Question

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 Σ_1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δ_n 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 A^B for the set of functions from B to A is really confusing, and I find it much easier to write this set as B→A.

• 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.)

Answer 1

$\sin^{-1}(x)$ as opposed to $\text{arcsin}(x)$. This encapsulates the fact that it is an inverse function.

Answer 2

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

Answer 3

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

Answer 4

$G \circlearrowleft X$ (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}
}}

Answer 5

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"...

Answer 6

The three-dot notation $f\mathrel{\scriptsize\vdots}A\to B$ to indicate that $f$ is a partial function from $A$ to $B$, meaning that $\text{dom}(f)\subseteq A$ rather than $\text{dom}(f)=A$. Partial functions are pervasive in logic, especially computability theory and set theory, and this notation is both compact and suggestive.

Answer 7

$a \vee b$ 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.)

Answer 8

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 $i \in \lbrace 1,2,\dots,n\rbrace$.

Answer 9

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

Answer 10

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: $$\mathbf 1_{n_k\in \{n: n\text{ prime}\}}.$$

Answer 11

$D_j f$ 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.

Answer 12

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.

Answer 13

I use the notation

$V \oplus^{\perp}W$

to denote orthogonal direct sum [Edit: direct sum of, say, subspaces of a given inner-product space].

Or

$(M,g) \times^{\perp} (N,g')$, or simply $M \times^{\perp} N$, to denote (orthogonal) cartesian product of Riemannian manifolds.

Answer 14

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).

Answer 15

If one needs to denote the fiber (not the stalk which is standardly denoted $\mathcal{F}_{x}$) of a sheaf $\mathcal{F}$ at the closed point $x$ of the $\Bbbk$-scheme $X$, one can write

$\mathcal{F}\mid_{x}$

After all, the fiber $\mathcal{F}\otimes_{\Bbbk}\;\kappa (x)$ is the restriction (pullback) of $\mathcal{F}$ to the point $x:\rm{Spec}\;\Bbbk\rightarrow X$.

The problem is that, when you identify vector bundles with locally free sheaves, the above notation clatches with the usual notation $E_x$ for the fiber of vector bundles. On the other hand almost always the context would be sufficient to clarify which of the two notations is being used.

Answer 16

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.

Answer 17

I recently saw the following notation in the context of divisors on algebraic varieties, and I liked it very much.

Suppose that $D$ and $E$ are reduced divisors on a normal algebraic variety $X$. One can use $D \vee E$ to denote the reduced divisor with support equal to $D + E$ and $D \wedge E$ to denote $(D + E) - (D \vee E)$. I could imagine variants on this if $D$ and $E$ are non-reduced (involving taking max's, respecitvely mins, of the divisors component-wise).

EDIT: I'm slightly curious as to why this was downvoted. I guess it's too common to be interesting?

Answer 18

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 $[a,b[$.

Answer 19

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.)

Answer 20

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 $\alpha=(\alpha_1,\dots,\alpha_n)$.

Answer 21

Diagrammatic notation for tensors (Penrose diagrams, birdtracks, etc.). It makes many things like the invariance of tr(A B C) under cyclic permutation into empty statements.

Answer 22

Bourbaki dangerous bend symbol to mark dangerous or difficult ideas.

Answer 23

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.

Answer 24

The lack of a nice obviously symmetric notation for $\binom{a+b}{b}$ has bothered me; Dijkstra suggested in EWD 782 the notation $P(a,b)$, generalizing it also to $P(a_1,\ldots,a_k)$ for $\binom{a_1+\ldots+a_k}{a_1,\ldots,a_k}$. (Though I certainly disagree with him about $\binom{n}{k}$ being useless - you certainly do want to think about it that way a lot of the time.) I haven't actually had any reason to use this since I saw it but I can certainly think of times I would have.

Also the double-parentheses multichoose notation $\left(\!\binom{n}{k}\!\right)$ is nice because it lets you say "...and this is n multichoose k (which is equal to this binomial coefficient)" instead of just jumping directly to a binomial coefficient whose relevance may not be immediately obvious. But I suppose that's not really on the level of giving you a better way to look at things.

Answer 25

Using $(a, b, ... )$ is handy to denote a column vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.

Answer 26

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.

Answer 27

The notation for transversality:

$M \pitchfork N$

Answer 28

The ever-controversial reverse Polish notation for functions: $f(x) = xf$. Thus in composition, the order makes sense: $(g \circ f)(x) = x f g$ (this point is moot for the fortunate Hebrew- and Arabic-speaking mathematicians). I hate this notation in practice but I can't deny that it is objectively right and "just makes sense" in more or less the same way that the original post discusses writing $B^A = A \to B$. Please no one vote this up.

Answer 29

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.)

Answer 30

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..