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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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    $\begingroup$ In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. $\endgroup$ Oct 20, 2010 at 20:54
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    $\begingroup$ I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... $\endgroup$ Oct 20, 2010 at 23:18
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    $\begingroup$ Yes, among other things. Also $A^B\times A^C=A^{B+C}$, where $+$ is disjoint union. But all the great reasons for it don't help for our mind thinking that maps start with the source and end with the image, not the other way round. $\endgroup$ Oct 20, 2010 at 23:20
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    $\begingroup$ Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. $\endgroup$ Oct 21, 2010 at 3:29
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    $\begingroup$ 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. $\endgroup$ Oct 21, 2010 at 3:38

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Multi-factorials are handy. Sometimes results can be expressed compactly by introducing a double factorial or possibly higher factorial. For example

$$\int_0^{\pi/2} \sin^{2n+1} \theta \:\: d\theta = \frac{(2n)!! }{ (2n+1)!!}$$

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    $\begingroup$ I wish there were a notation that didn't scream "iterated factorial", though (not that one sees this very much). I forget: does $n?$ mean anything? The question mark is handy because it suggests having to make a choice, as in "even or odd?" $\endgroup$
    – Ryan Reich
    Nov 16, 2010 at 10:48
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    $\begingroup$ The question mark has a meaning in C and programming languages derived from C. The notation a ? b : c; means to do b if a is true, otherwise do c. The question mark also means "optional" in regular expressions. For example, the regular expression ab?c matches abc or ac. I don't know whether either of these notations would make sense imported into math. On a related note, sometimes I would like to import C's % operator into math notation. $\endgroup$ Nov 16, 2010 at 23:30
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    $\begingroup$ True, but C also doesn't have a factorial operator, and ! means something entirely different again. There's not much reason to make mathematical notation agree with programming design choices. As for %, we always have "mod". $\endgroup$
    – Ryan Reich
    Nov 29, 2010 at 9:08
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    $\begingroup$ The problem with "mod" is that it is usually an equivalence relation and not a function. That is, you see "a equiv b mod m" more than "a mod m". I'm not sure the latter is common notation or that people agree in detail what it means. $\endgroup$ Nov 29, 2010 at 15:22
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I think that Inuit numerals are cool. (http://en.wikipedia.org/wiki/Inuit_numerals) They are useful for vigesimal type things.

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    $\begingroup$ Now I wonder where in mathematics you would like to do actual vigesimal calculations? $\endgroup$ Dec 19, 2011 at 10:44
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In type theory, the notation $(x : A) \to B (x)$ instead of $\Pi_{x : A} B (x)$ for the dependent product and $(x : A) \times B (x)$ instead of $\Sigma _{x : A} B (x)$ for the dependent sum. For non-type-theorists, a dependent product is a set-indexed product of a collection of sets, and a dependent sum is a set-indexed disjoint union of a collection of sets[1]. The former notation is used in the Agda, the latter is less widely spread but both are used for example in cubicaltt. Some advantages of this notation are:

  • It is consistent with notation for non-dependent sums and products $A \times B$ and $A \to B$ (the latter is in fact an instance of the third example of good notation given in the original question, and is already widespread among type-theorists as a notation for the set of functions from $A$ to $B$)

  • The fact that $A$ does not appear as a subscript certain complex types easier to follow, as in the equation

$$ (x : A) \times ((y : B (x)) \times C (x, y)) \cong (u : (x : A) \times B (x)) \times C (\pi_1 u, \pi_2 u) $$

  • The fact that $A$ does not appear as a subscript makes it easier to represent this notation in plaintext, as is frequently done when programming with dependent types.

[1] Here and in the rest of my answer, I am ignoring the distinction between a set and a type.

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Cauchy-Binet as a generalized Pythagoras theorem.

Let $X$ be an $ n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1 < ... < i_k \le n$, there is some advantage to denote by $X_I$, the determinant of the $k \times k$ submatrix of $X$ with rows indexed by $I$. For any two such $X,Y$, we can state the Cauchy-Binet formula as a pairing $$ \det (X^TY)= \sum_{I} X_I Y_I $$ where the sum is over all $n \choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since it says that the the volume-squared of the parallelepiped spanned by the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of the volume of the projections on the $n \choose k$ $k$-dimensional coordinates.

For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices $I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times k$ submatrix of $A$ with rows indexed by $I$ and column indexed by $J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by Cauchy-Binet twice, $$ \det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where $A^I$ is the $k \times n$ matrix given by the rows of $A$ indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and $(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.

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    $\begingroup$ This is not really about notation, but it's a very good point. $\endgroup$ Nov 2, 2010 at 0:09
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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.

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  • $\begingroup$ I'd like to know why this was downvoted: you simply don't like the symbol, or there's some deeper reason? $\endgroup$
    – Qfwfq
    Oct 23, 2010 at 22:13
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    $\begingroup$ I didn't downvote it, but I don't like the notation because it is redundant and emphasizes the wrong thing. For vector spaces, there is no intrinsic inner product with respect to which a direct sum can be either orthogonal or not. If you have two inner product spaces, then the direct sum is always orthogonal unless specified otherwise, because there is no one way to do it otherwise. Likewise for the direct product of two manifolds with a Riemannian metric. It is better to have a notation for when the sum or product is not orthogonal, and to specify how. $\endgroup$
    – Ryan Reich
    Nov 16, 2010 at 11:00
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    $\begingroup$ I have seen that some authors use the notation $V\obot W$ for this purpose (provided by mathabx package in latex), see tex.stackexchange.com/a/61882/39306. $\endgroup$
    – Name
    May 27, 2015 at 17:50
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Instead of writing $$|x-y|\le \varepsilon,$$ I used to write $$x\lessgtr y\pm \varepsilon.$$ You may read it as $x$ is more-or-less $y$ plus-minus $\varepsilon$.

One may also write something like $$x\lessgtr e^{\pm\varepsilon}\cdot y$$ which is much better than $$|\ln(y/x)|\le\varepsilon$$

It is easier to read, especially if instead of $x$ and $y$ you have long expressions.

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    $\begingroup$ What would you write for $|x-y|\geq \epsilon$? $\endgroup$ Jan 24, 2013 at 23:24
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    $\begingroup$ I'm not happy about the "principle" in Anton's reply to Nate's comment. In the body of the question, the upper inequality (with $<$ and $+\varepsilon$) and the lower inequality (with $>$ and $-\varepsilon$) are to be understood as combined by "and", whereas in the comment, the upper and lower inequalities are intended to be combined by "or". Allowing both uses of the notation seems to be inviting confusion. $\endgroup$ Jan 26, 2013 at 16:22
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    $\begingroup$ Andreas, you are right, I made this reply without much thinking :) $\endgroup$ Jan 30, 2013 at 19:10
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    $\begingroup$ Actually, I like $x =^\epsilon y$ for this concept. $\endgroup$
    – Lee Mosher
    Feb 2, 2013 at 23:43
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    $\begingroup$ @LeeMosher, as a $p$-adic analyst I like this, but surely those poor folks for whom $=^\epsilon$ is not transitive will not be so happy. :-) $\endgroup$
    – LSpice
    May 18, 2015 at 19:44
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In computations in differential geometry, when we need to omit one entry of a list, I like to write $x^{\hat\imath}$ instead of $(x_1,\dots,\hat{x}_i,\dots,x_n)$, and similarly write $x_{\hat\imath}$ instead of $(x^1,\dots,\hat{x}^i,\dots,x^n)$. This makes the Einstein convention work, and simplifies the computations of relations between exterior and Lie derivatives of differential forms, for example. Similarly, $x^{\widehat{\imath\jmath}}$ to omit two entries. Careful that $x^{\hat\imath\hat\jmath}\ne x^{\widehat{\imath\jmath}}$

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    $\begingroup$ Is $x^{\hat\imath\hat\jmath} \ne x^{\widehat{\imath\jmath}}$ because of the index shift, e.g., starting with $x = (x_1, x_2, x_3)$, we get $x^{\hat1\hat2} = (x_2)$ but $x^{\widehat{12}} = (x_3)$? Could this be remedied simply by not shifting the indexing, so that, e.g., we still call $x_2$ the $2$nd (not the $1$st) entry of $x^{\hat1} = (x_2, x_3)$, or does that come with separate disadvantages? $\endgroup$
    – LSpice
    May 19, 2021 at 12:55
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    $\begingroup$ @LSpice: exactly the problem; I am not sure what would be the best approach, thinking of sign problems that arise when we work with permutations of vectors entering differential forms. $\endgroup$
    – Ben McKay
    May 19, 2021 at 13:44
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    $\begingroup$ I don't know how it'll work for you, but I've often found it helpful explicitly to "reify" the indexing. That is, an element $x \in \mathbb R^n$ isn't just an $n$-tuple, but a function $\{1, \dotsc, n\} \to \mathbb R$; so, for $x \in \mathbb R^3$, we have that $x^{\hat2}$, though a $2$-tuple, is not a function $\{1, 2\} \to \mathbb R$ but a function $\{1, 3\} \to \mathbb R$. (This convention works best if, for natural $n$, you let $n$ stand for $\{0, \dotsc, n - 1\}$, so that $\mathbb R^n$ becomes literally a function space … but $0$-based indexing confuses many!) $\endgroup$
    – LSpice
    May 19, 2021 at 19:33
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UMBRAL NOTATION (courtesy of Blissard)

$$(q.)^n = q_n$$

Concise, elegant, and suggestive, it often allows for both brevity and comprehensibility of presentation and short derivations of operational results.

Examples of umbral variables:

$q.^n = q_n = d$, a constant quantity, maybe a single variable or polynomial,

$q.^n = q_n = d^n $, a constant quantity raised to powers,

$q.^n = q_n = x^n$, a variable raised to a power,

$q.^n = b.^n = b_n$; an element of a sequence of numbers, such as the Bernoulli,

$q.^n = B.(x)^n = B_n(x)$, an element of a sequence of polynomials, .e.g., Bernoulli,

$q.^n = (.)!^n = (n)!$, an expression containing an integer parameter,

$q.^n = \binom{x}{.}^n = \binom{x}{n} $, ditto,

$q.^n = M.^n = M_n$, a sequence of matrices, differential ops, ... whatever.

The umbral maneuver in more complex expressions:

  1. reduce an expression that contains the umbral entity to an analytic series in monomial powers of $q.$ when possible

  2. then lower the positive integer exponent to a subscript

  3. when in doubt apply

$$e^{q.xD_y} \; f(y) \; |_{y=0} = f(q.x +y) \;|_{y=0} = f(q.x) \;$$

  1. sensible results can often be obtained for functions not analytic at the origin from

$$f(q.x) := e^{-(1-q.)x D_y} \; f(y) \; |_{y=x} = f(x-(1-q.)x) = f((1-(1-q.))x) $$

(See below on Newton interpolation. E.g., with $f(x)=x^s$, then $f(q.x) = (q.x)^s = q_s\; x^s \; := (1-(1-q.))^sx^s \; $ when convergent.)

Basic umbral operations:

  1. binomial convolution for commuting quantities:

$$c_n = c.^n = (a. + b.)^n = \sum_{k=0}^n \binom{n}{k} \; a.^k \; b.^{n-k} = \sum_{k=0}^n \binom{n}{k} \; a_k\; b_{n-k},$$

  1. formation of e.g.f.s and o.g.f.s:

$$E(t) = e^{q.t} = \sum_{n \geq 0} \; q.^n \; \frac{t^n}{n!} = \sum_{n \geq 0} \; q_n \; \frac{t^n}{n!}$$

and, with $q_0 =1$,

$$O(t) = \frac{1}{1-q.t} = \sum_{n \geq 0} (q.t)^n = \sum_{n \geq 0} q_n \; t^n = e^{\bar{q}.t}= \bar{O}(t)$$

with $\bar{q}_n = n! \; q_n$.

Then multiplication of formal series is

$$EA(t) \; EB(t) = e^{a.t} \; e^{b.t} = e^{(a.+b.)t} = e^{c.t} = EC(t) \; .$$

We can even disregard convergence of both series if we are interested only in formal compositional or multiplicative inverses or the formal Laplace transform of an e.g.f. to an o.g.f. since the production of the $n$-th coefficient of the target series depends only on the coefficients of order $n$ or less of the source series for these transformations.

Brevity and intelligibility in umbral versus conventional presentations:

The Euler transformation for e.g.f.s in umbral notation

  1. $$ e^{a.x} = e^x \; e^{-(1-a. )x}$$

versus conventional notation

  1. $$ \sum_{n \geq 0} a_n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} \; a_k \; ] \; \frac{x^n}{n!} \; ,$$

and derivation of the conventional formula via the umbral,

  1. $$ e^{a.x} = e^x \; e^{-x} e^{a.x} = e^x \; e^{-(1-a. )x}$$

$$ = e^x \; \sum_{n \geq 0} (-1)^n \; (1-a.)^n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} \; (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} a_k \; ] \; \frac{x^n}{n!} \;.$$

(23 keys tapped for eqn. 1 versus 102 keys for 2, not counting the appreciable difference in taps for blank spacing and latex spacing for readability. I could have saved a good fraction of a lifetime of math work by being able to use umbral notation in symbolic apps, study, and publications rather than sigma gymnastics.)

Natural generalizations via umbralization--monic to partition polynomials:

Stirling polynomials of the first kind, related to cyclic permutations, have the e.g.f.

$$e^{St1.(x)t} = e^{x \ln(1+t)} = (1+t)^x\; ,$$

and Stirling polynomials of the second kind, related to partitions of sets,

$$e^{St2.(x)t} = e^{x (e^t-1)} \, .$$

Flexibility and precision is afforded if some notation is used to indicate at what level the umbral exponent lowering is to occur. For example, with $\langle \cdots \rangle$ denoting evaluation of the expression between the angular brackets after reduction to a series in the umbral quantity and with $q_0=1$,

$$\langle \; e^{-\ln(1-q.t)} \;\rangle = \langle \frac{1}{1-q.t}\rangle = \langle \sum_{n \geq 0} q.^n \; t^n \; \rangle = \sum_{n\geq 0} \langle q.^n \rangle \; t^n =\sum_{n \geq 0} q_n \; t^n $$

and

$$e^{-\langle \ln(1-q.t)\rangle} = \exp[\langle \sum_{n \geq 1} q.^n \frac{t^n}{n}\rangle] = \exp(\sum_{n \geq 1} q_n \frac{t^n}{n}) ,$$

which is the e.g.f. for the cycle index partition polynomials of the symmetric groups, a.k.a. the Stirling partition polynomials of the first kind, OEIS A036039.

The natural generalization of the Stirling polynomials of the second kind is via the umbralization, for $q_0 = 1$,

$$e^{\langle e^{q.t}-1 \rangle} = \exp[ \; \sum_{n \ge 1} \; q_n \frac{t^n}{n!} \; ], $$

giving the complete Bell polynomials of the Faa di Bruno composition formula, a.k.a. the Stirling partition polynomials of the second kind, A036040.

Including an extra variable as in

$$e^{x\langle e^{q.t}-1 \rangle} = e^{Bell.(q_1,..,q.;x)t}$$

and

$$e^{-x\langle\ln(1-q.t)\rangle} = e^{Cyc.(q_1,..,q.;x)t}$$

allows better tracking of block sizes and combinatorics within each partition polynomial. (Note $q_n$ could be a commuting sequence of wedge products, curvature forms, or matrices, just as well as integers, giving basic formulas for the Chern and Pontryagin characteristic class polynomials.)

Now for a little umbral mojo,

$$ e^{St1.(St2.(x)) \; t} = e^{St2.(x) \; \ln(1+t)}=e^{x \; (\exp(\ln(1+t))-1)} = e^{xt} \; ,$$

so the two sets of binomial Sheffer polynomial sequences are an inverse pair under umbral composition; i.e.,

$$St1_n(St2.(x)) = x^n = St2_n(St1.(x)) \; .$$

This is reflected in their lower triangular coefficient matrices being a matrix inverse pair. The same umbral compositional inverse and matrix relationships hold for any Sheffer polynomial sequence. For a pair of inverse/reciprocal Appell Sheffer sequences, this is due to their moment e.g.f.s being multiplicative inverses rather than compositional inverse. The generic Sheffer sequence has a mix of these conditions, belonging to the semidirect product of the Appell and binomial Sheffer subgroups.

Actions of diff op often become more transparent with umbral calculus and/or underlying combinatorics are revealed:

With $a.$ treated as independent of $x$ and $D = d/dx$ for $f(x)= e^{b.x}$,

$$e^{a.D_x} \; f(x) = f(a.+x) = e^{b.(a.+x)}.$$

For the Bernoulli number and polynomials (or any Appell Sheffer sequence),

$B_n(x) = e^{b.D} \; x^n = (b.+x)^n$,

$D \; B_n(x) = D \; (b.+x)^n = n \; (b.+x)^{n-1} = n \; B_{n-1}(x)$

$e^{a.D} \; B_n(x) = B(x + a.) = (x+b.+a.)^n = (B.(x)+a.)^n = (B.(a.)+x)^n \; .$

Adopt the additional notational convenience $(:AB:)^n := A^nB^n$. Alternatively, we could use $q.^n = q_n = A^nB^n$, but this would veil aspects/intuitions inherited from the properties of the derivative.

From our previous feat of umbral mojo,

$$St1_n(xD) = n! \; \binom{xD}{n} = ST1_n(ST2.(:xD:) = \; :xD:^n \; = x^nD^n,$$

consistent with the actions

$xD \; [x^n] = n x^n$,

$(1 + a.)^{xD} \; [x^n] = (1 + a.)^{n} \; x^n$,

and

$(1 + a.)^{xD} \; f(x) = \sum_{n \geq 0} a_n \; \binom{xD}{n} \; f(x) = e^{a.:xD:} \; f(x) = f((1+a.)x).$

The operational definition of the Stirling polynomials of the second kind is

$$(xD)^n = (St2.(:xD:))^n = St2_n(:xD:).$$

From this the e.g.f. given above for the polynomials can easily be derived, and, consequently,

$$e^{txD} \; f(x) = e^{t \; St2.(:xD:)} \; f(x) = e^{(e^t-1):xD:} \; f(x)$$

$$ =f((e^t-1)x + x) = f(e^t x),$$

a dilation, consistent with $e^{txD} \; x^n = e^{tn} \; x^n$.

The Lah polynomials $L_n(x)$ (the normalized, unsigned Laguerre polynomials of order -1, related to partitions of sets into ordered sets or lists) have several useful Stokes-Rodrigues diff op reps,

$$ x \; :Dx:^n \; x^{-1} = x \; D^n x^n x^{-1} = x^{-n+1} (xDx)^n x^{-1} = x^{-n} (x^2D)^n = L_n(:xD:) ,$$

and the e.g.f.

$$e^{L.(x)t} = e^{x \frac{t}{1-t}},$$

so

$$ e^{tx^2D} \; f(x) = e^{t \; :xL.(:xD:):} \; f(x) = e^{:\frac{tx^2}{1-tx}D:} \; f(x)$$

$$= f(\frac{tx^2}{1-tx} + x) = f(\frac{x}{1-tx}), $$

a special linear fractional, or Moebius, transformation.

(The inverse ops to the translation, $e^{tD}$; dilation, $e^{txD}$; and special linear fractional, $e^{tx^2D}$, transformations of SL2 are given by negating $t$, which is equivalent to additive ($t+(-t)=0$), multiplicative ($e^t \cdot 1/e^t =1$), and compositional inversion ($\frac{x}{1-tx}|_{x \to\frac{x}{1+tx}}=x$), respectively. SL2 becomes associated to the underlying combinatorics of the binomial Sheffer polynomials $St2_n(x)$ and $L_n(x)$ via the umbral symbolic calculus.)

Interpolation methods and results of integral transforms are naturally suggested and expressed in umbral notation:

Laplace transform of e.g.f. to o.g.f.s with $q_0=1$:

$$\int_{0}^\infty \; e^{q.xt} \; e^{-t} dt = \int_{0}^\infty \; e^{-t(1-q.x)} dt = \frac{1}{1-q.x}$$

Mellin transform interpolation to continuous indices (related to Ramanujan's master formula, e.g., the Bernoulli polynomials are essentially discrete samples of the Hurwitz zeta function):

$$ q_{-s} = (q.)^{-s} := \int_{0}^\infty \; e^{-q.t} \; \frac{t^{s-1}}{(s-1)!} \; dt$$

Newton interpolation to continuous indices:

$$q_s = (q.)^s = (1-(1-q.))^s = \sum_{m=0}^\infty \; (-1)^m \; \binom{s}{m} \;\sum_{k=0}^m \; (-1)^k \; \binom{m}{k} \; q_k$$

$$= \int_{0}^\infty \; e^{-(1-(1-q.))t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt= \int_{0}^\infty \; e^{-t} \; e^{(1-q.)t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

$$= \int_{0}^\infty \; e^{-q.t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

via analytic continuation. If $q_n = D_x^n$, the action of the integral on $H(x) \frac{x^\alpha}{\alpha!}$, where $H(x)$ is the Heaviside step function, gives a classic fractional calculus in terms of an analytically-continued (AC) Mellin convolution (essentially that for the AC Euler beta function integral) for the interpretation of $D_x^{s}$ for real or complex $s$.

See Boole, Blissard, Sylvester, Cayley, Appell, (Jensen?), Heaviside, Steffensen, Sheffer, Pincherle, Bell, Riordan, Rota, Taylor, Roman, Ray, and Lenart for contributions to and influences on the development of the umbral symbolic calculus.

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When I teach topology, I write $A\stackrel{\textrm{o}}{\subset} X$ to mean that A is an open subset of the topological space $X$, and I write $A \sqsubset X$ to mean that $A$ is a closed subset of $X$. I think it would drive me crazy to write the words open and closed constantly.

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  • $\begingroup$ I believe this notation (or something similar) is due to Chan-Ho Suh, who used it in his homework in a course Marshall Cohen taught at Cornell, and then Marshall started using it in his classes. $\endgroup$
    – Dan Ramras
    Dec 19, 2021 at 5:08
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    $\begingroup$ One could do something more symmetrical for closed sets, like $A\stackrel{\textrm{c}}{\subset} X$, but I suspect it would be easy to mistake c for o or vice versa, maybe especially in handwriting. $\endgroup$
    – Dan Ramras
    Dec 19, 2021 at 5:08
  • $\begingroup$ Personally I put a small circle / small diagonal dash on the lower bar of the $\subset$ to denote an open / closed subset. One can do the same for inclusion maps $\hookrightarrow$, this is a relatively common notation in algebraic geometry for open / closed immersions. $\endgroup$ Dec 19, 2021 at 9:59
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To resolve the ambiguity when using multiple instances of $\pm$ within a single expression, I use $\pm_n$ and $\mp_n$. $\pm_a$ must be equal to $\pm_b$ iff $a=b$.

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Since this one is on the front page again: In my personal notes, I have started writing sums/integrals over complicated index sets as $\sum \left( \text{summand} \mid \text{condition} \right)$, rather than a subscript. EG $$\sum {\large (} \log p \mid p \ \text{prime}, p \equiv 1 \bmod 4,\ p \leq N {\large )}$$ instead of $$\sum_{\substack{p \ \text{prime} \\ p \equiv 1 \bmod 4 \\ p \leq N}} \log p.$$

I find it a lot more readable. Additional benefits (1) it is reminiscent of set builder notation like $\{ \log p \mid p \ \text{prime}, p \equiv 1 \bmod 4,\ p \leq N \}$ (2) it means that the pairing between differential forms and cycles actually looks like a bilinear pairing.

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In algebra, it is very useful to write $J\cap A$ for the inverse image of an ideal $J$ in a ring $B$ under a homomorphism $f:A\to B$, rather than $f^{-1}(J)$. I normally also omit the morphism and write $IB$ for the ideal generated by the image in $B$ of an ideal $I$ in $A$, rather than $f(I)B$.

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I find the standard notation $X \times_G Y := \frac{X \times Y}{G}$ for balanced products of $G$-spaces annoying, because it conflicts with the well-established notation for fibre products. But on the other hand writing out the full quotient $(X \times Y)/G$ is cumbersome, especially if you're dealing with associated bundles to principal bundles, or if you have spaces with compatible left and right actions of different groups, and you need to write something like $X \times_G Y \times_H Z$.

So I came up with that following notation which combines the product symbol "$\times$" and the quotient symbol "$\_$" into a single binary symbol: enter image description here

The last one, without the group decorating it, is used in the same way as tensor products $\otimes$ when the ring involved is obvious (e.g. if you're working with associated bundles to a $G$-principal bundle $P$, and you're not planning on taking any reductions of the structure group).

It's just a rotated semidirect product, so it's easy to implement in LaTeX, and looks similar enough to the usual notation that it's easy to introduce in a talk. Here's my LaTeX code:

enter image description here

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    $\begingroup$ I learned from Indranil Biswas to use $X \times^G Y$ for $(X \times Y)/G$. I think this notation is becoming standard. $\endgroup$
    – Ben McKay
    Jul 15, 2017 at 10:26
  • $\begingroup$ I also think the notation $X\times^G Y$ is the standard one. $\endgroup$
    – Qfwfq
    Jan 14, 2018 at 0:49
  • $\begingroup$ Unfortunately not standard in differential geometry yet---everywhere I look people are still using the ambiguous fibre-product notation. $\endgroup$
    – ಠ_ಠ
    Jan 14, 2018 at 13:09
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Many textbooks say that the normal distribution is the one whose density is $$ x\mapsto \frac1{\sqrt{2\pi}} \times \frac1\sigma \exp\left( -\frac12 \left( \frac{x-\mu}\sigma \right)^2 \right) $$ and the gamma distribution is the one whose density is $$ x\mapsto \frac1{\Gamma(\alpha)} \times \lambda^\alpha x^{\alpha-1} e^{-\lambda x} \text{ for } x>0$$ and the Pareto distribution is the one whose density is $$ x\mapsto \frac{\alpha\kappa^\alpha}{x^{\alpha+1}} \text{ for } x>\kappa $$ etc.

I prefer to say that the normal distribution is $$ \frac1{\sqrt{2\pi}} \times \exp\left( -\frac12\left( \frac{x-\mu}\sigma\right)^2 \right)\, \frac{dx}\sigma $$ and the gamma distribution is $$ \frac1{\Gamma(\alpha)} \times (\lambda x)^{\alpha-1} e^{-\lambda x} (\lambda \, dx) \text{ for } x>0 $$ and the Pareto distribution is $$ \frac\alpha{(x/\kappa)^{\alpha+1}} \, \frac{dx}\kappa \text{ for } x>\kappa, $$ and so on.

This shows that each scale parameter, or its reciprocal, the rate parameter or intensity parameter, goes with an $\text{“} x\text{,”}$ and suggests a substitution to be used when thinking about integrating.

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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?

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    $\begingroup$ It is not me , I do not know algebraic variety, but as a guess may be there is some order there and so you have an inf and sup or even a lattice that would fully justify these notation and be rather common at that. $\endgroup$ Nov 23, 2010 at 23:54
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Has anyone come across any more similar examples of good notation that should be better known?

Some interesting glyphs:

  1. Combinatorial Principles in Set Theory:

  2. Bisimulation:

    • Given two states p and q in S, p is bisimilar to q, written p ~ q, if there is a bisimulation R such that (p,q) is in R.
  3. Boxplus operator in Coding Theory

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    $\begingroup$ I always considered 1. to be a prototypical example of bad notation. $\endgroup$ Dec 19, 2011 at 13:27
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    $\begingroup$ @EmilJeřábek, I think that (2) has to be a still more prototypical example, in the sense that, without additional context, no two people will agree which equivalence relation is denoted by $\sim$ (although I guess we all agree it is an equivalence relation?). $\endgroup$
    – LSpice
    Jul 7, 2019 at 12:31
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I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).

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    $\begingroup$ It looks like a typo. $\endgroup$ Jan 28, 2018 at 0:37
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The notation of $u \mathbin{[\wedge]} v$ for the product of Lie-algebra valued differential forms (I prefer this to another variant $[u \wedge v]$).

Writing the tangent natural tranformation as $\tau: T \rightarrow \operatorname{Id}$ which expands to $\tau_M : TM \rightarrow M$.

I've seen some category theorists write $\operatorname{Hom}_A[a,b]$ for the homset for a category $A$, which gets a bit long-winded when $A$ is a category with a long name like $\mathrm{Ring}$ or $R\text-\mathrm{Mod}$. It seems much more natural to write $A[a,b]$.

Personally, I like $\bar{\mathbb{R}}$ to write the order completions of the reals. It's less clumsy than $\mathbb{R} \cup \infty$.

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    $\begingroup$ Unfortunately, $u [\wedge] v$ u [\wedge] v by itself doesn't space well; you have to manually tell TeX you've still got a binary operator with $u \mathbin{[\wedge]} v$ u \mathbin{[\wedge]} v. I edited accordingly. $\endgroup$
    – LSpice
    Dec 17, 2021 at 21:53
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Many years ago, I invented the notation $\mathcal{Lex}_{T}(X)$ as the function equal to $1$ if and only if the consideration of $X$ in the theory $T$ entails no contradiction and $0$ otherwise. Lex is both "law" in Latin and a shortcut for "Logical existence", which in some sense is the only law of mathematics.

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    $\begingroup$ Why the down vote? Could someone familiar with the subject please explain. $\endgroup$
    – Ben McKay
    Jul 15, 2017 at 10:27
  • $\begingroup$ I'm afraid nobody works on this. I got interested in the subject more than 15 years ago, after a friend of mine starting to study maths told me something like "the integers of the empty set are even" and "the integers of the empty set are odd" were both equally true. I felt shocked and tried to figure out a way to forbid the consideration of "integers of the empty set" and other impossible concepts. $\endgroup$ Jul 15, 2017 at 10:37
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    $\begingroup$ Isn't this just $\operatorname{Con}(T \cup \{X\})$ or am I misinterpreting? $\endgroup$
    – user76284
    Oct 27, 2019 at 0:12
  • $\begingroup$ It seems $\operatorname{Con}$ is a unary predicate, while $\mathcal{Lex}$ is a map with values in $\{0,1\}$. But otherwise, the idea is the same. $\endgroup$ Oct 27, 2019 at 9:32
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I like the notation $f:A\cong\subseteq B$ for "$f$ is an embedding of $A$ into $B$." The idea is that the relation of embeddability is obtained by composing the relations "isomorphic to" and "substructure of."

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  • $\begingroup$ Hmm, others seem to disagree. What do you think of $\hookrightarrow$? $\endgroup$
    – David Roberts
    Nov 24, 2011 at 23:11
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    $\begingroup$ @David: I tend to use $\hookrightarrow$ for maps that are literally inclusions. $\endgroup$ Nov 25, 2011 at 4:00
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    $\begingroup$ I've mostly seen people use $\subset$ or $\subseteq$ for literal inclusions and $\hookrightarrow$ for embeddings (or whatever kind of injection is suitable). Reserving $\hookrightarrow$ for literal inclusions seems kind of pointless when $\subset$ exists. $\endgroup$ Nov 25, 2011 at 8:34
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    $\begingroup$ ... and then you can decorate the arrow with o or | to incorporate meaning like "open immersion" or "closed immersion". $\endgroup$ Dec 19, 2011 at 10:43
  • $\begingroup$ @KonradVoelkel, you use $\overset|\hookrightarrow$ for a closed immersion? Is that standard? $\endgroup$
    – LSpice
    Dec 17, 2021 at 1:59
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$x\gtrless 0$ to denote that $x\in\mathbb{R}\setminus\{0\}$. Or, if $x\notin\mathbb{R}$ but rather $x\in\mathbb{Z}\setminus\{0\}$ for example, then we write $x\in\mathbb{Z}_{\gtrless 0}$. I prefer the $\gtrless$ notation because it is simpler and more understandable when first met with the eye, and I hope it is set in stone sooner or later.

I also came across the following notation: $$\sum_p'f(x).$$ The little dash $'$ on top of the sigma $\Sigma$ denotes that $p$ is prime. I like this, but I don't feel this is necessary because we can write $$\sum_{p \ \text{prime}}f(x)$$ but when looking at both conventions, the former looks a lot neater.

Also, we have notation $>$, $\gg$ and $\ggg$ with $<$, $\ll$, and $\lll$ but I was thinking, is there a notation to denote that a value $a$ is not much less than a value $b$? I came up with $$a\overline{<} b\tag*{$a$ is not much less than $b$}$$ $$a\overline{>}b\tag*{$a$ is not much greater than $b$}$$

I have also seen that the symbol for concatenation is sometimes $||$, but if I was to see $$A \ || \ B$$ then my first thought would be $A$ is parallel to $B$, but that is just me. With some research, I discovered that there is however some kind of official notation, such that $A^\frown B$ but I don't like it.

I also went extreme and invented this: $$\mathop{\LARGE\Omega}_{\substack{x=k \\ \\ R}}^{x_0} (x, y)$$ to define a set of coordinates; a relation. Here, $x\to x_0$ and $R$ simply denotes the rule, $y = \cdots$. This way, we can write stuff like $$\mathop{\LARGE\Omega}_{\substack{x=36 \\ \\ y = 2x+1}}^{144}(x, y)$$ And then if we don't have a rule, but something like $y\geq0$ for example, then we can write a double index to refer to what value $y$ tends towards, namely $y_0$.

Oh and I almost forgot, I thought that maybe we can symbolise contradiction? For instance, I want to prove that $\sqrt{2}$ is irrational. I would first suppose it is rational, then come to a conclusion that contradicts this statement. Since most of us use $\Box$ in notation of completing a proof, I decided that I could use $\Diamond$ in notation of a contradiction. I thought about it because it is like a titled box $-$ a bit like approaching the proof $\Box$ on a different angle $\Diamond$, if you get what I am saying.

I am not taking this too seriously, for I was just being creative, but I believe there's nothing wrong in trying new things out. Any thoughts?

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    $\begingroup$ Re: your second-to-last paragraph, there are multiple symbols used for proof-by-contradiction - my personal favorite being "$\perp$," but "$\Rightarrow\Leftarrow$" is also common, as is some form of "lightning bolt." I would strongly object to "$\Diamond$," however, given the meaning of $\Diamond$ and $\Box$ in modal logic. $\endgroup$ May 13, 2019 at 18:08
  • $\begingroup$ Is your $\Omega$ thing any different than $ \{(x,y) \colon 36 \leq x \leq 144, y=2x+1\}$? $\endgroup$ May 13, 2019 at 19:18
  • $\begingroup$ @ZachTeitler nah, it's the same thing. The $\Omega$ thing is easier to write, but the "normal way of writing it" is easier to read ;) $\endgroup$
    – Mr Pie
    May 13, 2019 at 21:30
  • $\begingroup$ Instead of typing A\ ||\ B so that you see $A\ ||\ B,$ if you type A\parallel B then you see $A\parallel B.$ Is there anyone to whom that doesn't look better? $\endgroup$ Jan 21, 2022 at 2:59
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