# 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 $A$. –  Andres 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 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. –  Kallikanzarid Oct 21 '10 at 3:38
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If $$\underset{_{\text{k}\,=\,0}}{\overset{n}{\mathbf\Xi}}\ \Bigg(a_{_\text{k}}\ ,\ b_{_\text{k}},\ \frac1{N_{_\text{k}}}\Bigg)\ =\ \sqrt[^{N_{_\text{0}}}]{a_{_\text{0}}\ +\ b_{_\text{0}}\sqrt[^{N_{_\text{1}}}]{a_{_\text{1}}\ +\ b_{_\text{1}}\sqrt[^{N_{_\text{2}}}]{\ldots\ \sqrt[^{N_{_{n}}}]{a_{_{n}}}}}}$$ then $$\underset{_{\text{k}\,=\,0}}{\overset{n}{\mathbf\Xi}}\ \Big(a_{_\text{k}}\ ,\ b_{_\text{k}},\ -1\Big)\ =\ \cfrac1{a_{_0}\ +\ \cfrac{b_{_0}}{a_{_1}\ +\ \cfrac{b_{_1}}{\ddots\ {a_{_n}}}}}$$ Also, $$\underset{_{\text{k}\,=\,0}}{\overset{n}{\mathbf\Xi}}\ \Big(a_{_\text{k}}\ ,\ 1,\ 1\Big)\ =\ \sum_{k\,=\,0}^n a_{_\text{k}} \qquad\qquad;\qquad\qquad \underset{_{\text{k}\,=\,0}}{\overset{n}{\mathbf\Xi}}\ \Big(0,\ a_{_\text{k}}\ ,\ 1\Big)\ =\ \prod_{k\,=\,0}^n a_{_\text{k}}$$ and $$\underset{_{\text{k}\,=\,0}}{\overset{n}{\mathbf\Xi}}\ \Big(a_{_\text{k}}\ ,\ x,\ 1\Big)\ =\ \sum_{k\,=\,0}^n a_{_\text{k}}\ x^k\ =\ P_n(x) .$$ etc.

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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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$$a^{\cdot \, n} = a\cdot a\cdots a$$ $$a^{\wedge \, n} = a\wedge a\wedge\dots\wedge a$$ $$a^{, \, n} = a,a,\dots,a$$ For example one could write $$\langle(x+10y-z)^{, \, 2}\rangle= \langle(x+10y-z),(x+10y-z)\rangle.$$ or $$\sin^{\circ(-1)}x=\arcsin x$$ or $$\sin^{\cdot(-1)}x=\frac1{\sin x}$$

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In contrast, the insistence of Calculus textbooks to use $sin^{-1}$ for inverse or arc sine has baffled me. It throws even my best students off as being some kind of reciprocal, especially since we are already stuck with the idiosyncratic notation for powers like $sin^2(x)$ –  Hans Schoutens Jan 24 at 22:15
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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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What would you write for $|x-y|\geq \epsilon$? –  Nate Ackerman Jan 24 at 23:24
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. –  Andreas Blass Jan 26 at 16:22
Andreas, you are right, I made this reply without much thinking :) –  Anton Petrunin Jan 30 at 19:10
Actually, I like $x =^\epsilon y$ for this concept. –  Lee Mosher Feb 2 at 23:43
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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
Knuth's <i> x to the n falling </i> is $x^{\underline{n}}$, not $x_{\underline{n}}$. –  Bruce Arnold Oct 21 '10 at 13:50
I don't like the cis(θ) notation because it depends on the non-canonical choice of the square root of 1 (i.e., i). The exp(iθ) notation is better because the non-canonical choice is clearly visible. –  Dmitri Pavlov Oct 21 '10 at 15:51
@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
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In commutative algebra with many variables, repeating lists of variables in polynomial arguments and various rings gets very tedious. I suggest using $X_{1..n}$ instead of $X_1,\ldots,X_n$.

Here's an excerpt from Bourbaki's Commutative Algebra, page 222:

For every formal power series $f\in A[[X_1,\ldots,X_n]]$, $$f(X_1,\ldots,X_n)-f(Y_1,\ldots,Y_n)=\sum_{i=1}^n (X_i-Y_i)h_i(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$$ where the $h_i$ belong to $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

And here's how it looks with my suggested notation:

For every formal power series $f \in A[[X_{1..n}]]$, $$f(X_{1..n})-f(Y_{1..n})=\sum_{i=1}^n (x_i-Y_i)h_i(X_{1..n},Y_{1..n})$$ where the $h_i$ belong to $A[[X_{1..n},Y_{1..n}]]$.

I think this notation is maximally succinct, and helps a reader from getting lost in long lists of variables.

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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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If $\mathcal{C}$ is a category and $X,Y\in\mathrm{obj}(\mathcal{C})$, I like the notation $\mathcal{C}(X,Y)$ to denote $\mathrm{Hom}_{\mathcal{C}}(X,Y)$.

So, $\mathcal{C}(X,X)=\mathrm{End}_{\mathcal{C}}(X)$.

What do you think of the notation $\mathcal{C}(X):=\mathrm{Aut}_{\mathcal{C}}(X)$ ?

This would be consistent with the notation (or similar notations) $\mathsf{DIFF}(S^1)$ (resp. $\mathsf{TOP}(S^1)$ ) for diffeomorphisms (resp. homeomorphisms) of the circle, i.e. the $\mathrm{Aut}$ in the category $\mathsf{DIFF}$ of smooth manifolds (resp. $\mathsf{TOP}$ of topological manifolds), sometimes used in topology (see e.g. here and here. And (see e.g. here) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.

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Is there any particular reason you've chosen to make your links single letters, instead of pointing them out more explicitly? –  S. Carnahan Dec 10 '12 at 3:25
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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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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 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 at 6:42
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All of the notations created to simplify writing category theory. For instance, the idea of drawing a circular arrow inside of a diagram to indicate that that diagram is commutative. As well as the idea of putting an angle in the top left or bottom right of a square diagram to indicate that it is a pushout or pullback. And finally, the notation of augmenting any of these notations with $\simeq$ to indicate that the diagram is only "up to homotopy".

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Has this already been mentioned? If a group $G$ acts on a commutative group $A$ by homomorphisms, $G \to Aut(A)$, then use $a^g$ to denote the action. Especially if the group multiplication on $A$ is written multiplicatively, where we can say things like $(ab)^g = a^g b^g$. This can come up especially in Galois theory; I remember Lang using this notation in his Algebra to prove Hilbert's Theorem 90, and I thought it was very neat, and enhanced the readability of notation as well.

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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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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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I always considered 1. to be a prototypical example of bad notation. –  Emil Jeřábek Dec 19 '11 at 13:27

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
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
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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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@David: I tend to use $\hookrightarrow$ for maps that are literally inclusions. –  Andreas Blass Nov 25 '11 at 4:00
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. –  Ketil Tveiten Nov 25 '11 at 8:34
... and then you can decorate the arrow with o or | to incorporate meaning like "open immersion" or "closed immersion". –  Konrad Voelkel Dec 19 '11 at 10:43
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This is probably the exact opposite of what the thread starter intended, but here's an instance where it might have been useful to have overloaded notation! I recently found that in propositional logic, $p \to q$ obeys much the same rules as exponentiation $q^p$: for instance, we have $(r^q)^p = r^{q \times p}$, and similarly, $p \to (q \to r)$ iff $p \land q \to r$. This is apparently due to the universal property for exponential objects, as applied to a Boolean algebra viewed as a poset category. I suppose it's also an instance of the Curry—Howard correspondence.

More generally, it seems like it isn't such a bad idea to conflate exponentiation and arrows - it looks nicer, to me at least, to write that a function of the type $A \to (B \to C)$ is naturally isomorphic to a function of the type $A \times B \to C$, than to write about $A \to C^B$. Even, as some have suggested, $A \to {}^B C$ or $C^B \leftarrow A$ would look nicer. On the other hand we'd lose the association with cardinal arithmetic if we do this...

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

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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
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A) Two notations I love are the rising factorial $x^\overline n$ and its falling factorial twin $x^\underline n$. They are used and advocated in the great book see http://en.wikipedia.org/wiki/Concrete_Mathematics . In passing this book uses great notations.

B) A general trick with binomials to reuse them with sets instead of numbers, here are some typical examples.

1) $\binom S k$ to denote the set of all $k$-sets of the base set $S$ .

2) $S^\underline 2$ to denote the pairs $(x,y)$ of $S$ where $x$ and $y$ are different.

3) $S^\underline k$ to denote the $k$- uplets of $S$ (each uplet has $k$ different elements).

C) Another notation I find useful when listing some (big) families of examples in a combinatorial setting. Use as variables the very numerals $1$ $2$ .. themselves instead of $x_1$ , $x_2$ ... . For example ( very untelling because too small an example) : the intersection of $123$ and $34$ is $3$.

D) I also often use {{ a,a,b,c}} for multiset. Any other standard or suggestion (or a way to avoid speaking about multiset) is welcome.

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D) Perhaps {$(a)^2, b, c$} for your multiset {{a,a,b,c}} –  Qfwfq Sep 24 '11 at 21:04
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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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$f_*$ 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
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I like $A^{\text{H}}$ for the conjugate transpose of the matrix $A$, ananlogously to how $A^{\text{T}}$ and $A^{\text{C}}$ means the transpose and the conjugate. You call it the Hermitian of the matrix for short. I learnt this notation from Rózsa Pál, but I can't tell who invented it.

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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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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?" –  Ryan Reich Nov 16 '10 at 10:48
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. –  John D. Cook Nov 16 '10 at 23:30
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". –  Ryan Reich Nov 29 '10 at 9:08
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. –  John D. Cook Nov 29 '10 at 15:22

The universal property of the univariate polynomial ring: For any commutative ring $A$, any commutative $A$-algebra $B$ and any $x\in B$, there exists one and only one $A$-algebra homomorphism from the polynomial ring $A\left[X\right]$ to $B$ which maps $X$ to $x$.

This is the so-called evaluation homomorphism at $x$. I denote this homomorphism by $\lim\limits_{X\to x}$. This has the advantage that we have $\lim\limits_{X\to 0}\dfrac{\left(X+1\right)^n-1}{X}=n$ and similar properties hold just as in classical analysis. The polynomial $\dfrac{\left(X+1\right)^n-1}{X}$ is well-defined (since $X$ is not a zero divisor in $A\left[X\right]$ and divides $\left(X+1\right)^n-1$), but if we would blindly replace $X$ by $0$ we would obtain a $\dfrac{0}{0}$ error.

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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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This is not really about notation, but it's a very good point. –  darij grinberg Nov 2 '10 at 0:09

(This would be a comment on notation for partial functions, but I don't have the reputation points, as I just joined MO.) Though this is by no means standard, for personal use I've adopted the following system of arrow decorations that captures many standard types of binary relations. For a relation f from A to B, use $\rightharpoonup$ to indicate $\forall x\in A ~\exists y \in B~~xfy$, $\rightharpoondown$ to indicate $\forall x\in A~\exists^{\leq 1} y\in B~~xfy$, $\leftharpoondown$ to indicate $\forall y\in B~\exists x\in A~~xfy$, and $\leftharpoonup$ to indicate $\forall y\in B~\exists^{\leq 1}x\in A~~xfy$. So, $\rightarrow$ is for functions, $\leftrightarrow$ is for bijections, $\leftharpoonup\hspace{-1em}\to$ is for injections, $\leftharpoondown\hspace{-1em}\to$ is for surjections, $\rightharpoondown$ is for partial functions, $\rightharpoonup$ is for serial relations, and so on.

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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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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
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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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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. –  Jérôme JEAN-CHARLES Nov 23 '10 at 23:54

In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f$ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.

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

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