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

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 2

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 3

Bourbaki dangerous bend symbol to mark dangerous or difficult ideas.

Answer 4

$D_j f$ 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 5

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

Answer 6

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

Answer 7

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 8

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 9

The notation for transversality:

$M \pitchfork N$

Answer 10

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

Answer 11

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 12

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 13

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 14

$a \vee b$ 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 15

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

Answer 16

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 17

The abstract index notation.

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.

Answer 18

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: $(p^* )^* = \frac{(2p)^*}{2} \neq p$.

Answer 19

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 20

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).

Answer 21

$G \circlearrowleft X$ (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 22

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 23

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 24

$f_*$ 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 $Y$.

Answer 25

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 26

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..

Answer 27

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 and 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 $f^{-1}$.

Answer 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

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

Answer 29

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 30

Dirac's bra-ket notation

This notation is very useful when applying the Hilbert spaces in Quantum Theory. It exploits some properties of duality, eigenvalues/eigenvectors, projectors and self-adjoint operators. In mathematics, perhaps it is difficult to adopt, because mathematicians are using notations that are more general, and cannot exploit these particularities. But if you know Hilbert spaces you can learn this notation in one minute, and then it makes visible many of these nice properties.