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

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3
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1answer
32 views

Completing the square (and variants thereof)

When dealing with quadratics, completing the square is ubiquitous, and I can summarise my interpretation of it as the formula: $$x^2-2ax=(x-a)^2-a^2$$ Likewise, when working with circles (and, more ...
1
vote
1answer
13 views

Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f $ ...
2
votes
0answers
25 views

Is there a term for a function where equal output values must come from only one contiguous range of input values?

I'm looking for a word to describe a function where every output is guaranteed to have come from exactly one contiguous range of input values. For example, a monotonic function has this property, but ...
2
votes
2answers
31 views

What is the property of addition called when you break 97 into 100 - 3?

Sometimes it's easier to add numbers when you recognise that they're close to some round number, and then add the differences separately. $$97+198$$ $$=(100-3)+(200-2)$$ $$=(100+200)+(-3-2)$$ ...
0
votes
0answers
45 views

Is there a name for the two parts of a complex number?

A complex number is the sum of a real number and an imaginary number. Is there a collective name for the two parts comprising a complex number, such that when used, it is (pretty) clear that the ...
2
votes
1answer
23 views

Is there a name for dividing a set into pieces, some of which may be empty?

Suppose that $X$ is a set and $V_{0}$, $J$, and $V_{1}$ are pairwise disjoint subsets of $X$ whose union is $X$. If the three subsets were nonempty it would be a partition of $X$. However, I wish to ...
1
vote
1answer
18 views

What is the name for the point where a non-smooth transition occurs

In the question Smooth transition between two lines (2d) there is an example of a composite curve which has a point where it is non-smooth. In general, what is the name for that transition point?
0
votes
0answers
21 views

Term for a “Cartesian union/intersection/difference” of set families

Let $A,B$ be two families of sets. What is a term for the following families: $$C = \{a\cup b|a\in A, b\in B\}$$ $$D = \{a\cap b|a\in A, b\in B\}$$ $$E = \{a\setminus b|a\in A, b\in B\}$$ Since ...
1
vote
1answer
30 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
2
votes
1answer
28 views

$n$th root of $x$ - technical term for $n$?

As you can see in the title, I want to know how the number before a root is called. For example, if you have the cubic root of 8, I want to know how the 3 before the roof is called. Actually, I ...
0
votes
2answers
28 views

“conjugate to/with” or “conjugated to/with”, a terminology question in group theory.

This is a terminology question from a non-native English speaker. Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying : $$b=cac^{-1} $$ I could say : the element $a$ is ...
3
votes
0answers
83 views

Name for categories with a certain property on coproducts

Is there a name for categories with the following property: The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map $$\hom(Y,\bigoplus_{i \in I} ...
1
vote
2answers
71 views

What's the name for the property for which $x + x = 0 \Longleftrightarrow x = 0$?

I have a set $\mathbb{S}$ for which I have defined an operation: addition ($+ : \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{S}$). The structure $(\mathbb{S}, +)$ is a group. I have shown that if ...
0
votes
2answers
32 views

What is the connection between $l_p$ norms and “$l_p$ metrics”?

In some textbooks metric spaces you sometimes encounter these "$l_p$ metrics", $d_1, d_2, d_\infty$ (I don't think $l_p$ metric is very standard usage) For example, $d_1(x,y) := \sum\limits_i^m ...
0
votes
0answers
48 views

Is there any difference between “for any” and “for all”?

When we prove something, we use mathematical symbol ∀ to stand for "for all." Does it make any difference if we use same symbol for "for any."?
32
votes
9answers
2k views

Definition of “well defined” in mathematics

I have encountered this term "well defined" in many places in maths like well-defined set, well-defined function, well-defined group, etc. What are the contexts in which we can talk about well ...
0
votes
1answer
21 views

Understanding a terminology in a special type of group

I am trying to understand the following terminologies, and the resulting group (found in this link). In the original reference also, I didn't find the meaning of the terminology I am looking. It is ...
0
votes
1answer
25 views

Part of a sigmoid function?

I revised a sigmoid function to use in my research. The function looks like this. $$ f(x) = 0.4 \cdot \frac{1}{1 + e^{-5x}}+ 0.3 $$ where $ x \in [-1,1] $. Is there a specific name to refer to this ...
1
vote
2answers
26 views

Finding the probability space of the given experiment.

Specify the probability space completely for the following experiment: tossing a fair coin till we see the first heads. Here is what I have done so far: The sample space is simply $T^n H$ where $n$ ...
2
votes
1answer
49 views

Seeking more information regarding the “hybriation function.”

Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I ...
0
votes
0answers
20 views

Name of a number that matches place in a list?

I'm pretty sure there is a term for a number that has a value that matches its place in a list but my googling is failing me. For example in 4 2 2 4 1 0 the second 4 would have a special name.
1
vote
1answer
24 views

What does “modulo equivalence relationship” mean?

I am reading something on completion of metric spaces and it says: Let $\hat S$ be $\mathcal{C}$ modulo equivalence relationship of co-Cauchy sequences. Where $\mathcal{C}$ is the set is all ...
1
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0answers
28 views

Definition of mathematical expression

According to wikipedia: "In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical ...
0
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0answers
25 views

Are there clear, formal definitions for “terms” in subtraction operation?

I tutor children of all ages in Mathematics and I've noticed so many different words thrown around regarding binary operations, particularly with subtraction. For example, when working with a 2nd ...
0
votes
0answers
25 views

Mathematics Terminology

I was reading a paper, and the paper stated: $Cov_t (\epsilon_{a,t+1}, \epsilon_{b,t+1} \epsilon_{c,t+1}) =0$, for all $a$, $b$ and $c$. Does this mean that this also applies for cases where ...
2
votes
1answer
30 views

What is the difference between disjoint union and union?

If $S = A \cup B$, then $S$ is the collection of all points in $A$ and $B$ What about $S = A \sqcup B$?, I think disjoint union is the same as union, only $A, B$ are disjoint. So the notation is a ...
0
votes
2answers
19 views

What do we call the set of elements fixed by an involution of the second kind?

If $A$ is an algebra over a field $F$, and $\sigma:A\rightarrow A$ is an involution of the second kind, then it seems natural to talk about the set $S=\{a\in A\mid\sigma(a)=a\}$. I am not finding any ...
0
votes
1answer
34 views

What does it mean for the empty set to be connected and totally disconnected?

I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected. Recall definition of connected. A set $S$ is connected ...
0
votes
0answers
17 views

“Equidecomposable”: informal meaning

I am having trouble understanding the definition of the term "equidecomposable". Is it like two sets are split into many sets and then these many sets can be joined together to make either of the two ...
0
votes
1answer
25 views

If I subtract a number (from a sequence) from the average of all the numbers in that sequence - what do I have?

If I have a number (from a sequence) and I then subtract that number from the average of that sequence- what do I have? I would describe it as a 'deviation from the average' - but is there a better ...
1
vote
0answers
22 views

Where does the name of the hypergeometric distribution come from?

I understand what it does and how to get there, but why is it called hypergeometric? All the other distributions I know of have rather self-explanatory names like "binomial" or "exponential", or are ...
0
votes
0answers
14 views

Characteristic polynomial of a graph and structure function of a graph?

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs ...
1
vote
1answer
44 views

What is this kind of cone called?

Consider the following cone: Assume a circular base (but I don't think that's really critical to my question). Note that the line connecting the vertex of the code to the plane of the base, where ...
6
votes
3answers
371 views

What does echelon mean?

When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?
0
votes
0answers
3 views

Name for a complex but consistently wound polyline loop?

So I have an algorithn which operates on a plane region defined by a directed polyline loop. This algorithm has the unusual property of working properly for self-intersecting polylines, but only if no ...
0
votes
0answers
25 views

Fourier Polynomials: standardly used term?

When teaching Fourier series to students, I realized that one of my references (only one or two I know that does this) calls the $n$-th partial sum of the Fourier series of an $L^2$ function $f$, the ...
1
vote
4answers
93 views

Is there a name for numbers between 0 and 1?

In the world of competitive esports, players often discuss kill/death ratios, where higher is better. My friends sometimes call a poor ratio, like 1 kill to 4 deaths, as 'negative', but that's not ...
5
votes
2answers
144 views

How to picture a first countable space?

I find myself forgetting what it means for a space to be first countable on a frequent basis. This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating ...
0
votes
1answer
26 views

Difference finitely many and arbitrarily many

Is there a difference between "finitely many" and "arbitrarily many"? Some notes I am reading are making a point of distinguishing between the two and I thought they meant the same thing.
0
votes
0answers
60 views

Is there an open / established ontology for mathematics?

I am currently thinking about how to bundle some of the efforts of students to get / create good educational material. One idea of this little project (wiki-ed - still in the very early phase) is to ...
1
vote
1answer
26 views

“The following are equivalent” for only two statements

I often see "The following are equivalent" in theorems that proceed to only have two statements, (a) and (b) be equivalent to each other. When is this appropriate to do rather than say "if and only ...
34
votes
6answers
2k views

Why is a linear transformation called linear? [duplicate]

$T(av_1 + bv_2) = aT(v_1) + bT(v_2)$ Why is this called linear? $f(x) =ax + b$, the simplest linear equation does not satisfy $f(x_1 + x_2) = f(x_1) + f(x_2)$. Thank you.
0
votes
1answer
50 views

Is it appropriate to refer to the definite integral of $f$ from $a$ to $b$ even though $f$ is just defined on $(a,b)$?

Suppose we have a function $f\colon (a,b)\to\mathbb R$ that is both continuous and bounded. It is a theorem of real analysis that any extension of $f$ to the closed interval $[a,b]$ is integrable and ...
0
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0answers
26 views

Is multiplicative inverse defined for ideal? Eg. $x^3 y\in \langle x^3 y^2\rangle$?

Definition. A subset $I\subset k[x_1,\ldots,x_n]$ is an ideal if i. $0\in I.$ ii. If $f,g\in I$, then $f+g\in I$. iii. If $f\in I$ and $h\in k[x_1,\ldots,x_n]$, then $hf\in I$. ...
7
votes
0answers
111 views

What can the reals of an inner model be?

This is probably a silly question. Call a set of reals $X$ a constructibility ideal (in analogy with a Turing ideal) if $X$ is closed under effective join $r\oplus s: n\mapsto 2^{r(n)}3^{s(n)}$ and ...
3
votes
1answer
44 views

Is there a general term for $A\oplus B = \{a \oplus b | a\in A, b\in B\} $?

Is there a general term that specifies that if an operator $\oplus$ is applied to two sets, it's actually applied to all possible pairs of elements of the two sets? Or is that always the case and ...
1
vote
0answers
18 views

Correct term for higher-dimensional analogue of a cyclic polygon?

A $d$-dimensional polytope whose vertices lie on the boundary of some $d$-dimensional sphere is called a cyclic polygon when $d=2$. Is there a succinct name for this in higher dimensions?
1
vote
1answer
28 views

Explain Multidegree of a polynomial

Definition (The book Ideals, Varieties and Algorithms, Cox et al, pages 59-60 on 2008 edition): Let $f=\sum_a a_a x^a$ be a nonzero polynomial in $k[x_1,\ldots, x_n]$ and let > be a monomial ...
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0answers
28 views

Name for “3D quadrilateral” shape?

I am interested to know the name of the following solid construction - kind of a deformed cube... but I don't know the name, or even a general name: Left and right faces parallel with the $yz$ axes ...
2
votes
1answer
37 views

What is the precise relationship between vector space and vecctor field?

I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them. On wikipedia A vector field is: Given a subset $S$ in $R^n$, a ...