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

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2
votes
2answers
54 views

Do “small” and “large” numbers actually exist in an absolute sense?

Numbers like $(10)^{-10^{10^{10}}}$ are generally regarded as small, whereas numbers like, for example, Graham's Number, are regarded as extremely large. My question is, are these numbers simply ...
3
votes
0answers
19 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
10
votes
1answer
506 views

The word “integral” in calculus unrelated to “integral” / “integer” in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
1
vote
0answers
13 views

Directed Acyclic Graph - root and leaf node terminology

I have found conflicting terminology regarding how to label nodes in directed acyclic graphs. Specifically, I am looking for a definition of root and leaf nodes (preferably something to cite). For ...
2
votes
1answer
34 views

How to read $[G:N]$?

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$ is written $G/N$. I could find from this link how to read $G/N$ ("$G$ modulo $N$" or "$G$ mod $N$"), but I couldn't ...
3
votes
1answer
48 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
0
votes
0answers
48 views

Confusion regarding continuous functions between topological spaces – a subtle but possibly important point

Let $T: V_1 \to V_2$ be a linear mapping. Show that $T$ is a continuous function between $(V_1, \tau_{V_1}) $ and $(V_2, \tau_{V_2}) $ A direct solution to the problem is not what I am looking ...
4
votes
0answers
45 views

Terminology in graph theory

Let $G$ be a finite graph with the following property: For any vertex $a$ and edge $\{b, c\}$ of $G$, there is an edge connecting them: there is one of $\{a,b\}$ or $\{a, c\}$ in $G$. Is there ...
1
vote
1answer
28 views

What do you call a graph where the vertices are signed?

Let $G = (V, E)$ be a graph, and $f: V \to \{1,-1\}$ be a function assigning a sign to each vertex. What is this system $(G, f)$ called? In my current research, we've been using "oriented graph" ...
0
votes
0answers
3 views

Successive divisibility of a sequence? Progressive divisibility? terminology or reference

Perhaps I say that an (infinite) sequence $(r_n)$ of positive integers is progressively divisible iff $r_n \mid r_{n+1}$ for all $n$. Is there some other terminology that is in use for this? I am ...
1
vote
1answer
13 views

Name for a collection of subsets of a set $E$ such that every element of $E$ is in some member?

Let $X$ be a set. A partition of $X$ is a collection of subsets $\{E_\alpha\}\subset\mathcal{P}(X)$ such that the following holds. For every $x\in X$, there is some $\alpha$ such that $x\in ...
1
vote
0answers
33 views

Vector calculus vs Vector analysis?

I was just wondering, is vector analysis the same as vector calculus? What about multivariable calculus? Because my multivariable calculus book (which I assume is the same as vector calculus?) covers ...
2
votes
1answer
32 views

Is there a name for the vector field that points along contour lines of a scalar field, proportional to the gradient?

Given a scalar field $G$ on $\mathbb{R}^2$ (say), the vector field $(\frac{\partial G}{\partial x}, \frac{\partial G}{\partial y})$ is called the gradient of $G$. Is there a standard name for the ...
0
votes
1answer
66 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
0
votes
0answers
19 views

The basis induced by the nilpotent linear transformation

In Halmos's book, there is a theorem regarding the nilpotent transformation: If $A$ is a nilpotent linear transformation of index $q$ on a finite-dimensional vector space $V$, then there exist ...
0
votes
1answer
34 views

What does “421 is the smallest prime formed by the powers of two in logical order from right to left” mean and if so is it correct?

I've seen this on number gossip and a few other places, but I'm not exactly sure what it means. The only possibilities I have thought of for what they mean are "421 is the smallest center squared ...
3
votes
2answers
32 views

What is the name for one side of a ratio?

Basic example: "If you are asked to put a ratio in the simplest form, make sure that you have found the smallest factor that goes into both [?]." I've tried searching for ratio diagrams in Google, ...
0
votes
0answers
53 views

What is a nontrivial graph?

I've been operating happily under the definition that a nontrivial graph is a graph with at least two vertices for some time. Today I came upon a source which defined a nontrivial graph as a graph ...
1
vote
0answers
24 views

What to call a polynomial that with no roots in $\mathbb{Q}$ but does in $p$-adics

As an example, consider the polynomial $f(x)=x^2-2$. This polynomial has no roots over the rationals $\mathbb{Q}$. However, $\sqrt{2} \in \mathbb{Q}_7$, so it does have roots in the 7-adics. Is there ...
3
votes
1answer
31 views

Name for a nonlinear version of bilinear form

A map $b:X \times Y \to \mathbb{R}$ is called a bilinear form if $b$ is linear in both arguments. Is there a name for a form $b$ which is linear in only one argument and may be nonlinear in the ...
2
votes
0answers
35 views

What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
0
votes
1answer
77 views

What does it mean by “explode in finite time”? [closed]

I read some stochastic analysis book in probability theory (for the chapter of existence of solution) and it states "explode in finite time". What is the mathematical definition for this?
0
votes
0answers
16 views

What does it mean an ideal is nilpotent modulo another ideal?

Reference:Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition", Quart. J. Math. Oxford Ser. (2) 22: 73–83, doi:10.1093/qmath/22.1.73 Let $R$ be an rng and ...
1
vote
1answer
22 views

Terminology help for a set relation: for sets $X, Y$, not necessarily disjoint, such that neither is a subset of the other.

Is there an existing term for pairs of sets $X, Y$, not necessarily disjoint, such that neither $X \subseteq Y$ nor $Y \subseteq X$? Would it be incorrect (or misleading) to call them something like ...
1
vote
1answer
17 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
1
vote
1answer
43 views

Name when two functions are equal under integration (expectation)?

What is it called when $E[X] = E[Y]$? That is, $$\int x f(x)\,dx = \int y g(y)\,dy.$$ What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent ...
4
votes
2answers
123 views

What do we call the covector associated to a vector?

Let $V$ denote an inner product space. Write $V^*$ for either the algebraic dual, or else the continuous dual. In either case, for each vector $v \in V$, we get a covector $v^c \in V^*$ given by: ...
4
votes
0answers
112 views

When do the zero divisors of a commutative ring form an ideal?

Let $J$ denote the set of zero-divisors of a commutative ring $R$. Since we automatically have $RJ \subseteq J$, hence $J$ is automatically halfway to being an ideal. Furthermore, its already ...
2
votes
0answers
35 views

Proper names for different representations of the same formula

I would like to know what to call formulas that are all on one line and what to call the same formulas that are on multiple lines. One line example: P ÷ TVD ÷ 0.052 Multiple line example: ...
3
votes
1answer
29 views

Does “data” in Cauchy data come before or after the coinage of data in computer science

Is the usage of data as in Cauchy data (i.e. initial conditions) borrowed or came before the usage of data in computer science and do both usages mean roughly the same thing (data ~ information)?
1
vote
0answers
13 views

Is there a term for irreflexive relations whose reflexive closures are equivalence relations?

Consider a relation $\sim$ which is irreflexive, that is: $$ \forall A \:A\nsim A $$ Further suppose the reflexive closure of $\sim$ is an equivalence relation. The reflexive closure of $\sim$, ...
1
vote
1answer
42 views

prior probability vs a priori probability

What is the difference between "Prior probability" and "a priori probability" Wikipedia have two distinct pages for them. As of my inference i thought "Prior" and "a priori" are same, i.e., P(y) in ...
3
votes
0answers
25 views

By or through for a rotation

"Rotated through pi rad" vs "Rotated by pi rad" I have heard both used and also heard mentioned that there was a mathematical difference between the two. Is this true or can they be used ...
7
votes
5answers
406 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
0
votes
0answers
31 views

Name (reference?) for lax monoidal functors that are 'full, or surjective, on monoids'?

A lax monoidal functor $F$ takes monoids to monoids. Is there a name for a lax monoidal functor that is 'full' or surjective with respect to this property? In other words, the functor $F : \mathcal{C} ...
2
votes
3answers
95 views

Is there any difference between 'all real numbers' and '$(-\infty, \infty)$'

I've just thought about this. All the textbooks I've been looking at for pre-calc, the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as ...
0
votes
0answers
27 views

Name of a vertex set of the same out-degree

I have a graph and it is very important to me to distinguish vertex sets of the same out-degree. For example, I have a set of all vertex of the out-degree 1, a set of all vertex of out-degree2, and so ...
1
vote
1answer
17 views

Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
0
votes
1answer
18 views
1
vote
1answer
52 views

What does “thread” mean in general topology?

I am studying from "R Engelking - General Topology". In p.98 it is written: Why the word "thread" is chosen? I mean what's the relation to everyday meaning of thread?
2
votes
4answers
102 views

What is the direction along the edge of a circle called (in English and by chance German)?

Note: I am actually also searching for the term in German. That is why I posted this here (as opposed to the language SE's), besides me looking for this term in a mathematical/technical context. ...
5
votes
0answers
53 views

a new(?) operation using products of multiplicities

Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere? Notice that $\odot$ is ...
1
vote
1answer
44 views

Graph theory - how to find nodes reachable from the given node under certain cost

I'm considering the following problem (very rough description): Assume we have a graph where edges are assigned some non-negative costs, a starting node s and some ...
6
votes
3answers
266 views

What is the name for a function whose codomain and domain are equal?

What do we call a function whose domain and co-domain are the same set? Edit: While i expressed my question in terms of functions, domains and codomains, i was actually interested in the most ...
1
vote
1answer
89 views

Should the empty set be called “half-open”? [closed]

Empty set is both open and closed in any metric space (also in any topological space). Consider $\mathbb{R}$ with usual metric. In this metric space, should we say that the empty set is half open?
1
vote
1answer
50 views

How to call a category with a single morphism between every two objects?

How to call a category where for every pair of objects $A, B$, there is a unique morphism $f\colon A\to B$? (A trivial category?)
1
vote
1answer
43 views

Why are free modules called “free”?

Let $R$ be a ring (not necessarily commutative) with multiplicative identity. A $R$-module $M$ is called free if $M$ has a linearly independent generating set $\beta\subseteq M$. That is, for any ...
0
votes
0answers
38 views

Confused about Graph Theory Language

I was confused about some of the wording in this definition I came upon: Let G be a control flow graph (a control flow graph you can imagine as a directed graph) for program P. A hammock H is an ...
2
votes
1answer
74 views

Is a function a derivative?

I'm reading introductory calculus and I find that 'function' tends to be defined by what it does rather than what it is. If $y = f(x)$, then surely the value of $y$ is dependent on that of $x$, i.e. ...
1
vote
0answers
48 views

What is a “note” (that might be published) in mathematics?

My background is not in mathematics but I am translating a novel from the French that has a few mathematicians as characters. A few times, the publication of a "note" is mentioned. This is the French ...