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

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2answers
91 views

Why do some sources call calculus, “the calculus”?

No need to cite specific sources since I think it's a fairly common thing to see. What's up with that? Thank you Edit: I've seen it in several places. Here's where I'm currently looking at it at: ...
2
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0answers
99 views

Alternative to sin and cos

I was reading something on the Internet the other day, and I swear I came across a reference to an alternative sine function [which I now cannot find any mention of]. The usual sine function starts ...
5
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1answer
116 views

Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
0
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2answers
42 views

What is the difference of an n-tuple and a permutation of n elements

My understanding of n-tuple and a permutation of n elements is, that both are ordered sequences of n elements. Are there differences in the objects correlating to these two terms ? I guess it ...
0
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1answer
51 views

Writing a chain of implications in English

How to write a theorem of the form $A\Rightarrow B\Rightarrow C\Rightarrow D$ where every $A$, $B$, $C$, $D$ are formulated with words (English) rather than with formulas? One idea: The next item of ...
2
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2answers
56 views

What is meant by a “structure map”?

The title is the question. Somehow I should know the answer, but I am by no means sure what is meant exactly by it. Perhaps it doesn't have a definite meaning and only in context, could someone ...
4
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0answers
48 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
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0answers
25 views

What is the sigmoid *squashing* function?

I've just read the following The basic unit ("neuron" i) performs the following computation to update its state $y_i$: it computes a weighted sum $v_i$ of its inputs $x:j$ which is passed ...
5
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1answer
68 views

Name for a property in a brutally elementary presentation of a monad

For evil reasons of my own, I'm trying to give a presentation of a monad in primitive terms, assuming only the notion of a category. More honestly, I looked at this post and got intrigued by the ...
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0answers
17 views

Name for “relative difference”

If we want to express that two numbers $x, y$ are not so far away from each other absolutely we use the absolute value function $|\cdot|$ with $0 < \epsilon \ll 1$: $$|x-y| < \epsilon$$ ...
0
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2answers
27 views

How do we emphasize that $\displaystyle x\mapsto\frac{1}{f(x)-y}$ “makes sense” if we know $y\notin\text{im }f$?

Please take a look at the following function $$x\mapsto\frac{1}{f(x)-y}$$ where $f$ is "some other function". Suppose we know $y\notin\text{im }f$, i.e. the expression in the denominator "makes ...
2
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2answers
26 views

Some basic terms from finite group theory, normalising and centralising

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, ...
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1answer
16 views

Difference between $C_0^{\infty}(U)$ with support in $A$ and $C_0^{\infty}(A)$

Let $A \subseteq U$ be open sets of $\mathbb R^n$. Is it true that $$ \lbrace f \in C_0^{\infty}(A) \rbrace = \lbrace f \in C_0^{\infty}(U) : \text{support of } f \subseteq A \rbrace \quad ? $$ I ...
1
vote
1answer
26 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
1
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2answers
43 views

Fundamental confusion on set theory and permutations

I am confused on the following: A set does not have any order. Now I read that a permutation is a bijection of a set. But doesn't this imply an order? I mean a bijection is a one-to-one function from ...
6
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1answer
55 views

Closed and Open Set - History of Terms

I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of ...
3
votes
1answer
49 views

Sets that are equal to closure of its interior

Is there a standard name for set $M$ for which $M = \overline{M^0}$? $M^0$ is interior and $\overline{M}$ is closure. Often I work with well behaved sets and functions. I have in mind continuous ...
0
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1answer
16 views

Is there any special name for an algebraic structure (set, equivalence relation)?

I've seen the term "real multiset" but it doesn't seem to be very appropriate so i wonder whether there are any others. My second question is about multisets. In most sources i've seen one of the ...
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0answers
73 views

Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...
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0answers
24 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
4
votes
2answers
407 views

Use of the word “solve”?

This is not a mathematical question, but just a matter of terminology. I don't understand why so many people (especially on MSE) want to solve integrals. It makes sense for me (linguistically ...
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1answer
38 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
0
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2answers
29 views

What's this equation called (one more each iteration, find total for given iteration)?

Say you have +1 on first iteration, +2 on second, and so on until N, and you want to know the total. That's easily calculate using (N * (N + 1) ) / 2. What's that equation or technique called?
0
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1answer
51 views

Has this topology a name?

let $(X,\tau )$ be the topological space where $\tau =\{\emptyset, X, \{x\}, X-\{x\}\}$ , $ x \in X$. Does this topology have a name? Thanks in advance!
2
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0answers
46 views

“Algebraic indistinguishability” [duplicate]

When people talk about Galois theory they often say that the basic idea behind it is that certain numbers are "algebraicaly indistinguishable". I never really understood what this means in a way that ...
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0answers
46 views

Is there a term for this property of magmas?

There exists an element of the magma c such that for all x: $ x*x=c $ The consequence of this is that the elements on the diagonal of the Cayley table are all the same, e.a: $ * = \begin{bmatrix} 1 ...
1
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0answers
48 views

Why are frames called “frames”?

Definition: A frame $F$ is a suplattice such that for any $x_{i}, y\in F$ (for $i\in I$, $I$ a set), we have $$y\wedge\left(\bigvee_{i\in I}x_i\right)=\bigvee_{i\in I}(y\wedge x_i).$$ Why are ...
3
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1answer
40 views

Is there a term for an algebraic structure with two binary operators that are closed under a set?

For example, let's say we're using the operators +, and *, and the set {0,1,2} The Cayley tables look like this: ...
1
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0answers
19 views

Is there a name for the corresponding notion of inductive subset in the context of well-ordered sets?

This is a question of terminology. I can't avoid being a little verbose before getting to it. The principle of mathematical induction states that, for any subset $e$ of $\omega$ (the set of natural ...
5
votes
1answer
341 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
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1answer
33 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
36 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
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1answer
101 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, ...
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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
30 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
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0answers
110 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
24 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
70 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
77 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
45 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 ...
2
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0answers
21 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
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0answers
6 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
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3answers
583 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
99 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
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0answers
70 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
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1answer
59 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 ...