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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16
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255 views

Why is $J$ often used to denote $\mathbb{N}$ or $\mathbb{Z}$ in older texts?

In older books, I've noticed that authors tended to use $J$ to denote (usually) the natural numbers and (less commonly) the integers. Does anyone have any idea why that might've been? A few examples ...
12
votes
0answers
163 views

$\sin$ vs. $sin$ - history and usage

One thing newcomers to TeX or MathJax often get wrong is that they write something like $sin(x)$ instead of $\sin(x)$ - the point being that common mathematical functions with names consisting of ...
9
votes
0answers
172 views

Origin of $\mapsto$ notation

Who invented the brilliant $\mapsto$ notation for describing a function's action on a point, as in $x \mapsto x^2$? This is in some sense a counterpart to Who came up with the arrow notation $x ...
7
votes
0answers
85 views

Does anyone use $\subset$ for proper subset anymore?

I belong the the group of people who still write (not necessarily proper) subset as $\subseteq$ to avoid any confusion with proper subset, which I notate $\subsetneq$; I usually do not use $\subset$ ...
7
votes
0answers
76 views

Sets, that have $\operatorname{LCM}\left(|c_1|,\dots,|c_p|\right)=\sum_{k=1}^p |c_k|$

I found that the least common multiple of the sizes of conjugacy classes $c_k$ of the symmetric group $S_n$ is equivalent to $n!$ the order of the group. Equivalently the sum of all $c_k$ is also ...
7
votes
0answers
391 views

notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
6
votes
0answers
142 views

Why use Einstein Summation Notation?

Einstein summation convention dictates that repeated indices should be summed. Thus the equation $a_{ij} = b_{ik}c_{kj}$ is taken to mean $a_{ij} = \sum_k b_{ik}c_{kj}$ where in both cases the range ...
6
votes
0answers
120 views

Modern notational alternatives for the indefinite integral?

I like the Leibniz notation, and I think the reason it's survived for over 300 years and continued to be almost the only game in town is that in many respects it's a miracle of design. Nevertheless ...
6
votes
0answers
169 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 ...
6
votes
0answers
449 views

Typesetting imaginary unit and bessel functions

I know that this issue has been treated in many places, but I have yet to reach something conclusive, hence I am herein seeking your help. Following the 260.3-1993 - American National Standard ...
5
votes
0answers
92 views

Notation/terminology for “independent” subspaces/subalgebras

Let $V$ denote a vector space (or any other kind of algebraic structure). Question. Letting $I$ denote a fixed set and $X$ denote an $I$-indexed family of subspaces (subalgebras) of $V$, is there ...
5
votes
0answers
57 views

Where can I learn more about the “else” operation / “else monoid”?

(The set of natural numbers $\mathbb{N}$ starts at $0$ for me.) Let $X$ denote a set, and define $X_\bot = X \uplus \{\bot\}.$ Let $\mathbf{else}$ denote the binary operation on $X_\bot$ defined as ...
5
votes
0answers
50 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
5
votes
0answers
73 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
5
votes
0answers
2k views

What symbol expresses “less than approximately”?

Suppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is lesser than some value which ...
5
votes
0answers
337 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
5
votes
0answers
425 views

Is there a name or definition for this popular notation?

I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've ...
5
votes
0answers
249 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
4
votes
0answers
35 views

How to read an expression that is ambiguous?

(1) How should I parenthesize $\log n \log \log n$? Also: (2) What general rule/rationale is used to do this parenthesization? To elaborate; I see why $\log\log n$ is unambiguous, but $\log n \log ...
4
votes
0answers
96 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
4
votes
0answers
138 views

Why do Mathematicians use $u$ and $v$ as variables?

I'm sure this has happened to you as well: you are reading some hand-written work, the variables used are $u$ and $v$, and at some point the handwriting becomes unclear and you cannot distinguish the ...
4
votes
0answers
581 views

Notation for the pushforward measure

Given a measure space $(X,\Sigma,\mu)$, a measurable space $(Y,\Xi)$ and a measurable map $f\in\Sigma/\Xi$, the pushforward measure $\nu:=\mu\circ f^{-1}$ is given by $$ \nu[A] = \mu[f^{-1}(A)] $$ ...
4
votes
0answers
158 views

Why is the Euclidean metric called the prime at infinity?

I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
4
votes
0answers
66 views

What is the name of the function that indexes Grothendieck universes?

Assume Tarski-Grothendieck set theory. Then Grothendieck universes form a well-ordered proper class, so we can let $U_\alpha$ denote the $\alpha$'th Grothendieck universe, where $\alpha$ is an ...
4
votes
0answers
292 views

Divisibility notation history

I'm writing a paper project for school about divisibility, so I'd like to include a bit of history about that subject. I'm mostly interested in notation of $|$ sign used in past, but everything else ...
4
votes
0answers
336 views

Is there a notation for $f(x,y) - f(y,x)$?

Suppose we have a function $f(x,y)$ in two variables. Is there an operator on the function, say, $$([]f)(x,y) = f(x,y) - f(y,x)?$$ In other words, I'm looking for a commutator in terms of function ...
4
votes
0answers
132 views

Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
4
votes
0answers
246 views

Better Tensor Notation

I am in a General Relativity class, and I am finding the usual tensor notation very difficult to think about -- it seems like there are too many names to express something simple. E.g., I think of ...
4
votes
0answers
233 views

Where does the 'divides' sign come from?

When $a$ divides $b$ we say $a | b$. Where does the $|$ sign come from? This is not homework, just personal interest in the history of mathematical language.
3
votes
0answers
41 views

Standard notation for indices in group theory?

I've seen three notations for indices in group theory, namely $(G:H)$, $[G:H]$ and $|G:H|$. Is there any of these notations that is standard?
3
votes
0answers
26 views

Unique Conway notation for knots?

Is the Conway notation for a knot unique? Here are two rational tangles whose closures give the trefoil knot. However the Conway notation written for the trefoil knot is usually presented as 3 in ...
3
votes
0answers
31 views

Regarding the construction of the tensor bundle

Recall the construction of the tangent bundle: we write $$TM = \bigsqcup_{p \in M}T_p M$$ and define it as the prevector bundle with local trivializations $[\gamma] \mapsto (\gamma(0), (x\gamma)'(0))$ ...
3
votes
0answers
163 views

In differential geometry, is there established notation for the stuff that $\mathbb{R}^n$ is equipped with?

Given $v,p \in \mathbb{R}^n$, I will write $T_p(v)$ for $v$ viewed as a vector based at $p$. So: $$ T_p(v) \in T_p\mathbb{R}^n$$ Question 0. Is there an accepted notation for what I'm denoting ...
3
votes
0answers
46 views

Is there an established notation for this “replacement” operation?

If $S$ is a set, define $$(x \to y) \cdot S := \begin{cases} (S \setminus \{x\}) \cup \{y\} & \text{ if } x \in S \text{ and } y \not \in S; \\ S & \text{ otherwise.} \end{cases}$$ In other ...
3
votes
0answers
29 views

Interpretation of composite of random variable

Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$. How do I interpret the following statement: $$ Var(f \circ ...
3
votes
0answers
36 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
3
votes
0answers
49 views

Writing a Series of Equalities and Inequalities across Several Lines

What is the convention for writing a series of successive equalities and inequalities across multiple lines? Let me explain. Let $E_k$ denote an expression; for example, $E_0$ could be a sum or an ...
3
votes
0answers
39 views

What does $x_e$ mean in $I(x_e,y_0)$?

A book I am using has a problem which includes two points on the graph of $y=\ln x$, $M_1(x_1, y_1)$ and $M_2(x_2, y_2)$ and identifies the middle of the chord $M_1 M_2$ between them as $I(x_e, y_0)$. ...
3
votes
0answers
45 views

Why is the boundary of a topological space $M$ denoted $\partial M$?

Why is the boundary of a topological space $M$ often denoted $\partial M$? Is there any connection between boundary and partial derivative?
3
votes
0answers
43 views

Formulating equation for distances between atoms

I'm trying to formulate notation to describe code that calculates the distances between protein atoms (represented as points in 3D space). Fragments consist of residues (amino acids) and each residue ...
3
votes
0answers
62 views

$\times$ as symbol for multiplication, how common is this?

I never encountered the $\times$ symbol denoting multiplication (of real numbers or real-valued functions) since middle school until I found it recently in the measure theory notes by D H Fremlin. ...
3
votes
0answers
72 views

Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation ...
3
votes
0answers
49 views

Why is the slope-intercept form of the equation of a line often written $y=mx+b$? Why $m$ instead of $a$?

After a quick google search, I read something about Conway suggesting the $m$ having to do with "modulus" ... This seems odd to me, but perhaps there is some mathematical reason? I've heard of the ...
3
votes
0answers
32 views

What is the product of an empty family of similiar algebras, that is $\prod\langle \mathbf{A}_i \ | \ i \in I \rangle $, where $I = \emptyset$?

What is the product of an empty family of similiar algebras, that is $\prod\langle \mathbf{A}_i \mid i \in I \rangle $, where $I = \emptyset$? The family $\langle \mathbf{A}_i \mid i \in \emptyset ...
3
votes
0answers
53 views

Notation $(a,b)$ for $]a,b[$.

Is there any logic or justification for the notation $(a,b)$ to represent $]a,b[$? To me this notation is very ambiguous and confusing because it looks like a couple of numbers and not an interval. ...
3
votes
0answers
81 views

Can $\mathbb A=\{f(x)\mid x\in\mathbb R\}$ be shortened as $\mathbb A=f(\mathbb R)$?

Can $\mathbb A=\{f(x)\mid x\in\mathbb R\}$ be shortened as $\mathbb A=f(\mathbb R)$? I saw this notation in the IMO olympiad training materials (the solution to the Problem 16 (IMO 1999 Problem ...
3
votes
0answers
29 views

Equivalent to proportionality sign for additive constants

Short question Is there an equivalent to the proportionality sign $\propto$ for additive constants? The proportionality relation $y\propto x$ implies that $y=kx$ for some constant $k$. Is there a ...
3
votes
0answers
69 views

Etiquette for proper usage of Greek letters and other notation

I've progressed to the "output" point in my mathematics career and have run into a slightly embarrassing problem while writing a paper. Clearly, certain Greek letters are suitable for some situations ...
3
votes
0answers
80 views

A name for something like a CW complex?

The class $\mathcal{CW}_n$ of finite $n$-dimensional CW complexes can be defined recursively: $\mathcal{CW}_0$ consists of finite sets; If $X \in \mathcal{CW}_n$, $\phi:S^n \amalg \cdots \amalg S^n ...
3
votes
0answers
79 views

Good confusion-avoiding notation for gluing toric varieties from fans?

$ \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \DeclareMathOperator\Cone{Cone} $I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A ...