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

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2
votes
2answers
125 views

Is the standard scalar product in a coordinate space basis independent?

Would you say that the standard scalar product in $K^n$, $\left< x,y \right>=\sum_i x_i y_i$, is basis-independent or not ? I would argue that it is, because we don't use the components of the ...
2
votes
1answer
67 views

Question about the wording of a topology problem.

I was asked to show that the topology $\mathcal{T}_{X\times Y}$ is the smallest topology for which the functions $$f_X:X\times Y \rightarrow Y , f_X((x,y))=x $$ and $f_y$ are continuous (where $f_Y$ ...
0
votes
1answer
59 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
1
vote
0answers
30 views

Are maps and operators between two sets the same?

I have been reading on up on the definition of maps and operators, specifically reacting to sets (rather then the more restricted vector spaces) and their definitions seem to be identical. So are all ...
37
votes
6answers
3k views

What do mathematicians mean by “equipped”

I am a mathematical illiterate so I do not know what people mean when they say equipped. For example, I say that Hilbert space is a vector space equipped with a inner product. What does that ...
0
votes
2answers
39 views

Is there a non-ambiguous name for the “square of a function”?

Given a function $f$, I want to refer to $f \circ f$ other than by a formula. Is there any name for this other than square of $f$, which has the problem of being ambiguous? In analogy to the ...
3
votes
3answers
120 views

What is the element produced under a generic binary operation called?

For instance, for addition this is called the sum: $\underbrace{x+y}_{\text{summands}} = \underbrace{z}_{\text{sum}}$ But what is this called for a unspecified operation? $\underbrace{x\circ ...
0
votes
1answer
41 views

Small question: Name for the x of function f such that f(x)=x?

Background When doing maths and chemistry problems, I often came across things like $$x-\frac{x}{2}=\frac{x}{2}$$ It might seems trivial, but I found that it is often the presence of expressions like ...
2
votes
1answer
59 views

Why stronger norm defines weak local minimizer? [closed]

Why the stronger norm defines weak local minimizer, while the weaker norm defines strong local minimizer? For example, when minimizing a functional on $C^1[a,b]$, one can also consider the weaker ...
2
votes
0answers
21 views

Is this a bound variable?

If I write $\left \{\begin{array}{llll} & y = z \\ & z = x + 2 \end{array} \right.$ could I make the argument that $z$ is a "bound" variable. I've seen it referred to as a ...
2
votes
1answer
46 views

What exactly constitutes a 'term'?

From what I understand when I looked up the definition on wikipedia, a term is a monomial with a coefficient. However, I was taught in high school that a term could also be an expression depending on ...
1
vote
0answers
14 views

Is there a name/notation for coordinate-wise identical function?

Let's define $g: \mathbb{U}^n \rightarrow \mathbb{V}^n$ where $\mathbb{U}$ and $\mathbb{V}$ are arbitrary sets as $$g(u) = \left[f(u_1), f(u_2), \ldots, f(u_n) \right]^T$$ for some $f: \mathbb{U} ...
0
votes
1answer
20 views

Terminology for vectors in ''positive angle'' position

I would like to know whether there is a standard terminology for the following situation: Let $H$ be a complex Hilbert space and $\xi, \eta \in H$ are two vectors such that $(\xi, \eta)_H \ge 0$. Do ...
2
votes
2answers
61 views

Is there a name for a point on the circumference of a circle?

Is there an eloquent name for a point located on the circumference of a circle?
63
votes
38answers
8k views

Unusual mathematical terms [closed]

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
1
vote
2answers
45 views

Looking for the name of polynomials obtained as integrals over a simplex

I'm looking for the name of the following polynomials: $\mathrm{p}_1 = 1$ $\mathrm{p}_2 = x - \frac{1}{2}$ $\mathrm{p}_3 = \frac{1}{2} x^{2} - \frac{1}{2}x +\frac{1}{6}$ $\mathrm{p}_4 = \frac{1}{6} ...
1
vote
0answers
34 views

Name for a nowhere constant function?

Is there a pithy name for a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that there is no non-degenerate interval $I \subseteq \mathbb{R}^n$ such that $f$ is constant on $I$ (by '$f$ is ...
0
votes
0answers
23 views

Ask for the name of a condition on commutator of two operators

Let $T, S$ be two bounded linear operators on a Hilbert space. I wonder whether there is a standard way referring the following condition: $$ \text{The commutator $[T, S]$ is in the Hilbert-Schmidt ...
7
votes
4answers
251 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
1
vote
4answers
67 views

Is $S$ a monoid, or is $(S,*)$ a monoid?

If I have a set $S$ with operation $*$ as a monoid. Would I say I have a monoid $S$ with the binary operation $*$ or would I say I have a monoid $(S,*)$ where the binary operation $*$ does ...
1
vote
1answer
38 views

Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
0
votes
2answers
47 views

How should one rank a combination of positive and negative numbers from smallest to largest?

I thought size was the distance from zero, in which case ordering {-1, 2, -3} would be {-1, 2, -3}, but I saw it ordered as {-3, -1, 2}. Which method is correct?
1
vote
1answer
45 views

General name for interpolation and extrapolation

I would like to know if there is a technical term to cover both interpolation and extrapolation. The reason why I am asking is that I am writing a computer program to do interpolation and ...
0
votes
1answer
27 views

Area of set-difference of special sets

In a topological space, call a set $X$ special if it is equal to the closure of its own interior (is there a standard term for this?): $$X = \text{Cl}[\text{Int}[X]]$$ Let $X$ and $Y$ be two special ...
1
vote
1answer
59 views

Notation about cardinals

Does a cardinal $\mathcal{k}$ such that $2^\mathcal{k}=k^+$ have any special name? I never encountered any name for this property, but I think it is possible they have one.
1
vote
1answer
41 views

Meaning of a probability distribution being dominated by a measure

The following comes from Ghosh & Ramamoorthi (2003) Bayesian Nonparametrics. In terms of notations, $\Theta$ is a parameter space with Borel $\sigma$-algebra $\mathcal B(\Theta)$. For ...
2
votes
1answer
45 views

For which $n \in \mathbb{N}$ is it the case that every element of $\mathbb{Z}/n\mathbb{Z}$ is strongly associate to an idempotent?

Definition. Call two elements of a commutative ring associates iff each divides the other. Call them strong associates if there exists a unit that can be multiplied by the first to yield the second. ...
1
vote
1answer
30 views

Standard name for ideals generated by a subset of indeterminates?

I have been working on a problem in the polynomial ring $k[x_1,\ldots,x_n]$, where I've been dealing with ideals generated by subsets of the indeterminates, i.e., ideals of the form $$\langle x_i\mid ...
2
votes
1answer
52 views

What is an open sphere?

What is an open sphere? I recently came across this phrase in these notes on complex analysis (pg. 12, Lemma 2.1.3). I know what an open ball is, but this phrase is confusing. Any help would be ...
1
vote
1answer
74 views

Is there a name for graphs that only contain cliques?

I'm wondering if there is a name for graphs such that if there is an edge between vertices A and B and a second edge between vertices B and C then there must be an edge between vertices A and C. My ...
1
vote
1answer
60 views

Is there a name for functions that can not be described by sets?

Functions are usually (though not always) defined as a relation, which is a set of ordered pairs. In this sense, functions are merely sets. However, some ''functions" can only be described by proper ...
1
vote
1answer
52 views

What does being diffeomorphic mean in the context of configuration spaces?

A sphere space can serve as a "model space" for any configuration space that is diffeomorphic to the sphere space. This is a quote from my text book (Principles of Robot Motion: Theory, ...
2
votes
0answers
56 views

Name for classes of algebras closed under products and quotients

A class of algebras closed under products, quotetiens and subalgebras is a variety. Is there a name for a class of algebras closed under products and quotients? Could you refer me to any theorems ...
1
vote
1answer
105 views

The word “onto” - adjective?

According to Oxford English dictionary, the word "onto" is preposition only but I see it used as an adjective in mathematical writings. I think it is grammatically correct to say that "$f$ maps $X$ ...
3
votes
1answer
57 views

Is there a standard notation for building sets up form a given one?

In ZFC each set $S$ has a well-founded membership tree building $S$ up from the empty set $\emptyset$. You could attach the membership tree for any given set $A$ on each of the bottom nodes for the ...
0
votes
1answer
133 views

Where is the border between functional analysis and real analysis?

I always thought that real analysis deals with analysis on the real line, eventually on the Euclidean space $\mathbb R^n$. But why does someone have to label a course as real analysis when it is ...
1
vote
0answers
44 views

Cohomology calculation or computation?

I have a terminology question: Does one compute the cohomology of a group, or does one calculate it? Is it more common to speak of cohomology calculation or cohomology computation? Thanks for your ...
0
votes
0answers
27 views

Space of functions in the upper half-plane

Let $f(\tau)$ be a (say, holomorphic) function in the upper half-plane. Consider $$ A={\rm Span}_{\mathbb{C}} \{ f(\gamma \cdot \tau) : \gamma \in SL_2 (\mathbb{Z}) \}. $$ Is there a standard name ...
6
votes
1answer
124 views

Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

In category theory, I have seen "weakly initial object" used as follows: $X$ is weakly initial iff for all objects $Y,$ there is at least one arrow $X \rightarrow Y$. Of course, another way of ...
2
votes
1answer
53 views

What is the terminology for “lemma of lemma”

Let's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows: Lemma 1 Proof Lemma 2 Proof Lemma 3 Proof ...
3
votes
1answer
198 views

Do you decline a multiplier in reading a mathematical formula in Russian?

How do you read "Порядок определителя равен $2n$"? Is it "двум эн" or is it "два эн"? And in a sum, do you read $c = a_5 + a_6$ as "це равно а пятому плюс а шестому"? Or does the plus sign interfere ...
2
votes
3answers
70 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
2
votes
1answer
54 views

Notation for the number of times one element divides another.

Let $R$ denote a commutative ring with unity. Consider elements $a,b \in R$. Is there an accepted notation (like $a \| b$ or some such) for the number of times that $a$ divides $b$? Explicitly, we can ...
4
votes
0answers
103 views

Reading mathematical formulas in Russian & German

The book Russian for Mathematicians by Glazunova has a very useful section with examples of how formulas are read in Russian. (Most mathematical dictionaries don't seem to have this, as I suppose they ...
0
votes
3answers
80 views

Elementary “binomial theorem” in English

In German schools, the identitiy $(a+b)^2 = a^2 + 2ab + b^2$ is called the first binomial formula (literally translated). However, it seems to me this English term only refers to the more general ...
6
votes
1answer
183 views

Name for Theorem 3.27 from baby Rudin?

Rudin rarely gives names to the theorems in this book. Theorem 3.27 states if $\{a_n\}$ is a monotonically decreasing sequence of positive reals, then $$\sum_{n=1}^\infty a_n\,\text{ ...
-1
votes
1answer
125 views

What is the difference between helix and spiral?

The words spiral and helix are both used for curves that "wind around". For example, both searches "DNA spiral" and "DNA helix" (with quotation marks) result in many thousands of Google hits. Is ...
1
vote
0answers
33 views

Mathematical principles and therems - difference?

For me, intuitively, a mathematical principle is simply an influential theorem. Still, I am not clear on how and who decides (or decided) if a theorem/statement is a mathematical principle. Can you ...
0
votes
1answer
23 views

Confusion about the definition of upper bounds of a set

I am confused about upper bounds of a set. Consider a set: $A = ${$1, 2, 3, 4, 5, 6, 7$} How many upperbounds are there? Does the upperbound need to be in the set? Also about supremum. What is ...
3
votes
1answer
54 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...