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

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1
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0answers
49 views

Why is it called a primitive root?

I am looking for a paper or reference that explains why primitive roots are called primitive roots. I know what they are but was wondering if there was a reason?
3
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1answer
79 views

What is a 'disjunct' of a union called?

Say I have a set $C = A \cup B$ and I want to refer to $A$ in natural language. Had the expression been a Boolean formula with a disjunction, then I would call $A$ the first disjunct. Is there a ...
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0answers
35 views

Different names for “function”

Quoting a book, "functions can also be named: Mappings, Transformations, Operators, Arrows or Morphisms" I have the idea that these different names are used depending on different contexts. But I ...
2
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0answers
64 views

Please identify this equation: $\nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A$

Is this equation $$ \nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A $$ somehow named? F and A are vector fields. I guess inhomogeneous sign reversed Helmholtz equation isn't appropriate ...
1
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1answer
32 views

Definition of a geodesic ball?

I think it goes along the lines of: a ball made of a series of flat sides. Also is a geodesic ball and geodesic dome the same thing?
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0answers
39 views

Name for Number of Ancestors/Descendants of Vertex in Directed Acyclic Graph

Let $G = (V, E)$ be a directed acyclic graph. For each vertex $v \in V$, define the ancestors of $v$ to be the set of vertices $u \in V$ such that there exists a directed path from $u$ to $v$. ...
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2answers
66 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
0
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0answers
23 views

What is the edge called that converts a tree to a directed acyclic graph?

Neither Wikipedia nor mathworld gave the answer: What is the name of the edge (or multiple edges) without which a DAG would be a tree? Or maybe instead: What is the name of the subgraph such that ...
1
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1answer
40 views

A simple, yet non-superficial explanation of what “paramodulation” means in the context of automated theorem proving?

Modern automated theorem provers seem to be paramodulation-based. I only have a superficial understanding of what this means: we derive a proposition whose truth is implied from the truth of [two?] ...
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0answers
38 views

What do we call those functions that can be obtained from term operations by partial evaluation?

Let $T$ denote an algebraic theory and suppose $X$ is a $T$-algebra. Then a term operation of $X$ is a function $f : X^n \rightarrow X$ that is definable by an expression in the language of $T$. ...
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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
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1answer
58 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)?
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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 $$ ...
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0answers
62 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 ...
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2answers
297 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
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1answer
88 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
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1answer
44 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
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1answer
51 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
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2answers
607 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 ...
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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
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2answers
143 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
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0answers
47 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
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2answers
66 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
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1answer
83 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
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3answers
698 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
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1answer
103 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
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2answers
52 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
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1answer
70 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
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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
50 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
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1answer
30 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
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0answers
51 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
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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
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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
86 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
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2answers
48 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 ...
0
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0answers
65 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
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2answers
117 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
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1answer
57 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
56 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
177 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
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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
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1answer
48 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
93 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
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2answers
72 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 ...
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0answers
24 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
49 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
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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
32 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
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0answers
31 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, ...