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

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5
votes
2answers
437 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
vote
1answer
136 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
58 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
131 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
349 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 ...
8
votes
5answers
739 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
votes
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
133 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 ...
1
vote
0answers
61 views

Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
1
vote
1answer
162 views

Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
0
votes
1answer
451 views

Pairwise non-integral numbers

I have a set of complex numbers a_1 through a_n which are said to be "pairwise non-integral numbers". Could someone explain to me exactly what this means? Thanks. From comment below: I should also ...
2
votes
2answers
603 views

What are the primitive notions of real analysis?

My dad introduced my to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition? ...
2
votes
3answers
294 views

A question about a certain way to define mathematical objects

It is common in mathematics to see definitions of the following form: we begin with a certain object $A$. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for ...
1
vote
1answer
67 views

Does this function have a “name”, somewhat linked to Euler totient

If $\varphi$ denotes the Euler totient and $n=p_1^{k_1}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ we have $ \varphi(n)= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})= ...
1
vote
2answers
131 views

Elaborate on $A^{c}:=\{p\in\mathbb Q : 0<p<\sqrt{2}\}$ not open and not closed in $\mathbb R$

I know that $\sqrt{2}\not\in\mathbb Q$ and $\sqrt{2}\in\mathbb R$ but it is not obvious to me why $\{p\in\mathbb Q : 0<p<\sqrt{2}\} \subset \mathbb R$ is not open. If it is not open, it means ...
1
vote
3answers
77 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...
-1
votes
2answers
268 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
57
votes
9answers
4k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
6
votes
1answer
609 views

Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting ...
1
vote
3answers
4k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
1
vote
1answer
291 views

What is the definition of an induced cut

I am reading A simple min-cut algorithm (Stoer & Wagner, 1997), and the proof uses some terms I don't understand. Specifically I am unclear on what is meant by "$C_v$ the cut of $A_v \cup \{v\}$ ...
6
votes
2answers
205 views

What is $\mathbb{N}^{<\mathbb{N}}$?

What is the definition of this symbol $\mathbb{N}^{<\mathbb{N}}$? How is it related to the infinite product $\mathbb{N}^{\mathbb{N}}$?
4
votes
1answer
210 views

What is the difference between $\mathrm{E}[Y|X = x]$ and $\mathrm{E}[Y|X]$ and between $\mathrm{Var}(Y|X = x)$ and $\mathrm{Var}(Y|X)$?

I am slightly confused about the different between $\mathrm{E}[Y|X = x]$ and $\mathrm{E}[Y|X]$ and similarly for Variance. It seems to me the first should be a scalar, because we first pick a ...
2
votes
2answers
312 views

What is the product of the empty set?

Give: $fn(S)=\prod_{x\in S}x$ what is: $fn(\emptyset)$ I can see reason that it would be defined as 1 or 0. Note: I thought about restricting the domain of $S$ but that would make the problem ...
0
votes
2answers
96 views

Definition clarification on ideals

Suppose $P$ is the set of all subsets of a set $X$ and $P$ is a ring. Let $p$ be an element in $P$ (so that $p$ is a subset of $X$). What does it mean to say "an ideal generated by $p$"? And suppose ...
2
votes
1answer
279 views

Is Gödel's completeness theorem a representation theorem?

In general a representation theorem is — according to Wikipedia — a "theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure". ...
2
votes
1answer
310 views

What is a Gauss sign?

I am reading the paper "A Method for Extraction of Bronchus Regions from 3D Chest X-ray CT Images by Analyzing Structural Features of the Bronchus" by Takayuki KITASAKA, Kensaku MORI, Jun-ichi ...
4
votes
1answer
330 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
2
votes
1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
1
vote
0answers
96 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
2
votes
1answer
46 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
0
votes
1answer
167 views

Motivation behind the definition of reflections in affine hyperplane

What is the motivation behind the definition of the reflection map in affine hyperplanes? $R: x \to x-2(x\cdot u-c)u$ where $u\cdot x=c$ defines the affine plane. Of course one requirement is for it ...
2
votes
1answer
73 views

Definition clarifications : on adjectives of functions

Could somebody please explain what are the differences between a differentiable function and a holomorphic function analytic function and conformal function? (Am I right to think that all analytic ...
3
votes
1answer
322 views

Perfect Hash Function just an Injection?

I just read up on the concept of perfect hash functions on a set $S$. I am quoting: "A perfect hash function for a set S is a hash function that maps distinct elements in S to a set of integers, with ...
2
votes
2answers
304 views

Solvability and Simplicity

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says ...
7
votes
1answer
607 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
6
votes
3answers
1k views

How to write “let” in symbolic logic

How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is: $$ x := a ...
5
votes
1answer
450 views

Understanding the homotopy extension property

I'm reading Chapter 0 of Hatcher right now, and there's something in the definition of homotopy extension property that I don't understand. Suppose one is given a map $f_0:X\to Y$, and on a ...
2
votes
2answers
825 views

What is the differentiation operator

On Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives ...
1
vote
3answers
102 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ ...
0
votes
2answers
712 views

How many 'supremum(s)' and 'infimum(s)' can a set have?

I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum ...
1
vote
2answers
175 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
0
votes
1answer
245 views

definition: dual of a vector field

Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$ be a vector field, what is the definition of its dual ? I know that the set of vector fields on $\mathbb{R}^3$ forms an $\mathbb{R}$-vector space.
16
votes
2answers
561 views

What is it to be normal?

I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all. ...
0
votes
1answer
285 views

Why is this definition of an additive inverse significant

In the process of learning Real Analysis, I encountered a definition of an additive inverse of a cut $\alpha$ to be $$\text {add inv of } \alpha \colon= \{p:\exists r>0 \text{ s.t.} (-p-r)\notin ...
6
votes
1answer
2k views

Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
2
votes
0answers
218 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
3
votes
1answer
4k views

Relationships among the terms “slope”, “parameter”, and “coefficient”?

In $y=mx$, is $m$, are there different implications of referring to $m$ as a "slope", a "coefficient", a "parameter"? Or perhaps the "slope coefficient" or "slope parameter"? For context, I am ...