Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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37
votes
17answers
4k views

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend ...
2
votes
1answer
51 views

How to call a tree with a single branch?

How do you call a tree with only one branch (in other words, where every vertex has maximum one direct successor)?
1
vote
0answers
41 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
1
vote
0answers
59 views

Is it “group axiom” or “group definition”?

Some text books of group theory use "group definitions" when introducing group, and some other text books use "group axioms". But it is obvious that terms "definition" and "axiom" are different. Which ...
1
vote
2answers
269 views

Why is a random variable called so despite being a function?

According to my knowledge, its a function $P(X)$ which includes all the possible outcomes a random event.
4
votes
1answer
80 views

What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
1
vote
1answer
41 views

Definition of null space

I have two definitions of null space. One by Serge Lang Suppose that for every element $u$ of $V$ we have $\langle u,u\rangle=O$. The scalar product is then said to be null, and $V$ is called a ...
2
votes
1answer
48 views

In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
7
votes
2answers
536 views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
1
vote
0answers
21 views

What's the name for a polygon with exactly two sets of side lengths?

Is there a name for the shape similar to a regular polygon, but using exactly $2$ side lengths (or $n$ side lengths) instead of one side length?
5
votes
2answers
132 views

What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
3
votes
0answers
45 views

Objects without extensions

How do you call an object $X$ for which every monomorphism $i : X \hookrightarrow Y$ has a retract (i.e.\ a morphism $r : Y \rightarrow X$ such that $r \cdot i = 1_X$)? I think of Y as an extension ...
4
votes
2answers
59 views

Why are stochastic processes with decreasing expected value called supermartingales?

I am curious to know why a process which has decreasing expected value is called a supermartingale. From a beginners perspective it would seem reasonable to have the following picture: ...
7
votes
1answer
72 views

Is there a name for those commutative monoids in which the divisibility order is antisymmetric?

Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows. $$x \mid y \leftrightarrow \exists a(ax=y)$$ Is there a name for those commutative monoids such ...
21
votes
3answers
675 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
5
votes
1answer
98 views

Additive non-abelian group?

Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So ...
2
votes
2answers
43 views

What are a geometric system and a finite geometry?

Wikipedia says A finite geometry is any geometric system that has only a finite number of points. I wonder what a geometric system is? Is it some set system $(E, F)$, where $E$ is a set and $F ...
6
votes
1answer
66 views

Name of a shape that is intersected once by each ray that starts at a given point

Is there a particular name for a shape that is intersected exactly once by each ray that starts at a given point? To illustrate: I'm looking for a name for shapes like the left one in this image: ...
1
vote
2answers
247 views

*Presume* and *Imply*

I am not sure about the usage of the word presume. For example, is the sentence differentiability implies continuity equivalent to differentiability presumes continuity or to continuity presumes ...
2
votes
1answer
35 views

Matrices with the same characteristic polynomial

For all the $n \times n$ matrices, let's define an equivalent relation that two matrices are in the relation iff they have the same characteristic polynomial. How can we characterize the matrices ...
1
vote
1answer
25 views

In a set, what is the term to describe the number of unique values divided by the total number of values?

The closest word I can think of would be "uniqueness" although I know there is a more specific mathematical term. Say we have a set/table of data with two columns that describes cars. One column is ...
1
vote
0answers
49 views

Soft question (Etymology - Flatness)

Why where flat modules named "flat"? Is it because they are necessarily torsion free so in a "not convoluted" or circular like $\mathbb{Z}/n\mathbb{Z}$ is as a $\mathbb{Z}$-module?
2
votes
1answer
36 views

Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
0
votes
1answer
40 views

theorems that depend on the embedding of an affine variety into the affine space

Let $\mathcal{T}$ be a theorem regarding an affine variety $Y$ of $\mathbb{A}^n$. Question 1: What does the phrase "$\mathcal{T}$ does not depend on the embedding of $Y$ in $\mathbb{A}^n$" mean? ...
3
votes
1answer
80 views

Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
1
vote
2answers
46 views

If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ “have the same distribution”?

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution? In a way, this seems correct: both $X$ and $-X$ have the same probability density ...
1
vote
0answers
62 views

Name of an aglebraic structures $(A,*,\cdot)$ weaker than semirings.

I have a set $A$ with two binary operations on it $(A,*,\cdot)$ STRUCTURE A $(A,*)$ is not commutative, is not associative, it has not an identity $(A,\cdot)$ is a commutative group $(a*b)\cdot ...
0
votes
2answers
107 views

LambertW: $ x=W(x\cdot e^{x}) $ for $ x \ge -1$ but not for $x \lt-1$. How do I express my formula/my text?

I just found by numerical heuristics for some systematic numbers $q(x)_\text{heuristical}$ depending on $x=1,2,3,4,\ldots$ using WolframAlpha the suggested interpretation in terms of the ...
1
vote
1answer
43 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
2
votes
3answers
55 views

What's the name of the set of products of equal to a given value?

Suppose we have the * operator on a set $A$ such that * is associative but not commutative. Given $a$, $b$, $c \in A$, \begin{align*} abc &= (abc) \\ &= (a)(bc) \\ &= (ab)(c) \\ &= ...
2
votes
3answers
144 views

Are the pre-image and the domain the same, or not?

Throughout school I thought that the pre-image was a subset of the domain, not that they were necessarily the same. When I spoke of a function f:R->R, I didn't think that this meant that f was defined ...
7
votes
6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...
1
vote
1answer
47 views

When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
5
votes
1answer
89 views

What is the name of a graph made of k copies of a 4-cycle connected end to end in a chain, possibly with leaves?

Do graphs of the following sort have a specific name? We've been calling them Cactapillars, as they're cacti that look a little like caterpillars (and the name Caterpillar already refers to a ...
1
vote
2answers
69 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
1
vote
0answers
23 views

Enunciation of $\partial$ as the boundary map

How is $\partial$ typically pronounced when it is used as the boundary map in homology theory? The answer to this question provides some good information on the enunciation of $\partial$, but more ...
2
votes
3answers
46 views

Terminology: geometric sequences and geometric means

(I'll post my own answer to this one, but that should not deter others, since my answer is a surmisal.) Why are geometric sequences called geometric sequences? Whare are geometric means called ...
2
votes
3answers
149 views

Name the property $f(x) \ge x$

It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property: $$f(x) \ge ...
2
votes
0answers
27 views

what does “modular” mean?

I find some similarity of the concept "modular set functions" to the cardinality function. But I don't see the cardinality function is also called "modular" or something else. I wonder what "modular" ...
5
votes
0answers
29 views

“Advective”, “diffusive”, “dispersive”, and related terms in the realm of PDEs

Whenever I read a paper involving PDEs, the discussion inevitably refers to “the dispersive term” or “the advective term” or similar. From context it is usually possible to figure out the antecedent, ...
1
vote
0answers
22 views

What does “modular” in “modulr functions” mean?

From Wikipedia If $\Omega$ is a set, a submodular function is a set function $f:2^{\Omega}\rightarrow \mathbb{R}$, where $2^\Omega$ denotes the power set of $\Omega$, which satisfies one of the ...
2
votes
3answers
245 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
1
vote
2answers
52 views

Complete vs Perfect infomation in Combinatorial game theory

In their book "Winning Ways for Your Mathematical Plays", Berlekamp, Conway, and Guy used as the 7th condition for a combinatorial game "Both players know what is going on; There is complete ...
0
votes
0answers
24 views

Terminology in universal algbera

(Fix throughout a functional language $\Sigma$.) Given an algebra $A$ with underlying set $\vert A\vert$, there is an obvious surjective homomorphism from $A$ to the free algebra generated by $\vert ...
2
votes
3answers
61 views

Standard terminology for infinite limits with opposite sign on the two sides?

Consider the following limits: $$ \lim_{x\rightarrow0}\frac{1}{x^2}$$ $$ \lim_{x\rightarrow0}\frac{1}{x}$$ As far as I can tell, most authors say as a matter of terminology that these limits don't ...
1
vote
0answers
40 views

Addition, multiplication, exponentiation… What is next function of this series?

Addition can be (informally) defined as the application of successor function $S$ on $a$ $b$ times, i.e. $a+b=S\stackrel{b}{\cdots}S a$. Multiplication can be defined as the addition of $a$ with ...
6
votes
1answer
57 views

What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
0
votes
0answers
41 views

A term for category where every loop of morphisms is an identity

"A category where composition of every loop of morphisms is an identity." Moreover, in the case I am thinking about, morphisms are bijective functions. Is there a name for this concept?
0
votes
1answer
40 views

Help to conceive a name

Filter $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ Ideal $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I ...
1
vote
1answer
28 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...