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Questions tagged [terminology]

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

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Name for a coordinate system defined on a [unit] sphere

One of the commonly used coordinate systems in 3D is a spherical one. Which would be defined by: a radius $r$ polar and azimuthal angles $\theta$ and $\phi$, where "mathematicians and physicists ...
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2answers
52 views

What is the difference between a definition and an equivalence class?

In what way is 'the definition of $x$ is $y$' ($x:=y$) not the same as '$x$ is equivalent to $y$' ($x=y$)? I can find no justification for making the distinction aside from 'it feels right'. It seems ...
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“Dimension of a vector” vs. “Directions of a vector”

A vector like $(7,5,4)$ is said to have the dimension of 3. But I have also heard that the "5" is the "second dimension of the vector". In the later case one would say the vector has three dimensions (...
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3answers
28 views

X is to vector like matrix is to linear operator?

In linear algebra texts there is usually a clear distinction between linear operators and matrices. A linear operator is a map between two spaces that fulfills a set of conditions. A matrix is a 2D ...
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32 views

What do you call a “factor” of a union

If we have numbers $a,b$, they are called summands of $a+b$ or factors of $a\cdot b$. Given two modules $M,N$, they are direct summands of $M\oplus N$. Likewise, given two sets $A,B$, they can be ...
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50 views

Subgroup of finitely-generated subgroup

Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm ...
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What was the motivation behind the fraction ideal's name?

The fraction ideal is the ideal $$(I:J)=\{x\in R:xJ\subseteq I\}$$where $I, J$ are ideals in $R$. My question is, why is it called a fraction ideal? (Sometimes it is called the "Ideal quotient of $I$ ...
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29 views

Map Vs. Mapping?

I am so confused in the usage of the words: Map and Mapping. For example, consider the mapping: $T:W^{2,p}(\Omega)\rightarrow L^{p}(\Omega)$ defined by $T(u):=\Delta u.$ Can we write: consider the ...
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What's the English terminology for a “section” set? [duplicate]

I'm translating the terminology from my native language into English. The closest placeholder I have to the word is a "section" set, but I couldn't find anything like that online. We say that $A \...
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Terminology: naming different data-set computations / operations types

intro: I'm translating a manual for some 3D graphic utility which modifies set of vertices for a provided set of meshes. So basically it works with sets of sets of vectors. And the thing is - the ...
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Larger percentages give a sample that is a superset of a smaller percent

I want to give a certain percentage of users access to a feature, and then grow that percentage over time. Using some deterministic algorithm, the same set of users will always be selected for the ...
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72 views

What is the difference between, a “square” and a “perfect-square”, number?

Is, "36", a perfect square? I know that, "4" is a perfect square. Similarly, "1","9","25", are "perfect-square"s.
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What does it mean for a change in one value to “proportionally exceed” the change in the other value?

For example, suppose we have two numbers $x,y$, and their average $r=\frac{x+y}/2$. Suppose we change $x$ by $\Delta x$. This will also change $r$. What does it mean for "the change in $x$ to ...
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36 views

structure-preserving: a history?

I am familiar with the occurrence of structure-preserving morphisms in Category Theory, but I would love to know more about the history of the concept, where it started and so on? I suspect it might ...
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Is there a name for this notion of “radius of compactness” in a metric space?

I was proving some result about Riemannian manifolds that led me to introduce the following definition: Let $M$ be a metric space and $x \in M$. Define the "radius of compactness" $RC(x)$ to be the ...
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1answer
37 views

Is the term *monotone* used fairly consistently to mean non-decreasing or non-increasing but not strictly?

In BBFSK, (~1960, Germany) at least in the section I am currently reading, the authors use the term monotone increasing (decreasing) to mean what I often see called strictly increasing (decreasing). ...
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24 views

Is there a general term for a mapping of a set to itself?

I often wish to describe a mapping of a set to itself as injective, or bijective, etc. The sentence $f$ is an injective mapping of the set $\mathcal{A}$ into itself seems unnecessarily wordy. I ...
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What notation is available for denoting points on a path?

I have a closed-path motion on the plane, it looks like A-B-C-D-A on the figure. Each segment is a straight line. The point A ...
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2answers
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Big O Vocabulary/Terminology: for all $x \geq x_0$

The Big O Wikipedia page says: $$ \lvert f(x) \rvert \leq Mg(x) \textrm{ for all } x \ge x_0 $$ In Big O, and especially in general, what is the difference between x and $x_0$? In my naive, computer-...
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1answer
63 views

Is there a mathematical sign for “or”?

Is there a mathematical sign for "or"? When I have to explain something in numbers, and I have two ways to explain it and I need to separate them by a sign.
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48 views

How you call the relationship between variables $X$ and $Y$ if $X=1-Y$

How you call the relationship between two variables where one is equal to 1 minus the other. For example, if my variables are $X$ and $Y$ and I have that $X=1-Y$ I what to say the $X$ is the ________ ...
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62 views

Are there any special names given to the structures $\left<\mathbb{N}_0,+,\times\right>$ and $\left<\mathbb{N},+,\times\right>$?

Following my grade school education, I will call the set $\mathbb{N}_0\equiv\left\{0,1,2,\dots\right\}$ the whole numbers, and the set $\mathbb{N}\equiv\left\{1,2,3,\dots\right\}$ the natural numbers. ...
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32 views

Terminology: arithmetic vs. expressible vs. represented

A function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is arithmetic iff its graph is arithmetic, i.e., there is a formula $\psi(\vec{x},y)$ in the language of Peano arithmetic such that for all $\vec{a}$ ...
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25 views

What do we call a “nearly” order-isomorphism?

Suppose $f:X\to Y$ preserves $x_1\geq x_2\implies y_1\geq y_2$ but does not necessarily preserve $x_1> x_2\implies y_1> y_2$. In other words, an $f$ satisfying this relation might, upon ...
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242 views

Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?

It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a ...
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Terminology - discrete probability distributions

What is the correct word for a discrete probability distribution that is of the form $(1,0,0...)$. In physics the matrix versions of these are called pure states (vs mixed states). What is the ...
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28 views

Is there an official terminology for functors that are adjoint up to a given functor?

Assume that we are given categories $\mathcal{A}, \mathcal{B}, \mathcal{C}$ and functors $U:\mathcal{A}\to\mathcal{B}$, $L:\mathcal{B}\to\mathcal{C}, R:\mathcal{C}\to \mathcal{B}$ such that $$\mathsf{...
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1answer
35 views

Notation for a Cartesian product or tuple except one element?

If $X$ is a set containing element $0$, and we want the set that contains all elements in $X$ except $0$, we can write $X-\{0\}$. If $X=\prod_{i=1}^n X_i$ and we want to denote the cartesian product $...
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35 views

G is [property] if and only if N and G/N are both [property]

Is there a name for a property that is inherited by subgroups and quotients of a group? How about a property that if true for both a normal subgroup N and the quotient G/N is true for G? Or a name for ...
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What's the name of a function with a variable in the base and exponent?

For example, $k^x$, where $k$ is a constant, is an exponential function. It is differentiated quite easily. $x^x$ is also an exponential function. It takes much more effort to differentiate it. Is ...
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Antonym of the word nullity in the context of Boolean algebra

Depending on what axioms one uses to define a Boolean algebra, one result one can often show for a Boolean algebra $(S,+,*)$ is that for all $x \in S$, $x + 1 = 1$ and $x * 0 = 0$. I have seen some ...
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Is there a name for when we extend the codomain of a function? (sort of like the opposite of the restriction)

If I have a function $f: X \to Y$ and a subset $A \subseteq X$ then I can define the restriction $f|_{A}: A \to X$ by $f|_{A}(a) = f(a)$ for all $a \in A$. This can be interpreted as composing with ...
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62 views

Converting tabulated data to equation of multiple variables

I've come across a mathematical problem while working on my school project. I've got tables for airplane takeoff distance which is dependent on airport altitude, temperature and mass of an airplane. ...
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What is the name for a 3 pointed figure with two straight sides and an arc?

For example, if one draws a circle inside of a star so that the circumference touches all of the inner points --- it would be the opposite of a sector --- the red part in the image below: I would ...
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What are these types of mappings called?

Functions which do not change when iterated are called idempotent. Those which yield the identity after one reapplication are called involutions. What are those which yield the identity after a ...
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What is the name for this generalization of a category?

Is there a name for an object which is made of a set of objects, and a set of arrows which can be from objects/arrows to objects/arrows (all four combinations)? Equivalently, this is a category ...
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1answer
34 views

Is there such a term as a “Borel measurable set”?

Not sure if this is the right place to post such a rookie question, but I'd appreciate some quick clarification. Is there such a term as a "Borel measurable set"? I've seen it used all over the place ...
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1answer
63 views

What is a standard name for a “relation” as a subset of $X\times \mathcal P(X)$ rather than of $X\times X$

(Binary) relations on $X$ are formalized as subsets of $X\times X$. But there are also times when a "relation" is a subset of $X\times \mathcal P(X)$. For example, in topology, we may say that $x$ is ...
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What is the name of this hexagon/pentagon polyhedron?

What is the name of this convex polyhedron?                     $(V,E,F)=(14,36,24)$. The top and bottom vertices are degree-$6$, spanning ...
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Is there a more formal way of saying “side of an equation”?

I've been thinking about it, and the term "side of an equation" (in reference to an equation such as $x^2 + x + 1 = 0$) doesn't seem like a very formal way of trying to describe the expressions on ...
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2answers
66 views

what is the absolute value of a set?

Let $S$ be the equivalence relation defined on $\wp(\{1, 2, 3, 4\})$ defined by: $$XSY\text{ if and only if } |X|\equiv|Y|\;\mod 2$$ Write down the equivalence classes of S. I understand that ...
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38 views

Operation producing a set of ordered pairs from a pair of ordered sets

Consider two indexed sets $A=\left(a_{1}, a_{2}, \dots, a_{n}\right)$ and $B=\left(b_{1}, b_{2}, \dots, b_{n}\right)$. From a mathematical standpoint, what would be the name of the operation $\odot$ ...
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Superscripts in handwritten mathematics [closed]

Recently I first encountered using superscripts as abbreviations for inflectional morphemes usually combined with abbreviated variants of words. For example a spoken sentence: A family of ...
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27 views

Ordering relations : alternatives to the somewhat confusing terminology “ a precedes b and conversely b dominates a”

If R is an ordering relation, the fact that aRb ( or that (a,b) belongs to R) is often expressed as (1) a precedes b and conversely (2) b dominates a. ( Example, Lipschutz, Theory and Problems of Set ...
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50 views

What does it mean for a field to be generated by some set of elements?

I've read up online, but I'm having trouble understanding what it means for a field $F$ to be generated by a set of elements $S = \{\alpha_1, ..., \alpha_n\}$ over another field $K$. What are the ...
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102 views

Is it correct to say that every bijection of a set onto itself is a permutation?

Typically permutations are discussed in the context of finite sets. I seem to recall, however, at least one source saying that every bijection of a set onto itself is a permutation. That was in the ...
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Rank of tensor like operation?

What is this matrix $M$ called? What is its rank? Take linearly independent vectors $v_1,\dots,v_n\in\mathbb R^m$ and consider the matrix $M$ with $n^n$ rows $v_{i_1}\dots v_{i_n}\in\mathbb R^{mn}$ ...
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Position Vector valued functions and Vector Fields

What is the actual difference between these two: Vector valued functions and Vector Fields? As far as I can see, both these concepts are alike, they take in $x$ and $y\,\,$(simple case) and give out ...
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Why are elliptic points called elliptic?

Points on the upper half plane $\mathbb H := \{ z \in \mathbb C : \Im(z)>0 \}$ are called elliptic with respect to some $\gamma \in \operatorname{SL}_2(\mathbb Z)$ if they are fixpoints of the ...
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Doubt about proposition order on set theory proof

I have a doubt trying to understand thoroughly part of this set theory proof: $B\times\bigcap\limits_{i\in I} A_i = \bigcap\limits_{i\in I}(B\times A_i) $ Let $(a,b)\in B\times\bigcap\limits_{i\in I}...