# Suggestions for good notation

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:

• Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δn becomes [n=0]. (A similar convention is used in the C programming language.)

• The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

• I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing BA for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

• Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.

Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)

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this question is broadly useful, so perhaps better as community wiki? – Suvrit Oct 20 '10 at 20:09
In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. – Andrés E. Caicedo Oct 20 '10 at 20:54
I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... – Kevin H. Lin Oct 20 '10 at 23:18
Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. – Chandan Singh Dalawat Oct 21 '10 at 3:29
Isn't $x \mapsto f(x)$ commonplace? As for homomorphisms, they are not simply maps, and $\mathrm{Hom}(A, B)$ denotes the whole class, while $A \to B$ denotes a single mapping. – Alexei Averchenko Oct 21 '10 at 3:38

Among recent introductions, I like the notation and names (introduced by Kenneth Iverson and popularized by Donald Knuth) for the ceiling function $\lceil x\rceil$ and floor function $\lfloor x\rfloor$. Compare with the heavy "approximation by excess/defect"...

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According to Knuth, this notation and these names were introduced by Iverson in his book "A programming language" in 1962. – Richard Borcherds Oct 20 '10 at 20:25
Thanks for correcting! (fixed now) – Pietro Majer Oct 20 '10 at 20:36
But I have also seen some people call these "Gaussian Brackets" --- any reason why that is so? – Suvrit Dec 15 '10 at 13:56
@Suvrit: I have learned (in a German school) that Gauss used ordinary brackets [x] to denote the floor function, thus called Gauss bracket. However, I know of no source but guess that it is easily found in Gauss' works. – Konrad Voelkel Dec 19 '11 at 11:10
They are life-savers... – Felix Goldberg Jan 24 '13 at 12:15

I like notation such as $2^X$ for the set of subsets of $X$ and ${X\choose k}$ for the set of $k$-element subsets. Also $[x^n]F(x)$ for the coefficient of $x^n$ in the power series $F(x)$, and multivariate notation like $x^\alpha$ for $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, where $x=(x_1,\dots,x_n)$ and $\alpha=(\alpha_1,\dots,\alpha_n)$.

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I find the projection $[x^n]F(x)$ indeed very useful. – Quadrescence Oct 21 '10 at 17:23
And, in the spirit of the latter notation, the multivariate version $[x^\alpha]F(x)$ for polynomials in several variables. – Pietro Majer Oct 23 '10 at 19:14
I particularly like $X \choose k$ when X is say an algebraic variety and so $\chi {X \choose k} = {\chi(X) \choose k}$, where $\chi$ is the Euler characteristic... – Vivek Shende Oct 29 '10 at 2:29

Bourbaki dangerous bend symbol to mark dangerous or difficult ideas.

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Haha. I didn't know this was due to Bourbaki, I learned it from Knuth's "The TeXbook". – Andrés E. Caicedo Oct 21 '10 at 17:33
Perhaps this should be on every line in the Bourbaki text. – Daniel Spector Feb 3 '13 at 11:28

I like $A \hookrightarrow B$ and $A \twoheadrightarrow B$ for "$A$ injects into $B$" and "$A$ surjects onto $B$" respectively.

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I like this but I am confused what to use for bijections. – Marcel Bischoff Dec 18 '10 at 12:59
Maybe $A \leftrightarrow B$ for bijections. – Adeel Khan Dec 18 '10 at 18:22
A two-headed hooked right arrow. I don't know how to do it in normal LaTeX but you can easily do it using the xymatrix package, and obviously it's easy to write on a chalkboard. – Tom LaGatta Dec 18 '10 at 23:25
@Marcel - $\stackrel{\sim}{\to}$, a combination of the map and the isomorphism sign. – David Roberts Nov 24 '11 at 23:12
Add to this the notation $\looparrowright$ for immersions. – Turion Mar 25 '14 at 16:27

$D_j f$ to denote the partial derivative of a function between Euclidean spaces, w.r.t. the $j$'th coordinate. For some reason Jacobi's notation $\frac{\partial f}{\partial x_j}$ has become more popular. Jacobi's notation tends to cause much ambiguity and confusion, a point which is emphasized in the book "Multidimensional Real Analysis" by Duistermaat & Kolk. For instance (this example is taken from their book), let $e_1,e_2$ be the standard basis for $\mathbb{R}^2$ and define a new basis by $e'_1 = e_1 + e_2, e'_2 = e_2$. The passage from one basis to another is as follows: If $x_1 e_1 + x_2 e_2 = y_1 e'_1 + y_2 e'_2$ then $y_1 = x_1, y_2 = x_2 -x_1$. Now the meaning of $\frac{\partial y_2}{\partial y_1}$ is ambiguous: If one interprets $y_1$ and $y_2$ as independent coordinate functions, then $\frac{\partial y_2}{\partial y_1} = 0$. On the other hand, $\frac{\partial y_2}{\partial y_1} = \frac{\partial (x_2 -x_1)}{\partial x_1} = -1$, right? This was the source of much confusion for me when I was taught multivariate calculus and the notation $D_j f$ would have eliminated this confusion entirely.

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Another reason to prefer $D_j$ or $\nabla_j$ over $\partial/\partial x_j$ is that when you're expressing tensors using the Einstein "index gymnastics notation," the derivative $\nabla_j$ has a lower index, as it should, whereas the same operator in Leibniz notation is $\partial/\partial x^j$, which looks like it transforms as an upper-index tensor. – Ben Crowell Oct 15 '12 at 20:39
Regarding differential geometry: If $f:M\to\mathbb R$ is a smooth function on a manifold and $x:M\to\mathbb R^n$ is a chart, I prefer $\left(\frac{\partial f}{\partial x}\right)_j$ or $\left(\frac\partial{\partial x}\right)_j f$ (or even $\partial_j f$ if the choice of the particular chart is clear or irrelevant). Because the notation $\frac{\partial f}{\partial x^j}$ suggests that $\frac{\partial f}{\partial g}$ could be defined using only $g$, and in fact you need to know that you are restricting to the curve along which the other coordinates $x^i$ are constant. – Marcos Cossarini Jan 24 '13 at 21:03
@Ben, the index in the coordinate expression $\frac{\partial f}{\partial x^j}$ for the 1-form $df$ is clearly in the low position! In fact, this is the main reason that I see for having to put the indexes of the coordinates in the high position as we do, instead of doing everything in the opposite way, which would be better in some way: we could write $f=x_1^2+x_3$ instead of $f=(x^1)^2+x^3$. – Marcos Cossarini Jan 24 '13 at 21:14

String diagram-notation makes for example adjoint functors, monads, tensor categories,... much clearer.

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And it also connects monoidal categories to braids, knots, links, tangles... – Qiaochu Yuan Oct 20 '10 at 20:45

The notation $M^{\oplus n}$ and $M^{\otimes n}$ to denote, respectively, nth direct sum and nth tensor product.

The notation $X \mathbin{\pi} Y$ to denote product of objects in an abstract category, and the analogous with the "upside down $\pi$" for coproduct. I once have seen this being used by B.Keller in a talk. It'd be nice to have a smaller $\Pi$ (resp. $\amalg$) symbol instead.

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$\sqcap$ resp. $\sqcup$? – Max Nov 8 '10 at 18:39
You can get your smaller Π sign with a bit of a hack. This isn't exactly robust, though, since the size is constant relative to the surrounding size (but can be fixed if one looks up the right command): $X \mathbin{\scriptsize{\Pi}} Y$ and $X \mathbin{\scriptsize{\amalg}} Y$ – Quadrescence Nov 24 '11 at 21:08
I prefer to use the same notation as in arithmetic: $\times$ for binary product, $\prod$ for product of an arbitrary family, $+$ for binary coproduct, $\sum$ of coproduct of an arbitrary family. A small pi-like symbol for binary product seems to me to be a step in the wrong direction. – Tom Leinster Dec 10 '12 at 0:28

The notation for transversality:

$M \pitchfork N$

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I like $f\colon\thinspace M\looparrowright N$ to denote an immersion of smooth manifolds.

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I use exactly this arrow to denote an operation of a group M on a space N. – Jan Weidner Oct 20 '10 at 21:39
I prefer the notation $\alpha: G \curvearrowright X$ for an action $\alpha$ of $G$ on $X$ (see my answer). – Qfwfq Oct 20 '10 at 22:57
... And I also like the "self intersecting arrow" to denote immersions that are possibly not embeddings. – Qfwfq Oct 20 '10 at 22:58

I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed). e.g.

• $X = O_k(Y)$ (or $X \ll_k Y$, or $Y \gg_k X$) means that $|X| \leq C_k Y$ for some $C_k$ depending only on $k$.
• $X = o_{n \to \infty; k}(Y)$ means that $|X| \leq c_k(n) Y$ for some function $c_k(n)$ of both $k$ and $n$, which goes to zero as $n \to \infty$ for fixed $k$.
• (Rarer) $X = O_{n \to \infty; k}(Y)$ means that $|X| \geq C_k Y$ whenever $n \geq N_k$, for some $C_k$ and $N_k$ depending only on $k$.

Of course, if there is a parameter that influences all the constants (e.g. the ambient dimension) then it is better to explicitly state at the beginning that all constants will depend on this parameter so that one does not have to put in the explicit subscripts in all the time.

It can be instructive to rewrite some basic notions in analysis in this sort of notation, just to get a slightly different perspective. For instance, if $f: {\bf R} \to {\bf R}$ is a function, then:

• $f$ is continuous iff one has $f(y) = f(x) + o_{y \to x; f,x}(1)$ for all $x,y \in {\bf R}$
• $f$ is uniformly continuous iff one has $f(y) = f(x) + o_{|y-x| \to 0; f}(1)$ for all $x,y \in {\bf R}$
• A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is equicontinuous if one has $f_n(y) = f_n(x) + o_{y \to x; F,x}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$ (note that the implied constant depends on the family $F$, but not on the specific function $f_n$ or on the index $n$)
• A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is uniformly equicontinuous if one has $f_n(y) = f_n(x) + o_{|y-x| \to 0; F}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$
• $f$ is differentiable iff one has $f(y) = f(x) + (y-x) f'(x) + o_{y \to x; f,x}(|y-x|)$ for all $x,y \in {\bf R}$;
• (similarly for uniformly differentiable, equidifferentiable, etc.)

(These formulations are close to the nonstandard analysis formulations of these concepts, which uses similar but not quite identical asymptotic notation, but that is another story.)

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The subscripts seem a little bulky. Couldn't you just write, e.g. $f(y) = f(x) + o_{y \to x; f,x}(1)$? I guess it's not really changing much. – Ryan Reich Oct 21 '10 at 17:18
Maybe it's time for you to go to really big-O notation: $X=\underset{n\to\infty}{\overset{k}{\LARGE\mathcal O}} Y$ for $O(\cdot)$ and then $f_n(y)=f_n(x)+\underset{|y-x|\to 0}{\overset{F}{\LARGE\mathrm O}} 1$. for $o(\cdot)$. :D – Quadrescence Oct 21 '10 at 17:34
The "really" big-O notation is a little bit confusing; since normally we write summation like this with $Y$ depends on the parameter $k$, but here we have the constant $C_k$ depends on it instead. – Hsien-Chih Chang 張顯之 Oct 22 '10 at 1:07
Chang: definite sums can depend on their upper index. – Quadrescence Oct 22 '10 at 6:08
I think O-o notation is one of the worse. It does not worth to improve --- better to start from scratch. – Anton Petrunin Jan 30 '13 at 19:18

I once came across the notation $\underline{n}$ for the set $\lbrace 1,2,\dots,n\rbrace$. It came in very handy to write $i \in \underline{n}$ instead of $1\leq i \leq n$ or $i \in \lbrace 1,2,\dots,n\rbrace$.

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Sometimes people write ${}[n]$ for this. I agree, a shorthand of this kind is quite useful. – Andrés E. Caicedo Oct 20 '10 at 20:51
In set theory, every ordinal number is equal to the set of smaller ordinal numbers (a definition due to von Neumann), and this makes for infinitely many conveniences similar to the ones you mention. Thus, n={0,1,...,n-1} and in general $\alpha=\{\beta | \beta\lt\alpha\}$. – Joel David Hamkins Oct 20 '10 at 21:04
In Ramsey theory, sometimes it actually matters to use $\{1,\dots,n\}$ rather than $\{0,\dots,n-1\}$, so the ${}[n]$ notation is useful (even though I would much like to just use the set theoretic $n$) but I have found ${\bf n}$ may be better sometimes. For example, one also uses interval notation, so ${}[a,b]=\{i\in{\mathbb N}\mid a\le i\le b\}$, and using both may look confusing. Plus, the beloved $[X]^k$ (the collection of $k$-sized subsets of $X$) now gets really confusing. I've seen Di Prisco use $X^{[k]}$, probably to avoid this issue. – Andrés E. Caicedo Oct 20 '10 at 21:32
In numerical linear algebra MATLAB notation is sometimes (ab)used, e.g. 1:n. – J. M. Oct 21 '10 at 2:42
This can be confusing though, since the topologist's simplex $[n]$ is the ordered set $\{0<...<n\}$. The rationale is that $n$ counts the number of arrows. – Harry Gindi Oct 21 '10 at 4:52
1. For rising and falling factorials: $x^{\overline{n}}$ and $x^{\underline{n}}$ à la Knuth. Much better than the traditional way to write the Pochhammer symbol: $(x)_n := x^{\overline{n}}$. In a book I'm writing, I use the notation $x^{\uparrow n}$ and $x^{\downarrow n}$, which I find much less clumsy (consider $(2x+1)^{\overline{6k-2}}$ vs $(2x+1)^{\uparrow6k-2}$). Anyway, the utility in either of these notations is seen in the umbral calculus; it makes the connection to "ordinary" calculus much more apparent, such as with $$\Delta x^{\uparrow n} = n x^{\uparrow n-1}\qquad\text{compared to}\qquad D x^n = nx^{n-1}.$$

2. The simple idea of omitting parentheses for function application: $f\,x$ as opposed to $f(x)$. I think this often makes some mathematics look cleaner, especially when the argument isn't especially complex. It also allows for some nice (= convenient) abuse of notation, such as in $$\left[ (-1)^{p - m - n} z \prod_{j = 1}^p \left( z D_z - a_j + 1 \right) - \prod_{j = 1}^q \left( z D_z - b_j \right) \right] G(z) = 0,$$ where $D_z:=d/dz$. Note this equation isn't a product (entirely); upon expansion, we'd have $D_z G(z)$ terms.

3. Do fractions count? Imagine having to write $$\sqrt{(x^2 + 2x + 1)\div (5x^3 - 3x^2 + 2x - 7)}$$ instead of $$\sqrt{\frac{x^2 + 2x + 1}{5x^3 - 3x^2 + 2x - 7}}.$$

4. Big-O notation. Though often abused, this is a much less clumsy way to express boundedness and asymptotics and errors and even lets you begin to do some algebra with them (provided you're careful). I don't think doing such is as obvious when you write it all out manually.

5. $\square(x)$ for the square wave, $\triangle(x)$ for the triangle wave, $Ш(x)$ for the Dirac comb (seriously, see Appel's "Mathematics for Physics and Physicists"). These are more cute than explicitly useful.

6. Notation used with musical isomorphisms as a way to do raising and lowering of indices. We have $X^\sharp$ which raises the index (in the context of Einstein summation) and $X^\flat$ which lowers the index. Here, $\flat$ and $\sharp$ are isomorphisms between tangent $TM$ and cotangent bundles $T^*M$: $\flat:TM\to T^*M$ and $\sharp:T^*M\to TM$.

7. Using $\operatorname{cis}\theta = \cos\theta + \mathrm{i}\sin\theta$ (cosine i sine), which is nice for obvious reasons (yes, $\omega = e^{\mathrm{i}\theta}$ is nice too) and $\operatorname{cas}\theta = \cos\theta + \sin\theta$ (cosine and sine), which is used in e.g., the Hartley transform.

8. Notations for hypergeometric functions $${}_pF_q \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) = {}_pF_q(\mathbf{a},\mathbf{b};z)$$ and Meijer-$G$ functions: $$G_{p,q}^{m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right)=G_{p,q}^{m,n} \!\left( \left. \begin{matrix} \mathbf{a} \\\\ \mathbf{b} \end{matrix} \; \right| \, z \right)$$

9. Notation for general continued fractions: $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$ The $\mathrm{K}$ comes from German's "Kettenbruch", which is "continued fraction."

I think that's good for now. There are probably lots more. :)

To end, I'll say one notation I do not like: the use of fraktur. Most of the time it just looks ugly and no one can actually write fraktur letters.

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9. I am a fan of this. :D – J. M. Oct 21 '10 at 4:53
You're wrong that nobody can write Fraktur letters. If you put some effort into it then you absolutely can draw them. – KConrad Oct 21 '10 at 18:13
@KConrad: Sure you can, but you shouldn't. As with other scripts (say Latin or Cyrillic), the old script for writing German has a version for typesetting and a version for handwriting: Sütterlin (en.wikipedia.org/wiki/Sütterlin). If you like Fraktur, you will love it. – Maxime Bourrigan Oct 22 '10 at 15:00
@Bourrigan: I whole-heartedly disagree. Fraktur letters aren't that hard to write, and are easy to read. Sütterlin is illegible spaghetti, and even worse, the letters more often than not aren't what you expect them to be (the 'h' looks like an 'f', the 'e' like an 'n', etc.). It would be severely unpedagogical to use the script for anything other than to demonstrate how not to make a legible script. – Ketil Tveiten Nov 16 '10 at 9:26
"Bruch" is also the standard word for "fraction" in German, so the literal translation would be "chain fraction". – Henry Cohn Apr 17 '11 at 15:41

Instead of $[X]$ one often sees $\mathbf 1_{X}$ (especially in probability work?). This is neat because it literally is 1 on $X$. Also it has the advantage over $[X]$ that you can write things like $(2+\mathbf 1_{X})^2$ for the function that is $9$ when $X$ occurs and $4$ otherwise; $(2+[X])^2$ would be less appealing here. On the other hand, if there is a lot of notation replacing "$X$" this is not so good: $$\mathbf 1_{n_k\in \{n: n\text{ prime}\}}.$$

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so how about $1[X]$? – Hsien-Chih Chang 張顯之 Oct 21 '10 at 2:25
$\mathbf{1}_X$ is conventionally used for characteristic function of a set. – Alexei Averchenko Oct 21 '10 at 3:42
The problem with 1_{X}, as you observed, is that it pushes the main stuff to the small font of the subscript. I don't understand the advantage you wrote of: what's wrong with (2+[X])²? IIRC, expressions of this sort are used without hesitation in Graham-Knuth-Patashnik (Concrete Mathematics). – shreevatsa Oct 28 '10 at 6:51

$a \vee b$ and $a \wedge b$ to denote the maximum and minimum of the numbers $a$ and $b$. (This seems to be well-known only among probabilists.)

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This notation is also used in a lattice or Boolean algebra for least-upper-bound and greatest-lower-bound, which agrees with your meaning in a total order, such as the reals. – Joel David Hamkins Oct 20 '10 at 21:01
Sometimes I have seen $a \cup b$ and $a \cap b$ for the greatest common divisor and least common multiple of positive integers $a, b$. Much better than $(a, b)$. – Arend Bayer Oct 21 '10 at 3:59
AByer, it's by analogy with $\cup$ and $\cap$ that I remember that $\vee$ is maximum and $\wedge$ is minimum. (Otherwise I tend to think that since $\vee$ points down, it must be the minimum.) – Michael Lugo Oct 21 '10 at 4:20
Actually, I find this notation incredibly frustrating for the reason Michael mentions in his comment; $a\vee b$ is "clearly" a cartoon of a point lying below both $a$ and $b$. – JBL Oct 22 '10 at 13:19
$[a \vee b] = [a] \vee [b]$ – Arend Bayer Oct 22 '10 at 20:03

A good notation and a bad notation (in my opinion).

Good: $p' = (1 - \frac1p)^{-1}$. It is commonly enough used in analysis (Holder inequality) that it is good to have a shorthand, and it makes clear that the conjugate exponents are dual pairs: $(p')' = p$.

Bad: $p^* = \frac{np}{n-p}$ the Sobolev conjugate in Sobolev inequalities. It hides the dependence on the spatial dimension $n$, and overloads the $*$ for something that does not have a duality: $(p^* )^* = \frac{(2p)^*}{2} \neq p$.

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Having shorthand for the dual to a norm is indeed convenient $\left|\mathbf{A}\right|_{p^{\prime}}$. :D But maybe there is a better superscript than the already heavily used prime? – J. M. Oct 22 '10 at 1:22
@J.M.: both of these are historical (so don't blame me for them); also, when else do you see a prime applied to a number? – Willie Wong Oct 22 '10 at 10:03
@Willie, it is a convenient way to denote zero! – Mariano Suárez-Alvarez Oct 22 '10 at 16:43
Ah, so we could write Euler's formula as $\sum_{n \geq 0} n^{-s} = \prod_p (p')^s$! – Todd Trimble Oct 30 '10 at 18:22
I like Todd's pun very much, although it should be $\sum_{n\geq 0} n^{-s} = \prod_p (p^s)'$, shouldn't it? Anyway "Zeta(s) is the product over p prime of p to the power s prime" reads nicely! – David MJC Nov 8 '10 at 22:09

The original problem with the indices was that they were used to label coordinates, so mathematicians preferred more and more coordinate independent operators, while physicists continued to use indices. Then, Penrose realized that it has to be something beyond the indices that makes them useful - mainly the Einstein summation convention - and proposed the abstract index notation. This notation is almost identical in form with that of coordinate indices, but it is invariant, like the notation used by mathematicians, and maintains the simplifications due to the use of indices. The indices are not interpreted as labeling coordinates, but as representing the type of vectors and tensors and how they act on each other.

I think that there are advantages and disadvantages in both notations. Though, many tensor operations, especially contraction and type change, are easier to define and perform by using indices.

The following fields can benefit of this notation: Linear Algebra, Representation Theory, Group Theory, Differential Geometry.

This notation can naturally be related to Penrose's diagrammatic notation.

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I agree that abstract index notation was a brilliant observation. However, the problem with tensor calculus in general and (abstract) indices in particular, is that it does not conceptualize the calculations being performed. This can be very pragmatic, as a tensor expression admits multiple interpretations, but it can also be confusing. For instance a contraction could be indicating composition, an inner product, or a trace, while skew symmetrization could be indicating a wedge product or a Lie bracket. – David MJC Oct 22 '10 at 21:07
@David MJC: That's one reason because I like it. Some operations like those you mentioned are, from a viewpoint, indeed "equivalent". We can see this notation like a forgetful functor, or as a polymorphism in object oriented programming. It introduces a higher layer of abstraction, which allows us to see relations between apparently unrelated things. You are right that it hides some concepts, but it reveals others. At a much smaller scale it is like Category Theory, exposing hidden connection and general patterns. But I do not prefer it over using operators, I view them somehow complementary. – Cristi Stoica Oct 23 '10 at 11:02

The notation $\perp$ to denote either orthogonality, or to indicate independent random variables, or perhaps even to indicate relatively prime numbers.

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Wait, if $n \perp m$ means that n and m are relatively prime, then does $(n,m)$ represent their inner product? Far out... – Ryan Reich Oct 20 '10 at 20:19
;-) --- anyhow, it seems that Knuth advocates the $\perp$ notation for relative primeness on p.126 of Concrete Mathematics, a book wherein several other pieces of great notation are described by him. – Suvrit Oct 20 '10 at 21:38
With Knuth's proposal to write $k \perp n$ for k relatively prime to n' we can write by the Moebius inversion formula $\sum_{1 \le k \le n} \left[ k \perp n \right] = \sum_{1 \le k \le n} \left[ k \mid n \right] \mu\left(\frac{n}{k}\right).$ Note that by the conventions of the Iverson bracket when the Iverson-bracketed statement is false, it annihilates anything it is multiplied by - even if that other factor is undefined'. – Bruce Arnold Oct 22 '10 at 15:46
@David: I think it's appropriate for the "inner product" to be a (bi-)multiplicative function rather than an additive one, since "orthogonality" is a multiplicative concept. Note: "far out" was a term of appreciation! – Ryan Reich Oct 22 '10 at 22:03
There is also a nice use of $\bot$ in logic to mean absurdity, i.e. the negation of $\top$. – Jon Beardsley Oct 15 '12 at 19:28

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} dx dy = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$

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David, you forgot the $dydx$ on the left. – Chandan Singh Dalawat Oct 16 '12 at 3:30
Thanks Chandan. – David Speyer Oct 16 '12 at 3:41
This can also be done by writing $\int_x^{2\pi} dx\, \sin x$ etc., which is a little bit shorter. I have often seen this in physics. – rem Jan 24 '13 at 18:32
Do you still need the $dx dy$ on the left if you specify the integration variable in the left part of the symbol? – Federico Poloni Jul 9 '13 at 6:42
@Federico: the "$dx$" is very expressive in integrals for a variety of reasons. It makes clear that what you are integrating is actually a differential form, which is useful even in one dimension: for example $\int f(u)du=\int f(u(t))u'(t)dt$, and $\int_{t=a}^b udv=uv|_a^b-\int_{t=a}^b vdu$. Also, in Physics it makes physical quantities add up correctly ($\int F\cdot ds$ for the work done by a force makes it clear that work is measured in units of force times units of length). – Qfwfq Dec 25 '13 at 0:41

I really like the arrow notation for limits: $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0} 1.$$ I've seen people use this on the blackboard, but I don't think I've seen it in print. The right-hand side of an arrow expression can be decorated with a "+" or "-": $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0^+} 1^-.$$ Arrow expressions can be treated as propositions (e. g., $x\rightarrow 0$ implies $\frac{\sin(x)}{x}\rightarrow 1$), but this is usually less succinct than the stacked arrows. However, it's easier to chain limits this way:

If $f$ and $g$ are continuous [in the sense of elementary calculus], then so is $f\circ g$: if $a$ is fixed and $x\rightarrow a$, then $g(x)\rightarrow g(a)$ (since $g$ is continuous), so $f(g(x)) \rightarrow f(g(a))$ (since $f$ is continuous), QED.

This can be made rigorous, say, with nonstandard analysis, although there are probably more elementary ways.

Sometimes, we need to use a limit as a subexpression in a formula, rather than just stating that the limit equals something. For this, I like the notation $f(x)|_{x\rightarrow a}$ in favor of $\lim_{x\rightarrow a}f(x)$. To me, it's an obvious and intuitive extension of the notation $A(x)|_{x=a}$, which is commonly used to denote the expression that results when $x$ is replaced by $a$ in the expression $A(x)$ (in which $x$ occurs free).

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Interesting... could you tell where you saw this first? – J. M. Oct 27 '10 at 8:44
@J. M. - actually, I can't recall. :-( It's possible that I use it more than others, but I'm certainly not the inventor. I probably haven't seen "x -> a" treated as a proposition exactly as I did above, but it's basically equivalent to "x~a" (in nonstandard analysis, to mean that the difference between x and a is smaller than all standard positive reals). – Darsh Ranjan Oct 28 '10 at 5:35
As an undergraduate in Cape Town I saw and used "$\frac{\sin x}{x} \rightarrow 0$ as $x \rightarrow 0$" almost as often as the $\lim$ notation. – Max Nov 14 '10 at 11:33
This notation was always present in my education (in Bremen) and is especially popular in functional analysis where there is $\to$ for strong convergence and $\rightharpoonup$ for weak convergence. – Dirk Nov 25 '11 at 7:18
@Max, I hope your post-undergraduate education has taught you a different value for that limit! – L Spice Jun 14 '12 at 19:23

As a freshman, I "invented" the notation

$H \lhd ! \; G$

to say that $H$ is a characteristic subgroup of $G$, i.e. a subgroup invariant under any automorphism of $G$ (whereas a normal subgroup $N\lhd G$ is only invariant under the inner automorphisms).

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I know some people absolutely DESPISE using coordinates and components to do "tensor analysis", but sometimes there is no recourse, and then Einstein's summation convention is a big help.

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Have you ever seen Penrose's notation? It seems crazy at first, and I've never used it but it actually seems like a pretty good compromise between "abstract" notation and co-ordinates. – Deane Yang Oct 21 '10 at 1:55
I love both Penrose's notation (which is to a first approximation the same as the string diagrams mentioned in another answer) and the Einstein summation convention. Neither of them solves all problems; each is sometimes more efficient than the other. (In particular, I find sums of several terms get ugly quickly in string diagrams. So for proving an associativity, I'll typically try string diagrams first, but for proving a Jacobi identity, I'll go for Einstein summation.) – Peter LeFanu Lumsdaine Oct 21 '10 at 3:15

$G \circlearrowleft X$ (or $G \circlearrowright X$) to denote that $G$ acts on $X$.

Edit by A.H. :
Here are some latex definitions that produce the symbol that David Speyer describes in his comment:

\def\acts{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(7,6)(4.5,8.3)(2,7)
\qbezier(2,7)(-1.5,4)(2,1)
\qbezier(2,1)(4.5,-.3)(7,2)
\qbezier(7,6)(6.1,7.5)(6.8,9)
\qbezier(7,6)(5,6.1)(4.2,4.4)
\end{picture}
\hspace{.1cm}
}}


and

\def\acted{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(1,6)(3.5,8.3)(6,7)
\qbezier(6,7)(9.5,4)(6,1)
\qbezier(6,1)(3.5,-.3)(1,2)
\qbezier(1,6)(1.9,7.5)(1.2,9)
\qbezier(1,6)(3,6.1)(3.8,4.4)
\end{picture}
\hspace{.1cm}
}}

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I prefer $G \curvearrowright X$. – Mark Oct 20 '10 at 20:34
I use a notation like that, but I have not found it in latex: the closest I can find is $G\subset X$ but with the $\subset$ turned into an arrow (so that $G$ appears as the label on the arrow, you could say) I distinctly dislike using \circlearrowleft for this! :) – Mariano Suárez-Alvarez Oct 20 '10 at 20:35
In hand writing, I prefer a roughly 3/4 circle which starts at about 4 oclock and ends at about 2 oclock. That way the arrow starts at X and returns there. Important note: if draw the arrow going the other way, people may mistake it for a capital G. – David Speyer Oct 20 '10 at 22:36
I agree with David Speyer. You want the arrow to begin and end at $X$ and go by way of $G$. (Not that this answers the original question, which was to come up with notation that substantially improves understanding...) – JBorger Oct 21 '10 at 1:39
\lcirclearrowright in MnSymbol looks ok. – Hsien-Chih Chang 張顯之 Oct 22 '10 at 1:22

To denote an action $\alpha: G\times X \rightarrow X$ of a group $G$ on a space $X$, there is the nice piece of notation:

$\alpha: G \curvearrowright X$

or simply

$G \curvearrowright X$ (the latter when the action is understood from the context).

E.g. you can say something like: $\rm{GL}(V) \curvearrowright V$ linearly. Or, to say that $W$ is an invariant subspace for $G \curvearrowright V$, you just write: $G \curvearrowright W$.

Another example: $\rm{Ad}:G \curvearrowright \mathfrak{g}$, and so on.

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I prefer for the arrow to be a nearly-full circle, coming from X back to X. Warning: if you put the arrow on the lower part instead of upper, it looks like you've written G twice but in two different fonts. – Allen Knutson Oct 21 '10 at 2:31

I like to interpret $f(x)$ as meaning $f\circ x$, otherwise known as the pullback $x^*f$. For instance $x$ could be the standard real valued coordinate on a line. This makes rigorous sense of the concept of a "variable" and hence also dependent and independent variables ($y=f(x)$). In the example of functions on a line, $f'=dy/dx$ is simply a ratio of 1-forms.

Such an interpretation also answers the common complaint that $f=f(x)$ confuses a function with its values. Instead it represents the very common shorthand of omitting pullbacks!

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Did you invent it your-self? If not did you see this notation in some books? – Anton Petrunin Nov 8 '10 at 1:00
Any book which describes a function f as a relation y=f(x) between dependent and independent variables is using this notation, so in that sense I didn't invent it. Also for mathematicians such as E.Cartan, points were always variable, i.e., functions on some unspecified parameter space: $x\in X$ means "x is a function on an unspecified domain with values in X", which is Grothendieck's "functor of points". However, I don't have a reference for my interpretation, and it does generate a laugh (e.g. in a colloquium) to say that confusing a function and its values amounts to omitting pullbacks. – David MJC Nov 8 '10 at 22:00
This one is nice both because: a) it makes sense of the reverse-Polish notation for functions, $f(x) = xf$, and b) it makes sense of the term "random variable" in probability, which it took me a long time to understand the meaning of. – Ryan Reich Nov 25 '11 at 1:54
This point of view is espoused in Munroe's 1958 AMM article "Bringing calculus up to date" (jstor.org/stable/2308879). He gives a very little bit of history, mostly without references. – L Spice Jun 28 '12 at 22:01

As mentioned in a comment, $\lfloor x\rfloor$ is much better notation than $[x]$ for denoting the greatest-integer function. Most especially since it doesn't collide with the $10^6$ other things that $[]$ is used for, e.g. the $0,1$ function Richard Borcherds mentioned.

I very much like, though haven't had much use for, the notation $n{q\atop \cdot}$ for $|GL_n(q)/B|$, pronounced "$n$ $q$-torial". Famously, it extends to a polynomial function of $q$, and when $q=1$ we have $n{1\atop \cdot} = n!$

(Oops: I left out the $/B$ the first time, thanks Jim and David.)

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Allen, that's not quite right about the cardinality of $\mathrm{GL}_n(q)$. For instance, when $q=1$ you get $0$. (Maybe you want the number of points on $G/T$?) – JBorger Oct 21 '10 at 6:44
I'm pretty sure G/B is what Allen wants. – David Speyer Oct 22 '10 at 13:40

Since the standard notation for open interval $(a,b)$ can be confused with the coordinates, gcd, and other stuffs (open brackets have been used A LOT!), I've seem notations like

$]a,b[$

occurred in the book "Elementary Classical Analysis" by Marsden, and we can denote half-open half-closed interval like this:

$]a,b]$ or $[a,b[$.

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I have to -1 this one. Nothing annoys me more than seeing $[x,y[ \cup ]a,b[$ and such. – Quadrescence Oct 21 '10 at 4:36
I was a fan of this notation until I read Quadrescence's comment. – Ryan Reich Oct 21 '10 at 6:30
The point is that $]a,b[$ is the French way to write the open interval (this explains Harry's comment). It is still taught in high school, and students never learn about $(a,b)$, even at university. Only researchers adapt to this notation once they write in English. I agree that $]a,b[$ is clearer. – Denis Serre Oct 21 '10 at 6:48
@Ryan Reich: Quadrescence's example is only annoying when improperly typeset. Look at the difference in LaTeX between $[x,y[\cup]a,b[$ [x,y[\cup]a,b[ versus $\left[x,y\right[\cup\left]a,b\right[$ \left[x,y\right[\cup\left]a,b\right[. – Willie Wong Oct 21 '10 at 17:22
I prefer Knuth's notation, which uses $(a.\,.b)$ for the open interval and $[a.\,.b]$ for the closed one. – Zsbán Ambrus Oct 22 '10 at 19:38

$f_*$ and $f^*$ for direct and inverse image. We really should use this right from the beginning, for functions $f\colon X\to Y$, where $f_*\colon P(X)\to P(Y)$ ($P(X)$ being the power set) and $f^*\colon P(Y) \to P(X)$ instead of the awful notations $f(A)$ and $f^{-1}(B)$ for subsets $A$ of $X$ and $B$ of $Y$.

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Why do you think $f(A)$ and $f^{-1}(B)$ are awful? – Anton Petrunin Nov 9 '10 at 0:32
I would say they are extremely confusing for beginners. Sometimes they think there is an inverse map $f^{-1}$. – Leo Alonso Nov 9 '10 at 15:40
Per Leo, it is indeed confusing for beginners, and not surprisingly so: f is being used both for a function from X to Y and for a function from P(X) to P(Y). This can even be ambiguous as well as confusing e.g., when the domain is a set such as {0,{0}}. And I further agree that the notation $f^{-1}$ for inverse image is likely to suggest there is an inverse function. It is but a small consolation that when there is an inverse function, the notations are self-consistent (because $f^*=(f^{-1})_*$ in that case). – David MJC Nov 9 '10 at 20:36
I have also seen $f^{\rightarrow}$ and $f^{\leftarrow}$. – Max Nov 14 '10 at 11:39
I like $f^\rightarrow$ and $f^\leftarrow$ better, since when applying the inverse, it looks like: $(f^{-1})^\rightarrow = f^\leftarrow$. – Hsien-Chih Chang 張顯之 Dec 18 '10 at 15:25

I found the notation $K_\bullet$ for a complex (in with objects an abelian category or as an objects of the derived category) is very helpful. Otherwise people have to write something like $\cdots \to K_{n}\to \cdots \to K_{2}\rightarrow K_{1} \to K_{0}$ which just contains exactly the same amount of information.

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One can decorate a subscript or superscript by additional symbols to indicate what the subscript or superscript is doing. For instance, consider a truncation $f 1_{|f| \leq N}$ of a function to its values whose magnitude is at most $N$. One could of course call such a function something like $f_N$, but why not call it $f_{\leq N}$ instead? Then one can do things like "Decompose $f = f_{\leq N} + f_{>N}$, where $f_{\leq N} := f 1_{|f| \leq N}$ and $f_{>N} := f 1_{|f| > N}$." Notation of this type is sometimes used in PDE, particularly with regard to Littlewood-Paley frequency projections.

Similarly, one could imagine the operation of shifting $f$ by $N$ to be denoted something like $f_{+N}$ rather than $f_N$, etc..

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Hum, I would prefer if $f 1_{|f| \leq N}$ be written as $f|_{\leq N}$ to make clear that it is a spatial cutoff. $f_{\leq N}$ suggest so me a vertical cut-off: $=f$ if $|f| \leq N$ and $=N$ otherwise. This could be a nice notation when doing Calderon-Zygmund... – Willie Wong Oct 21 '10 at 17:41
I'm just making this up, but it seems you could be more consistent with the projection $[x^n]$ as in $[x^n]f(x)$ by doing something like $[\le N]f$: "Decompose $f=[\le N]f + [>N]f$", or if you're daring, "decompose $f=([\le N] + [>N])f$." – Quadrescence Oct 21 '10 at 17:42

Here is a notation in algebraic geometry that in my opinion is very useful and self-explanatory but not used widely.

For a birational morphism $f:X\to Y$ there exists an open dense set $U\subseteq Y$ for which $f$ induces an isomorphism $f^{-1}U\to U$. For a closed subset $Z\subseteq Y$ such that $Z\cap U\neq\emptyset$ the strict transform is defined as $$\overline{f^{-1}(Z\cap U)}\subseteq X,$$ i.e., the closure of the preimage of the part of $Z$ that lies on the part where the morphism is an isomorphism. This is a very important construction and there isn't a universally accepted notation for it.

János Kollár invented the following notation for this: $$f^{-1}_*Z:= \overline{f^{-1}(Z\cap U)}\subseteq X$$ The genius of the notation is that anyone familiar with basic notation in algebraic geometry should understand what it is:

1) As $f$ is birational, $f^{-1}: Y\dashrightarrow X$ exists as a rational map.

2) For any map $g$, it is common to use $g_*$ to denote push-forward of cycles.

The strict transform is really just the push-forward of cycles via the rational map $f^{-1}$.

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## protected by François G. Dorais♦Jul 9 '13 at 16:41

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