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

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2
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1answer
61 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
0answers
109 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 ...
2
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0answers
115 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 ...
2
votes
1answer
162 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
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1answer
69 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
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0answers
102 views

What do we call a Schauder-like basis that is uncountable?

In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?
2
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3answers
10k 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
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3answers
1k 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
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1answer
127 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
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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
101 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) ...
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2answers
2k views

Definition of “contradiction” and use of the term for “⊥”

If one looks in Internet for definition of “contradiction” (including respective words in other languages), one finds a mess. See for example this index of Wikipedia articles in various languages. The ...
1
vote
3answers
189 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
111 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
97 views

Does this object have a category-theoretic name?

I have morphisms: $$ f : A \to B \\ g : B \to C $$ The composition is: $$ g \circ f : A \to C $$ In the function $(g \circ f)$ we call $A$ the domain and $C$ the codomain (or range). I'm ...
0
votes
1answer
125 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 ...
10
votes
2answers
513 views

Injection and surjection - origin of words

Can anyone give me a good explanation of how and why words surjection and injection came into use in mathematical community? What do they exactly mean? Who introduced them? I have a feeling students ...
9
votes
1answer
444 views

Is “monotonous” ever used as a synonym for “monotonic” in math?

I saw a few questions and answers recently that wrote "monotonous" instead of "monotonic." Then I Googled and see a ton of usages of "monotonous" in M.SE instead of monotonic. It occurred to me this ...
9
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3answers
2k views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
8
votes
1answer
508 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 ...
6
votes
3answers
277 views

Does “monotonic sequence” always mean “a sequence of real numbers”

When we say a sequence is monotonic, does that imply the sequence is Real Number Sequence? And other propositions about monotonic, all real-valued? When I see some ...
6
votes
5answers
914 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
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
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1answer
264 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
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2answers
161 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
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3answers
78 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
111 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
108 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
202 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 ...
5
votes
1answer
109 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
5
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1answer
300 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
5
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3answers
2k views

Can the word “derive” be used to mean “take the derivative of”?

Back when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be ...
4
votes
3answers
131 views

What does “lower density” mean in this problem?

If $\mathscr{U}$ is a ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. This is an exercise on page 76 of Problems and Theorems in Classical Set Theory, ...
4
votes
4answers
412 views

Is it proper to say that two infinite sets are the “same size” if there is a bijection between them?

I get the fact that a set can be called countably infinite if it can be bijected with $\mathbb{N}$, but it feels wrong on many levels to say that they are the same size. Example: $A=\{x \in ...
4
votes
1answer
127 views

What is the correct terminology to say that $\small f(x)=a+bx+cx^2+…$ can be expressed by $\small g(x)=A(1-x)+B(1-x)(2-x)+C(1-x)(2-x)(3-x)+… $

Hm, I do not even know the best formulation for my question in the header. It is not for the math but for the proper writing/terminology. I've come across the term "base change" recently but the ...
4
votes
1answer
121 views

“belongs to” versus “contained in”

Let us consider a set $A$. let $B$ be an element of the set. Now what I want to know is that whether saying $B$ is contained in $A$ and $B$ belongs to $A$ means the same? Could anyone here cite any ...
4
votes
5answers
591 views

How can I succinctly but correctly say that a set is finite?

If I want to say that a set $A$ is numerable but infinite, I can do so like this: $$|A| = \aleph_0$$ What should I use instead to say that a set is finite? $|A|\in\mathbb{N}$? $|A|< \infty$? ...
4
votes
0answers
214 views

Is there some official name for this function?

$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$ Like half-circles connected to each other to look like waves: Its plot looks smooth, but the function is actually not ...
3
votes
1answer
146 views

The functor $\mathbf{D} \rightarrow \mathbf{Prof}$ obtained by “splitting” $F : \mathbf{C} \rightarrow \mathbf{D}$ at each object of $\mathbf{D}.$

(Write $\mathbf{Prof}$ for the category whose objects are categories and whose arrows are profunctors.) I'm pretty sure that every functor $F : \mathbf{C} \rightarrow \mathbf{D}$ yields a ...
3
votes
3answers
116 views

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in ...
3
votes
2answers
740 views

Validity vs. Tautology and soundness

I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula? They also ...
3
votes
3answers
7k views

In graph theory, what is the difference between a “trail” and a “path”?

I'm reading Combinatorics and Graph Theory, 2nd Ed., and am beginning to think the terms used in the book might be outdated. Check out the following passage: If the vertices in a walk are ...
3
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0answers
1k views

Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
3
votes
1answer
177 views

One-to-one mapping vs one-to-one correspondence

Does the phrase "one-to-one mapping" mean the same thing as "one-to-one correspondence?" I know that the latter refers to a bijection. Does the former refer to an injection (i.e. it is the same as ...
3
votes
3answers
679 views

What is a rational trigonometric function? Is $\cos x$ rational?

I am reading Trigonometry by Gelfand and Saul. On p.140 they discuss rational trigonometric functions and define one as: A rational trigonometric function is a function you can get by taking the ...
3
votes
2answers
1k views

difference between “minimal” and “minimum” edge cuts.

I was going through the topic about connectivity of graphs. There it was mentioned about the terms "minimum edge cut" and "minimal edge cut". I know both are the sets of edges if removed from the ...
3
votes
3answers
108 views

What are the sets $S_n=\omega-n$ called?

What are the sets $S_n$ where $S_n:=\omega-n$ called? I explain better: if ordinals are defined in this way $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
3
votes
2answers
265 views

Zorn's Lemma $\equiv$ Axiom of Choice

I'm confused a little bit about this, I've been told many times that Zorn's lemma is equivalent to the axiom of choice. Is it an axiom or is it lemma, I mean is there a proof of Zorn's lemma or we ...
3
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2answers
5k views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
3
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1answer
82 views

Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.

I will illustrate my question in the case of the definition of vector spaces. It is custom to define a vector space in the following way: "Let $K$ be a field. Then a $K$-vector space is a set $V$ ...