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

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5
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1answer
338 views

Term for: There Exists a Rational between every two Rationals?

The integers and the rationals have the same cardinality, but the rationals satisfy the property that: $$ \forall p,q\in\mathbb{Q},\quad \exists r\in\mathbb{Q}\quad \textrm{s.t.}\quad p<r<q, $$ ...
1
vote
1answer
30 views

What are the terms for the elements in the Euclidean algorithm $a = qb + r$?

In a ring $R$ with a Euclidean norm $N$, given $a,b \in R$ with $b\neq 0$, there are elements $q, r$ such that $a = qb + r$. Is there any special terminology for the elements $a,b,q,r$ in this ...
0
votes
1answer
16 views

What to call the Euclidian norm divided by a constant

I'm using the Euclidian distance $d_{2}$ divided by a constant $T$, i. e. $\frac{d_{2}}{T}$. However, I'm not sure what to call this. I'd like to keep things simple so I thought maybe "scaled ...
1
vote
1answer
33 views

Any name for a special matrix with only non-zero entry

Consider an $n\times n$ matrix $\mathbf{E}_{ij}$ which is 1 at entry $(i,j)$ and zero everywhere else. Is there any special name for this kind of matrices?
0
votes
1answer
98 views

Must different constant symbols denote different objects?

In first-order theory with equality and >=2 constant symbols (let's denote two of them by c and d), does it always happen that $\neg(c=d)$ is derivable (possibly stated as an axiom)? In other words, ...
0
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0answers
36 views

A graph with single circle

Is there a name for a (directed) graph that has exactly one circuit that goes through all of its vertices (in the same direction)? If so, what is it called? Example is as follows: $$A\to B, B \to C, ...
1
vote
1answer
29 views

Corresponding graph: Forest <-> Tree, “???” <-> Polytree

Is there a specific term for a graph, that consists of Polytrees, like a Forest consists of Trees? Or can a Polytree be disconnected?
2
votes
0answers
106 views

How mathematical theorems and concepts gain their names?

Cantor's theorem, Woodin Cardinal, Sacks Forcing and Martin's Axiom are just some of well-known theorems and concepts of mathematics which have the name of those mathematicians who introduced these ...
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0answers
14 views

Nominalization for being “not convex” and “not coercive”

Having a function f which is not convex or not coercive (coercive = |x| goes to infinity implies ...
0
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1answer
17 views

Nominalization for being “not convex” and “not coercive”

Having a function f which is neither convex nor coercive. What is the correct nominalization for these properties? I suppose something like ...
1
vote
1answer
68 views

What defines “triviality”?

I realize the title is perhaps not the most helpful. I am aware of several uses of the word "trivial," and I'm hoping that perhaps someone can provide some further insight. 1) Trivial sub-objects, ...
4
votes
1answer
71 views

Derivable doesn't exist in english?

I have a question about terminology. See this is what happens: someone says "this function is derivable", and then another, more experienced Anglo-Saxon mathematician goes on to correct this someone, ...
3
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1answer
42 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
1
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0answers
17 views

Arithmetic and geometric sequences: where does their name come from?

Where does the name of these two famous types of sequences come from? The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the ...
0
votes
0answers
5 views

Is there any relationship between co(ntra)-variance and active/passive transformation?

I have got that contravariance means that vector coordinates are corrected as $P^{-1}$ when basis vectors are transformed as $P$. Thereby, there are some objects that are attached to the axis and ...
12
votes
3answers
578 views

Why are groups “abelian” but rings “commutative”?

I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to ...
2
votes
1answer
96 views

What does it mean to say that a forcing “collapses cardinals”?

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension? If ...
6
votes
0answers
63 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
0
votes
1answer
56 views

Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ ...
0
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0answers
16 views

Perturbation factor terminology

This is a question about usage of English. I have an inequality $a^\textsf{T}x \leq b$, where $a$, $b$, and $x$ are vectors in $\mathbb{R}^n$. Now, I want to perturb this inequality by a small amount ...
1
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0answers
32 views

“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
0
votes
0answers
7 views

What is the term for vector of indicators of the presence of values in another vector?

(I hope this isn't off-topic for the site.) Suppose I have a vector $v$ of non-negative integers (or a finite sequence, if you like to think of it that way). The vector $h$ of integers, with $h(k)$ ...
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0answers
35 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
4
votes
2answers
80 views

Mathematical concept for formal languages

A formal language is defined as a subset of finite-length strings over an alphabet. It is similar to the mathematical concept "relation", but the lengths of its strings are not fixed. Since the name ...
3
votes
1answer
40 views

What is the central idempotent of a representation?

The article I am reading says Let $P_\lambda \in Z(G)$ be the central idempotent corresponding to the representation $\lambda$. Could someone explain what this sentence means?
0
votes
1answer
33 views

What is the name of a certain subset in a poset?

Is there a name for a subset $\{x_i\}$ of a poset $(P,\leq)$ satisfying $x_1 \leq x_2 \geq x_3 \leq \cdots \geq x_{n-1} \leq x_n$? (The subset could be infinite and the inequalities could be strict.) ...
1
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0answers
21 views

Standard deviation and related quantities

By definition, standard deviation is the square root of the variance. There is some common terminology for the quantities $$\mathbb{E}(|X-\mathbb{E}X|^p)^{1/p} $$ for $p \geq 1$? Or, they are just ...
2
votes
1answer
39 views

Difference between classification and characterization [closed]

What is the difference between classification and characterization in reference to mathematical objects ? Some examples will be appreciated.
0
votes
1answer
48 views

Is there a name for numbers that have 2 as their greatest common divisor?

Is there a name for numbers that have two as their greatest common divisor? Such as 8 and 130.
0
votes
1answer
16 views

Is there a name for this type of homogeneity

Given a polynomial $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$, there exist sequences of homogeneous components $f_{(j)},f^{(k)}:\mathbb{R}\times\mathbb{R}^n\to\mathbb{C}$ such that $f_{(j)}(x,y)$ ...
0
votes
1answer
34 views

What's the difference between “continued fractions” and “compound fractions”?

What should we call a fraction which includes another fraction in its numerator or denominator, like $${ab\over {c \over d}}$$?
2
votes
1answer
42 views

How many contiguous subsets of size $N$ does an infinite grid have?

Suppose I have an infinite grid. How many sets of grid points are there that contain $N$ contiguous grid points, and include the grid point at the origin? So for example, if $N$ = 2, then there are 4 ...
0
votes
1answer
68 views

What is the name of $a \mapsto b$?

$f: A \to B$ is called a mapping, where $A$ and $B$ are two sets. What is $a \mapsto b$, where $a \in A$ and $b \in B$, called then? Thanks. Note that $a↦b$ is not a function/mapping, since $a$ and ...
0
votes
4answers
58 views

Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this: (c) For any $r>0$, $\overline{B}(x;r)$ ...
1
vote
0answers
19 views

Is cocycle condition necesarry for coassociativity of coproduct and why is it called “cocycle condition”?

Let $\Delta_0$ and $\Delta$ be coproducts related by \begin{equation} \Delta h = \mathcal F \Delta_0 h \mathcal F^{-1} \end{equation} where $\mathcal F \in H\otimes H$ is Drinfeld twist and $h \in H$ ...
1
vote
0answers
26 views

Name for numbers with a single non-zero digit.

Given a base, is there a name for the numbers (positive or negative) that have only a single non-zero digit? For example: Decimal: 4000, -30, 0.0008 Binary: 1000 Base 5: 300, -0.1 Contrived ...
5
votes
2answers
578 views

Why the name “square root”?

Why do we say that $\sqrt{a}$ is a square root of $a$? Is this because $\sqrt{a}$ is a root of the function $f(x)=x^2-a$? Cubic root similarly? Thanks in advance
0
votes
1answer
18 views

Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
0
votes
1answer
17 views

What's the difference between a partial function and a relation?

My understanding of a partial function is that it is one which only maps a subset of some set $A$ to another set $B$ (where $B$ could be $A$). On the Wikipedia page, the below image is given as an ...
3
votes
1answer
80 views

The walk of a knife

"A knife is slowly moved parallel to itself over the top of a cake. At each instant the knife is poised so that it could cut a unique slice of the cake. As time goes by the potential slice increases ...
30
votes
6answers
1k views

Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
1
vote
1answer
136 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
6
votes
1answer
83 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
1
vote
1answer
30 views

The origin labelled on a graph: $0$ or O?

When one draws a graph, say in the x,y plane, we label the origin with a circular/elliptical symbol. Now is this a $0$ (zero), or is it O (for Origin), or simply just a circle/ellipse? Can it be ...
2
votes
1answer
20 views

Corrective terms for combinations

Take: $12$ people need to be split up into equal teams for a quiz. How many ways can this be done? The answer may initially seem to be $\displaystyle \frac{12!}{6!6!}$. but, since a single grouping ...
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0answers
23 views

Has an order with this property a special name?

If $a$ is an element in a preorder then you can eventually go 'a step back' (and repeat this) in the sense of finding an element with $b\leq a$ and not $a\leq b$. Is there a special name for ...
1
vote
1answer
60 views

Is there a difference between “unity” and 1 in applied mathematics?

Is there a difference between "unity" and 1 in applied mathematics? I know mathematicians have "roots of unity" and "partitions of unity", but at least those have become standardized. In ...
1
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0answers
25 views

term for a sum of diagonal and skew-symmetric matrix?

Is there a term for a matrix that is a sum of a diagonal and a skew-symmetric matrix? One particular example of this is a 2x2 matrix of the form $$ M = \begin{bmatrix} a & b \\ -b & a ...
0
votes
0answers
53 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
1
vote
3answers
49 views

Definition of homogeneous ODE

In my lecture notes, it gives this following definition of a homogeneous ODE: A differential equation is called homogeneous if it can be written in the form $x′=f(\frac{x}{t})$ Then in one of ...