference between tuples and sequences?

Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?
3
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2answers
1k views

$\{0,1\}^n$ and $[0,1]^n$ notations

Can someone please help me clarify the notations/definitions below: Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s? Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector ...
4
votes
2answers
132 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...