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

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1answer
59 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something $\...
0
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1answer
41 views

Meaning of “$r \to s$ is a tautology” in the definition of “implication” and “equivalence”

What does it mean to say the following: $$ r \to s\ is\ a\ tautology$$ I make the following truth table: $$\begin{array}{ l c c r } r & s & \lnot r & r \to s \\ \hline T & T &...
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2answers
29 views

Genral solution vs Fundamental Solution to ODE. The Difference is?

I've been reading intros to ODE and the problem of terminology has overwhelmed me. As far as I understand: n-Parameter family of solutions to ODE is a solution in a form $c_1y_1+c_2y_2+...+c_ny_n$ ...
5
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4answers
87 views

An antonym for “converse”

Suppose you are proving $p \leftrightarrow q$. In your first paragraph you prove $p \rightarrow q$. Your second paragraph begins, “For the converse, assume $q$ holds.” In this situation, we have a ...
3
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3answers
221 views

Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
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6answers
750 views

When can we not treat differentials as fractions? And when is it perfectly OK?

I am a first year calculus student so I would prefer if answers remained in Layman's terms. It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat ...
5
votes
3answers
97 views

Whats the difference between a series and sequence?

I was looking at a question earlier that involved sequences and found out that the sequence converged to 0 but the series diverged to infinity. How is that possible? for example the sequence was $a_n$ ...
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2answers
66 views

Name of the union of a set with its holes

Given an arbitrary connected and compact set $S$ with holes in it, is there a name for the simply connected set formed by the union of $S$ and its holes? For example, let $S = \{x\in \mathbb{R}^n\ |\ ...
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2answers
41 views

A matrix of a single 1 in each row and 0 elsewhere

Is there a particular name given to a matrix of m rows and n columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance: ...
0
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1answer
13 views

Is there a proper term for sets of interdependant functions?

I am looking for a term that would describe the sets of functions that are very closely related, such as: the trigonometric functions $\sin$ and $\cos$ the hyperbolic functions $\sinh$ and $\cosh$ ...
2
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1answer
32 views

Name of the modular group

I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's ...
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0answers
16 views

Is there a good term for pairs of related variables in a system?

(Non-mathematician here. Sorry). Suppose you have a problem with lots of unknowns. The problem allows many solutions (possibly infinite). Certain pairs of unknowns (you don't know which ones) ...
0
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1answer
29 views

What counts as a “Neighbor” in Conways' game of life?

I have looked everywhere but I cannot find an answer for this. Since I am bored, I am trying to create this game, but I can't seem to figure out what is considered a "Neighbor". Is it only directly ...
2
votes
1answer
38 views

What does the word “Comprehension” mean in the Axiom of Comprehension?

I understand roughly what the Axiom of Comprehension means, that any predicate can be used to construct a set of the elements that satisfy the predicate. But in English terms, where does the word "...
2
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2answers
36 views

How to rigorously state $P(x)$ is true for all but a finite number of $x$?

How to rigorously state that the predicate $P(x)$ is true for all but a finite number of $x\in\mathbb{N}$? My Attempt There is a finite set $\mathcal{S}\subset \mathbb{N}$ s.t. $s\in\mathcal{S}\...
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0answers
16 views

Terminology for asserting truth of equality/inequality based on symbolic equalities/inequalities

This may seem silly, but I am curious about algorithms used to computationally assert the truthiness (true, false, or unknown) of symbolic statements subject to a set of inequality constraints, for ...
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1answer
36 views

What is the term for relation whose inversion is a function?

Do we have a conventional term/name for such a relation $R$ (which is not necessarily a function) that $R^{-1}$ is a function? If not, what are your suggestions?
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2answers
26 views

Name for complex number with nonzero imaginary component

Complex numbers include all real numbers. Is there a name for the subset of complex numbers which does not include any real numbers (i.e. where the imaginary component is nonzero)?
0
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2answers
41 views

Name of the notation where number is expressed as a sum

I have the following general form of a number: Does this notation have a name? Here is the example of using the form:
6
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1answer
66 views

Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic?

Is there a geometric interpretation of second order linear partial differential equations which explains why they are classified as either elliptic, hyperbolic or parabolic, or is this just a naming ...
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0answers
16 views

Difference between modular equation and congruence equation

Is there any difference between a modular equation and a congruence equation, or are both the same thing?
0
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1answer
17 views

What is different between in a set and on a set?

I saw that in is sometimes used and so is on. For example, Let $f$ and $f_k$, $k=1, 2, \cdots$, be measurable and finite a.e. in $E$. If $f_k\to{}f$ a.e. on $E$ and $|E|\lt+\infty$, then $\{f_k\}$...
2
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1answer
25 views

what is the terminology of this form of equation $x^2 +1/x^2 + \sqrt{x}$

what is the terminology of this form of equation. It has only one variable, but with rational exponents, it can be positive, negative or fraction such as below: $ax^2 +b/x^2 + c\sqrt{x} =0 $ I ...
0
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1answer
40 views

What is it asking when it says define and then gives a function and some conditions?

For example, the question: Define $f$ on $[3,4]$ by $f(x)=x+5$. Using the definition of the Riemann integral, show that $f$ is integrable on $[3,4]$. or Let $E=\{x \in \mathbb{R} : x \ge 1\}$. Define ...
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0answers
27 views

What to call a term-in-context whose context contains exactly the variables occurring in the term?

In type theory, a term-in-context $\Gamma \vdash t : \tau $ is only well-formed when $\Gamma$ contains all the variables occurring in $t:\tau$. Is there a name for when it contains exactly the ...
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1answer
41 views

Math-english for non-natives: What does “supported in” mean?

As a non-native English speaker, I am struggling with the following sentence: "Fix a function $f:\mathbb{R}\to\mathbb{C}$ such that $f$ is supported in the unit Ball." Does this mean $\...
4
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0answers
58 views

Does this operation have a name?

For a field $F$, define the binary operation $\parallel :(F\mathbb{P}^1 \times F\mathbb{P}^1 \setminus\{(0,0)\}) \to F\mathbb{P}^1$ by $$a \parallel b = \frac{1}{\frac{1}{a} + \frac{1}{b}}.$$ This ...
1
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1answer
82 views

How do you pronounce $\preceq$?

I've been reading about partial orders and partially ordered sets and have come across sentences like "Suppose that $\preceq$ is a partial order on $X$" and "If $x\preceq y$ and $y \preceq z$ then $x \...
0
votes
1answer
60 views

Does a cylinder with equal height and diameter have a special name?

I'm working on z-calibration part for my 3d-printer and I'm wondering if this has a special name? cylinder({r: 5, h: 10}) Basically a cylinder that has a height ...
0
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2answers
30 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 ...
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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.."?
5
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2answers
66 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
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1answer
10 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) - c\...
2
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2answers
85 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 ...
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0answers
44 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
32 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 ...
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0answers
19 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
15 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, ...
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0answers
62 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
1answer
27 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 ...
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0answers
62 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
1
vote
1answer
13 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 ...
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1answer
73 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 ...
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0answers
44 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 ...
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0answers
20 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 ...
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3answers
35 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?)
4
votes
2answers
68 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?
2
votes
2answers
61 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 ...
2
votes
1answer
57 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 ...
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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 $\frac 2 3$.