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

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6
votes
5answers
533 views

What's the name of this algebraic property?

I'm looking for a name of a property of which I have a few examples: $(1) \quad\color{green}{\text{even number}}+\color{red}{\text{odd number}}=\color{red}{\text{odd number}}$ $(2) \quad ...
0
votes
2answers
58 views

Very simple notation question

What notation is it called when a number is represented as a series of additions, for example: 124 = 100 + 20 + 4 This is a very simple question obviously but I don't remember what it's called! ...
2
votes
1answer
31 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
2
votes
0answers
42 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
0
votes
1answer
28 views

Terminology: Alternatives for zero crossing

Is it correct to name the red and blue points hinge points, as an alternative to zero crossing? Or are their better terms to describe these points? Update I have several functions like these. I ...
1
vote
1answer
31 views

Is the colimit of finite tensor products a tensor product?

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. ...
4
votes
1answer
58 views

What is the name of the matrix that is created by a vector times its transpose.

I am looking for the name of the matrix created by the following operation: $Z = z*z^T$ I know it should create a symmetric matrix with an element $Z_{ij} = z_{i}z_{j}$
3
votes
1answer
99 views

What does “calculus” mean?

"calculus" and "formal system" From http://en.wikipedia.org/wiki/Propositional_calculus#Terminology a calculus is a formal system that consists of a set of syntactic expressions ...
1
vote
0answers
30 views

Why are compact and noncompact manifolds without boundary called closed manifolds and open manifolds, respectively?

Why not just call them compact and noncompact manifolds? Isn't the general assumption that manifolds have empty boundary unless stated otherwise?
0
votes
1answer
24 views

An English question for a logical term

Consider a tuple of logical expressions: $(P_1, \ldots, P_n)$ such that $P_i\Rightarrow P_{i+1}$ for every $i=1,\ldots,n-1$. An English question: Should I call it implications tuple or tuple of ...
0
votes
0answers
23 views

Meaning of abstractness and concreteness

Do abstractness and concreteness mean for formal systems and their models respectively? Do they relate to how big the theory is? For example, the theory of rings is richer than the theory of ...
1
vote
3answers
129 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
2
votes
1answer
42 views

When does intersection of measure 0 implies interior-disjointness?

If there are two "nice" shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their ...
0
votes
1answer
45 views

Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
0
votes
0answers
23 views

Canonical term for $\overline X / X$ where $X$ is a normed space.

Let $X$ be a normed vector space. Let $\overline X$ denote its completion. Is there a canonical name for the quotient space $\overline X / X$? Some authors seem to use "torsion" as a name, but I ...
1
vote
0answers
45 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
votes
1answer
64 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 ...
-1
votes
0answers
78 views

What is the name of formula?

Can someone help me to name this formula? $$ f(x) = \begin{cases} 1 + x & x \ge 0 \\ \frac{1}{1-x} & x < 0 \end{cases} $$ thanks.
1
vote
0answers
33 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
votes
0answers
62 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
vote
1answer
25 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?
5
votes
0answers
30 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 $v$. ...
1
vote
2answers
63 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
votes
0answers
20 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 ...
0
votes
1answer
30 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?] ...
0
votes
0answers
37 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 $T$. ...
34
votes
16answers
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
49 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
40 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
259 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.
3
votes
1answer
70 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
35 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
521 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
117 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 ...
-1
votes
0answers
25 views

What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...
4
votes
2answers
52 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
64 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 ...
20
votes
3answers
651 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
97 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
41 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
61 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
244 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
33 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
21 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 ...