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

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0
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2answers
24 views

Is there a name for generalized ellipsoids?

In two dimensions, we have the following series of generalizations: circle $\rightarrow$ ellipse $\rightarrow$ smooth, convex, closed curve $\rightarrow$ smooth, simple, closed curve And in three ...
0
votes
1answer
14 views

In Graph to tree: name of operation where edges removed and vertex/edge additions?

The graph has tree paths IN-1-OUT, IN-2-OUT and IN-3&4-OUT between IN and OUT in the left. I want to make each path to a branch like the right. What is the name of this operation or the name ...
0
votes
0answers
9 views

Meaning of the term “dense relative to..”

In understand the meaning of the mathematical term dense. In the lecturers solutions I came across a sentence which said "...dense relative to .." Does this just mean to say "..it is dense in.."?
1
vote
2answers
99 views

Why isn't area the same as surface area? [closed]

Doesn't a 2D shape have a surface? In that case, is it incorrect to call the area of it surface area?
34
votes
11answers
3k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall ...
5
votes
2answers
50 views

In layman's terms what is the difference between a model and a distribution?

The answers (definitions) defined on Wikipedia are arguably a bit cryptic to those unfamiliar with higher mathematics/statistics. I am a high school student very interested in this field as a hobby ...
1
vote
1answer
8 views

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$, where $dx$ is some tiny increment of $x$? What we know about $V$: $V(z) = U(z) - ...
2
votes
2answers
80 views

About the Words “recursion” and “recursive”

According to Wikipedia, Recursion is the process of repeating items in a self-similar way. On the other hand, the word "recursive" is an adjective and is often used as a synonym of "computable" when ...
3
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0answers
34 views

Is there a name for those concrete categories in which every subset / quotient set inherits the structure of an object in at most one way?

The following situation seems to occur a lot in abstract algebra: We have a category $\mathbf{C}$, concrete over $\mathbf{Set}$, that satisfies: For every object $Y$ of $\mathbf{C}$ and every set ...
2
votes
1answer
30 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
0
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0answers
16 views

Topology; difference between open subsets of $X$ containing $x$ and open neightborhood od $x$?

I see my lecture notes and some texts alternate between the two. What is the difference in saying that "an open subset of $X$ containing $x \in X$" and an "open neighborhood of $x \in X$"?
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0answers
8 views

What is cut space of directed graph (digraph)?

A cut is partition of vertices into two disjoint subsets. Digraph is a directed graph. Cut space is defined for an undirected graph as by Wikipedia where the definition for an undirected graph, ...
0
votes
0answers
27 views

Is it acceptable to refer to “the $\ell_2$-norm ball of radius $r$”?

Assume $r > 0$. Is it standard to use the expression "the $\ell_2$-norm ball of radius $r$" to refer to the set \begin{equation} B = \{ x \in \mathbb R^n \mid \| x \|_2 \leq r \}. \end{equation} ...
0
votes
0answers
5 views

Is there a name for differential operators with certain homogeneity

Is there a name for the ODEs of the following form ? $$\sum_{m_k+n_k= N, }a_{m_k,n_k}u^{m_k}u^{(n_k)}=0,$$ where $u^{m_k}$ denotes the $m_k-$th power of $u$ and $u^{(n_k)}$ denotes the derivative of ...
3
votes
2answers
117 views

What are “set-theoretic maps”? [closed]

Can someone explain to me what is the meaning of “set-theoretic maps”? I've encountered this term in real analysis in $n$ variables. Specifically, I encountered it in the following statement: Let ...
1
vote
1answer
12 views

What are those weighed graphs called?

Let $G = (V, E)$ be a directed graph, and define the weight function $f : V \sqcup E \to \mathbb{R}^+$ as follows: sum of weights of vertices is 1, if a vertex has edges coming out of it, their ...
4
votes
2answers
60 views

What's the order of a semigroup?

For a group the order, of an element is the smallest positive integer m such that a^m = e. But what's the order of an element of a semigroup? Or there isn't anything like that?
0
votes
1answer
68 views

What is an order of an element of a partition"?

I'm reading a paper, in which the set of all 3^3 mappings from {0,1,2} to itself (for instance {001,020,110,121,122}, {002,010,112,011}, {0,1,2}, ...) is partitioned, after which is written two ...
3
votes
1answer
99 views

Why is the nuclear norm called so?

A simple question. Why is the sum of the singular values of a matrix called its nuclear norm? What is the origin of, and motivation for, this term? Apparently the term nucleus is sometimes used to ...
2
votes
1answer
46 views

What is meaning of dyadic compactness?

What is meaning of dyadic compact space? Is it as the same as Cantor cube $D^\kappa$?
2
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0answers
102 views
+150

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
3answers
157 views

Where could I learn basic math terminology?

I am an english learner and I would like to learn the etymology of Mathematics. I would like to know the most common phrases in Algebra, and Geometry as well. I want to know at a level of UK's A+. ...
2
votes
1answer
2k views

What is the definition of a geometric progression?

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement? So, is $\{0, 0, ...
2
votes
2answers
601 views

What is the general term for a shape that is topologically equivalent to a sphere?

You have a chunk of ideal rubber and can stretch and shape it any way you want, but you are not allowed to puncture it. The chunk of rubber will always be a ____. shapes described by the term: ball, ...
18
votes
1answer
777 views

Is there such a thing as a mathematical thesaurus?

I want this for two reasons: When writing proofs, I am constantly in need of synonyms of basic words like thus, there exists, for all, such as, contains, etc. A lot of mathematical concepts have ...
0
votes
0answers
37 views

How to call a “non-strict” monoidal category?

A monoidal category is a category $\mathsf{C}$ equipped with a bifunctor $\otimes : \mathsf{C} \times \mathsf{C} \to \mathsf{C}$, a unit object, an associator, and right and left unitors satisfying a ...
27
votes
10answers
11k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
1
vote
2answers
51 views

What is a filled rectangle called, if anything?

In geometry, the set of points within a circle is called a disk (open disk if it excludes the boundary, closed disk if it includes it). Is there a similar notion for squares or rectangles? "A filled ...
1
vote
1answer
34 views

What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?

Let $\mathscr{F}$ be the set of all plane, closed Euclidean figures having positive perimeter, and let $\sim$ be the similarity relation on $\mathscr{F}$. Then, for any equivalence class ...
1
vote
2answers
390 views

Are pairwise mutually exclusive events the same as mutually exclusive events?

Larson (1982) defining the probability axioms talks about "mutually exclusive" events, while Poirier (1995) about "$A_1, A_2, \ldots$ as a sequence of pairwise mutually exclusive events events in the ...
8
votes
1answer
77 views

A function that crosses each horizontal line only finitely many times

Consider a real function $f(x)$ (not necessarily continuous) defined on a finite interval. Given a constant $C$, divide the interval to sub-intervals such that, in every sub-interval, either ...
3
votes
3answers
289 views

Why are “irrational numbers” not named “nonrational numbers”?

Possible that I'm misunderstanding the concept of irrational numbers, but seems like the term nonrational would be much more clear. Why is "irrational" more clear than "nonrational"? UPDATE: Just to ...
0
votes
3answers
33 views

Name for region of plane bounded by two rays?

Is there a name for e.g. the locus $$\pi/6 \leq \arg z \leq \pi/3$$ on an Argand diagram? (Perhaps something analogous to a half-plane?)
0
votes
0answers
19 views

Limit Terminology

From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the ...
10
votes
4answers
2k views

What is the name of the vertical bar in $(x^2+1)\vert_{x = 4}$ or $\left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$?

I've always wanted to know what the name of the vertical bar in these examples was: $f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$) $\int_0^4 (x^2+1) \,dx = ...
2
votes
1answer
95 views

If a tesseract is to a cube what a cube is to a square, what is to a sphere as a sphere is to a circle? What about rectangle, ring, triangle?

I'm trying to come up with sensible names for a programming library I'm putting together. One minor part of this library is the generation of shapes of varying dimensions. Basically I'm just trying ...
2
votes
1answer
55 views

Is there an adjective for rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Question. Is there an adjective for rings whose every non-zero prime ideal is maximal? Remarks: Every PID has this property; more generally, every ...
7
votes
2answers
404 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
0
votes
0answers
15 views

Term for “Remainder in the Whole”

If I have a proper fraction I want to know what the name is for the amount remaining in the whole. So given $\frac1 3$ I want the name of the term $\frac 2 3$.
2
votes
0answers
32 views

3D shaped matrices - how would multiplication work? [duplicate]

I've been thinking about vectors and matrices lately, and I got a little curious. Why don't we have cubic shaped matrices? After all, vectors are 1-dimensional matrices, so it follows that there ought ...
2
votes
1answer
30 views

Terminology for $[0,\infty)^n$

It dawned on me a couple of weeks ago that I had no idea what terminology was used for the sets $[0,\infty)^n\subseteq \mathbb{R}^n$ in general. In one dimension, it's just the half line; in two ...
0
votes
0answers
20 views

Number transformation

Does somebody know what would be the proper name of this number transformation: ...
3
votes
1answer
925 views

What does height of an algebraic equation mean?

I want to solve this question but I don't understand what "height" of an algebraic equation is: Find the number of solutions of the set of all algebraic equations of height 2.
0
votes
1answer
37 views

Word Form of an Expression [closed]

What is the word form of the expression? $$\sum \frac{1}{n^s}$$ That is exactly the way the expression appears in a paper which I am trying to read. It is $$\sum_{n=1}^\infty \frac{1}{n^s}$$
0
votes
0answers
7 views

Exterior algebra subspace of all grade-n wedge products of a vector

Let $V$ be a finite-dimensional vector space, and let $\Lambda(V)$ be its exterior algebra. Then if $S_k = \text{span}(k_1,k_2,...,k_n)$ and $\hat k = k_1 \wedge k_2 \wedge ... \wedge k_n$, there is ...
1
vote
0answers
15 views

Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
0
votes
0answers
11 views

Number of letters moved by a product of permutations

Let p and q be permutations in the symmetric group on n letters. p and q need not have the same cycle structure. Now compute q * inv(p) -- for inv(p) the inverse of p -- and count the number of ...
1
vote
0answers
20 views

(Partial/total/well-) order vs ordering

Order or ordering – what is the difference? Is either correct? Is one British and one American English? Not exactly a maths question but probably still the best place to ask.
0
votes
1answer
159 views

Name of the class of graphs obtained by deleting $\mathcal{Q}_d$ from $\mathcal{Q}_n$

Let $\mathcal{Q}_n$ denote the $n$-cube graph. I would like to know if there is a name for the class of graphs obtained by deleting a ${\bf single}$ arbitrary copy of $\mathcal{Q}_d$ from ...
2
votes
2answers
62 views

How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...