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

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7
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0answers
159 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
7
votes
3answers
384 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
7
votes
3answers
240 views

The usage of the term “family” in mathematics

In our lecture notes, the term "family" is used quite persistently and with no definition given. Some examples: (i) Let V be a vectorspace and $(v_i)_{i \in I}$ a family of vectors... ...
7
votes
2answers
796 views

Meaning of “a mapping factors over another”?

I was wondering what "a mapping factors over another mapping" generally means? Does it have something to do with commutative diagram in category theory? I have seen this usage in different ...
6
votes
1answer
329 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
6
votes
2answers
2k views

opposite of disjoint

Sets whose intersection is the empty set are called disjoint. What is the opposite of a disjoint set? For example the sets $\{1,2\}$ and $\{2,3\}$ satisfy this condition. I know that you can just say ...
6
votes
2answers
520 views

(k+1)th, (k+1)st, k-th+1, or k+1?

(Inspired by a question already at english.SE) This is more of a terminological question than a purely mathematical one, but can possibly be justified mathematically or simply by just what common ...
6
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5answers
873 views

How common is the use of the term “primitive” to mean “antiderivative”?

I don't know if this should actually be asked on the English stackexchange. It seemed like I would find better answers here. I have all but finished an undergraduate degree in mathematics in the ...
5
votes
1answer
138 views

Why do we traditionally use letter U for open sets?

Most of traditional usages of symbols in mathematics have origin in English, German or French words that start with that letter, for an example: $p$ for a prime number, $\mathbb{Z}$ for integers (ger. ...
5
votes
2answers
6k views

What is the difference between an axiom and a postulate?

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more ...
5
votes
1answer
488 views

Difference between elementary logic and formal logic

In Kelley book on topology, in the appendix on elementary set theory, he says in the second paragraph, that "a working knowledge of elementary logic is assumed, but acquaintance with formal logic is ...
5
votes
3answers
424 views

A 1-1 function is called injective. What is an n-1 function called?

A 1-1 function is called injective. What is an n-1 function called ? I'm thinking about homomorphisms. So perhaps homojective ? Onto is surjective. 1-1 and onto is bijective. What about n-1 and ...
5
votes
3answers
3k views

What is Modern Mathematics? Is this an exact concept with a clear meaning? [closed]

Motivated by this question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern ...
4
votes
2answers
349 views

Is there a meaningful distinction between “inclusion” and “monomorphism”?

The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case? Edit: ...
3
votes
0answers
170 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 ...
3
votes
0answers
163 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 ...
3
votes
1answer
85 views

Minimality in the case of partial derivatives and Sobolev spaces?

I am trying to understand this question here that considers Sobolev spaces apparently and hence partial derivatives. What is the definition of minimality there? Is the minimality defined by ...
3
votes
2answers
243 views

What should we call the 'sets' which don't exist under certain set theory axioms?

For example we know that the set of all ordinals does not exist in ZFC, so what should we call it? Set? Collection?
3
votes
2answers
2k views

What is the difference between the terms “classical solutions” and “smooth solutions” in the PDE theory?

What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? Especially,the difference for the evolution equations? If a solution is in ...
3
votes
3answers
3k views

Definition of “maximal” and “minimal” [duplicate]

Possible Duplicate: difference between maximal element and greatest element When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my ...
3
votes
2answers
1k views

name for a rational number between zero and one?

I'm searching for a unified name to convey for the concept that a number will always be between zero and one. Some info for context: in probability we've got a number between 0 and 1. Percentages ...
2
votes
1answer
67 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 ...
2
votes
1answer
183 views

Definition of the $\sec$ function

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ ...
2
votes
1answer
70 views

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this ...
2
votes
4answers
323 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...
2
votes
3answers
13k views

What is the meaning of equilibrium solution?

What are the equilibrium solutions for the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 0.2\left(y-3\right)\left(y+2\right)$ My Question: What does equilibrium solution mean in this ...
2
votes
3answers
2k views

Is there a special name for the operands of a multiplication?

Sometimes operands for a specific operation are given a special name. For example, in division the first operand is a quotient, the second is a divisor. Is there a word that means "one of the operands ...
2
votes
1answer
129 views

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in ...
2
votes
4answers
1k views

$y'''-y=x^{2}$ has solution — `“multiplicity”`?

The page 667 of the book (sorry not in English) claims $y'''-y=x^{2}$ to have the solution $$y(x)=C_{1}e^{x}+e^{-x/2}\left(C_{2} \cos \left( \frac{\sqrt{3}x}{2} \right)+C_{3} ...
2
votes
1answer
102 views

Jordan Measures without $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$?

I am trying to prove that Jordan measures satisfy with the following properties $A, B \subset \mathbb R$ and $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$, similarly for $B$: $$\bar{\mu} (A) ...
1
vote
2answers
194 views

Why do we call Functional Analysis like this?

Functional analysis is 'a kind of mathematical analysis' where the object of study are functions. The tool for studying functions are the operators. A specific type of operators are the functionals. ...
1
vote
3answers
196 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
1
vote
1answer
119 views

Divisible abelian $q$-group of finite rank

What does "finite rank" mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
0
votes
1answer
132 views

System, dynamic system and feedback system

Can a system be defined as a mapping from a set of mappings, called input signals, to another set of mappings, called output signals, where the two sets of mappings may or may not have the same ...
13
votes
3answers
420 views

Etymology of “topological sorting”

This may be a dumb question, but what's "topological" about topological sorting in graph theory? I thought topology was related to geometry and deformations.
12
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3answers
295 views

What do we call entities (like $\sum$) that bind variables?

In logic, we refer to entities like $\forall$ and $\exists$ as quantifiers, because they bind variables. However, variable-binding doesn't just occur at quantifiers. For example, the symbol $i$ ...
11
votes
3answers
779 views

What is the $x$ in $\log_b x$ called?

In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is ...
10
votes
2answers
5k views

Difference between root, zero and solution.

Can somebody precisely tell me what is the difference between a root, a zero and solution ? Is it correct to say that an equation has solutions, and a polynomial has zeros or roots?
10
votes
2answers
638 views

Usage of the word “formal(ly)”

This is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems to ...
8
votes
2answers
395 views

What is an element of a rng called which is not the product of any elements?

Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in ...
8
votes
1answer
527 views

Summation formula name

What is the name of the following summation formula? $$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f - \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f'w,$$ where $w$ is the “sawtooth” function, defined by ...
7
votes
3answers
968 views

What does a space mean?

In Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?
6
votes
5answers
1k views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
6
votes
1answer
283 views

What are “Lazard” sheaves?

Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example: Let $\mathcal{LH}$ be the category whose objects are topological ...
6
votes
2answers
167 views

History of the vocabulary for group extensions

In regular everyday English if you say something like "A was extended by B to get C", to me it means that A was in existence, B was added onto it, and now there is a larger object C. For example, ...
5
votes
1answer
45 views

Is there a name for a monoid with a distinguished absorbing element?

Let $M = (M,·,1,0)$ be a monoid $(M,·,1)$ together with an distinguished absorbing element $0 ∈ M$, that is such that $∀x ∈ M\colon 0·x = 0 = x·0$. Does such a structure $M$ have a nice name? ...
5
votes
3answers
82 views

Name for introducing negation with quantifiers

The rewriting of $\varphi\to \psi$ into the logically equivalent $\neg \psi\to\neg \varphi$ is called contraposition. Is there a similar word for rewriting $\forall x.\varphi$ into $\neg\exists ...
5
votes
1answer
120 views

What word means “the property of being holomorphic”?

As in the title, I am looking for a single word meaning "the property of being holomorphic". The obvious candidates are "holomorphy" and "holomorphicity" but both look wrong to my eye. ...
5
votes
0answers
112 views

Is there a name for graphs with the following property?

The property of the graph is the following: For any vertex, there is a hamiltonian path starting with this vertex, but the graph is not hamiltonian. The following graph is a small example: ...
5
votes
1answer
250 views

On proving $n = \sum_{d\mid n}\varphi(d)$

$\def\nset{\{1,\dots,n\}}$ I'm trying to work out my own proof1 of Euler's classic formula $$n = \sum_{d\mid n}\varphi(d)\;.$$ I'm looking for some pointers to the standard terminology and/or ...