Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.

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0
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3answers
104 views

What are the differences between: $\sqrt{(-3)^2}$, $\sqrt{-3^2}$ and $(\sqrt{-3})^2$.

First, is $\sqrt{-3}$ is equal to $-3$ or is it imaginary? What is the difference between: $\sqrt{(-3)^2}$ $\sqrt{-3^2}$ $(\sqrt{-3})^2$ Can I write $(\sqrt{-3})^2 = -3$? And, given the rule ...
1
vote
2answers
67 views

Standard notation for the transform that turns a function $A \rightarrow (B \rightarrow C)$ into a function $B \rightarrow (A \rightarrow C).$

Suppose we're given sets $A,B$ and $C$. Then to each function $f : A \rightarrow (B \rightarrow C)$, we can assign another function $F : B \rightarrow (A \rightarrow C)$ by defining: $$F(b)(a) = f(a)(...
54
votes
7answers
4k views

Why does the symbol for the multiplication operation change shape?

Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen ...
0
votes
0answers
6 views

Notation for arguments that maximize functions with priority

Suppose we have some functions $f_1(x), f_2(x), \ldots, f_n(x)$ with $x \in \mathbb{Z}^n$. We can denote the subset $X_1$ of $\mathbb{Z}^n$ that maximizes $f_1(x)$ as: $$X_1 = \underset{x \in \...
2
votes
0answers
42 views

In mathematics, what is a placeholder?

Google defines the word placeholder in the following image: . My question is: Is this the only definition of the word placeholder in mathematics? I am thinking along the lines: If $M = p^k m^2$ ...
2
votes
2answers
3k views

Is there a Math symbol that means “associated”

I am looking for a Math symbol that means "associated" and I don't mean "associated" as something as complicated as isomorphism or anything super fancy. I am looking for a symbol that means ...
1
vote
1answer
9 views

Sampling Distribution Notations

I'm reading a chapter on sampling distributions of a statistic and I don't seem to have an understanding of the notations used. From probability theory, a random variable is usually denoted by a ...
2
votes
0answers
86 views

Is this notation common in Calculus?

Okay this is going to be quite a stupid question, but to me this seems... wrong, or at the very least not completely correct. In the material I'm reading there's a part that states that $y$ evolves ...
3
votes
6answers
61 views

Is there a difference between $y(x)$ and $f(x)$

Oftentimes functions described by $f(x) = 2x+4$, and when this is mapped to the Cartesian plane, $f(x) = y$. This surely implies that $y = 2x+4$. Is there a difference between this and $y(x) = 2x+4$?
1
vote
0answers
19 views

The definition of the operator $\Delta_{\bar\partial}$

Preliminaries: Let $(X,h)$ be a Kahler manifold of complex dimension $d=2n$. Lets denote with $\mathscr A^{p,q}_X$ the sheaf of $C^\infty$ (complex) $(p,q)$-forms over $X$ and let $A^{p,q}(X):=\Gamma(...
0
votes
1answer
31 views

Vector in vector notation

I'm a bit confused as far as notationally differentiating between row and column vectors goes. Suppose I define a column vector $$\boldsymbol{a} = (a_{1}, a_{2})^{T}$$ and another column vector $$\...
3
votes
0answers
41 views

What does $x$ represent? [on hold]

$x$ can mean: A defined value in $x = 5$ An unknown, TBD. value, in $3x = 2+4$ A variable in $y(x) = x+4$ A sum of numbers in $\int_0^2 \mathrm{d}x$ Are there any other examples?
0
votes
2answers
32 views

Set notation for unordered cartesian product

In the question unordered cartesian product an shorthand notation for the unordered cartesian product was discussed but without any standard notation. So my question is what would be the explicit ...
1
vote
2answers
25 views

Summation Notation Question in McMillan's Theorem Proof

Let me preface by saying that this question does not pertain as much to coding theory, as it does to mathematical notation. Every symbol in this question is a natural number. Anyhow, I am currently ...
0
votes
1answer
28 views

Notation for set of unit vectors

Is there a standard notation for the set of unit vectors $\{\vec v\ :\ |\vec v|=1\}$?
17
votes
10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
0
votes
3answers
25 views

How to read this ArgMax definition in plain english

I was reading on Wikipedia about Argmax (https://en.wikipedia.org/wiki/Arg_max) and they gave the following equation. While I get most of this line, how would you read the following in plain English? ...
1
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3answers
41 views

How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way. What is a popular way? In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?
11
votes
3answers
457 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
-1
votes
1answer
32 views

If G and H are two gaphs then what does $G \Delta H$ indicate in graph theory?

I came across this notation in a book titled " Combinatorial Optimization Theory and Algorithms" by Bernhard Korte and Jens Vygen.
42
votes
12answers
4k views

Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
6
votes
1answer
252 views

A question about co-exponentials

An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$ or equivalently, as the terminal object of $...
1
vote
2answers
38 views

Notation for a sum over a set of variables

I have a vector of variables $y=(y_1, \ldots, y_n)$ whose elements are either zero or one. I would like to express the sum over all variables belonging to a subset $S$. For example, if $n=4$ and $S=\{...
3
votes
1answer
10 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
3
votes
3answers
51 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
0
votes
0answers
33 views

What does the '#' sign mean in elliptic curves?

My question is regarding specifically elliptic curves. I have seen the notation $\#E(\mathbb{F}_{q})$ used over and over again (especially in the description of Hasse's theorem). I know that sometimes ...
0
votes
1answer
33 views

Proper notation for motion integration

Say you have a projectile where at $t=0$, $ v = 0 $ and $ x = 0 $. Given $ \ddot x = -4$, in order to find $ \dot x $, we must integrate $ \frac{dv}{dt} $ as follows: $$ \frac{dv}{dt} = -4 \...
1
vote
0answers
21 views

derivative and curl vector notational difference

This must be a really trivial question, and I am not sure I am allowed to post these here. Just wanted to know but didn't know what to search. What is the difference between $\overrightarrow {dr}$ and ...
2
votes
3answers
62 views

How $f:[a,b]\rightarrow[c,d]$ should be read?

I found it in a book but I don't know what the ":" means. What does this expression mean?
3
votes
2answers
66 views

What does $\mathbb{Z}/n \mathbb{Z}$ mean in abstract algebra?

If you look at this wiki page under the image on top of right hand side, you see $\mathbb{Z}/ \mathbb{8Z}$. What does it mean and give example if possible please thanks.
0
votes
0answers
31 views

Distinguish between constant function and a constant. [closed]

The constant function $\begin{array}&y: &\mathbb{R}\to \{c\}\\&x \mapsto c\end{array}$ and the constant $y = c$ are often simply written as $y = c$ and it's not always ...
-1
votes
2answers
79 views

Why is $\wedge$ a minimum and $\vee$ a maximum? [closed]

Why does $\wedge$ denote a minimum and $\vee$ a maximum? Where did this notation come from? I keep getting them mixed up because to me, $\wedge$ should be a maximum: it's a hill, or a curve reaching ...
0
votes
2answers
25 views

Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
2
votes
1answer
23 views

Inverse relations notation (not a function) [closed]

Even if $f:X\to Y$ is not a bijective function, can I still notate the inverse relation of $f$ as $g:Y\to X$?
0
votes
0answers
20 views

Is there an accepted notation for the monoid of linear polynomials?

Is there an accepted notation for the monoid of linear polynomials (with addition as the operation) with coefficients from some ring R? Like $2p+3$, where $p$ and the identity generate the monoid ...
7
votes
4answers
141 views

Confused about notation “:=” versus plain old “=” [duplicate]

Relating to sets, I find the following in a text book: "...the set S := {1, 2, 3}". The book has an extensive notation appendix, but the":=" notation is not included. What exactly does ":=" mean, and ...
0
votes
1answer
15 views

Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$?

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though ...
0
votes
0answers
32 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
4
votes
4answers
64 views

I want to write “$x,y > 0$”.

I want to write "$x,y > 0$". Can I do this? Or do I have to write "$x > 0$ and $y > 0$"? Which one is the proper way to write in maths?
0
votes
1answer
15 views

Is it possible to unify these two expressions into one?

I have the following expressions $\forall n \in \Bbb N$: $E = f(n)-1$ if $n \gt 1$ $E = f(n)+1$ if $n = 0$ I would like to have only one expression like this: $E = f(n)+$(some nice notation able ...
-1
votes
1answer
61 views

Why is the notation for irrational number not mainstream? [closed]

In everywhere you see the symbol for the set of rational number as $\mathbb{Q}$ However, to find actual symbol to denote the set of irrational number is difficult. Most people usually denote it as $\...
2
votes
3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
1
vote
1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...
1
vote
1answer
25 views

Notation for minimum

I'm reading Huber's Robust Statistics right now, and at the beginning of Chapter 3, he writes the following notation: $\sum \rho(x_i;T_n) = \min!$ Similarly, a few lines down, he writes: $\sum \rho(...
18
votes
6answers
4k views

How to represent “not an empty set”?

I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians? My idea is: $|A| > 0$ However, using the emptyset $\emptyset$ ...
2
votes
1answer
34 views

Understanding the notation of a paper

I am reading a paper on Algebraic Number Theory that says If $p$ divides the discriminant of polynomial $f$ $r$ times and there is the factorization into irreducibles $$f(x)\equiv g_1(x)\dots g_r(...
1
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0answers
38 views

How do I write a function that maps a variable to a set?

I have a function $\Gamma$ that maps elements from $N$ to a (possibly empty) subset of $N$. The number of elements in the resulting subset depends on which element of $N$ we are dealing with, i.e. $\...
2
votes
0answers
35 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
0
votes
0answers
13 views

Meaning of notation $L_{subscript}$ in ridge detection.

In the wikipedia article on ridge detection, it says "let $L_{pp}$ and $L_{qq}$ denote the eigenvalues of the Hessian matrix \begin{pmatrix} L_{xx} & L_{xy} \\ L_{xy} & L_{yy} \end{pmatrix}...
2
votes
1answer
57 views

$T*T$ Notation and proof

Let $T:H\to H$ be compact where $H$ is a Hilbert space and let $T^*$ be the adjoint operator of $T$. Prove that $T^*T$ is compact and self adjoint and that the eigenvalues of $T^*T$ are nonnegative. ...