# Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any "$E=mc^2$-titles", as I think of them—how Einstein in retrospect might have entitled his 1905 paper (instead of "Zur Elektrodynamik bewegter Körper"!)—paper/book titles composed entirely of math symbols.

There are two close misses in the responses to that MO question: Connes et al.'s "Fun with $\mathbb{F}_{1}$", and Taubes's "${\rm GR}={\rm SW}$: Counting curves and connections." The only title entirely composed of math symbols with which I'm familiar is the delightful book A=B, by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger. Can you identify others?

Please interpret this question in a weekend-recreational spirit! :-)

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If Chaitin came out with a book called $\Omega$, that would be the last word in "$E=mc^2$-titles (sorry, couldn't help myself). –  David Roberts Jul 11 '11 at 5:55
On "Fun with $\mathbb{F}_1$" it's worth noticing that the French for "1" is "un".. :) –  domenico fiorenza Jul 11 '11 at 14:45
The "most memorable titles MO question" was only secondarily a request for examples of titles, but that secondary question was the only one that people answered, until after a large number of such answers had appeared. I think I posted at least two such examples that were favorable viewed, but then I posted something that was closer to the primary thrust of the question. I was severely and in fact abusively taken to task for not staying on topic, by someone who would have known that I was in fact on topic if he had read the question. –  Michael Hardy Jul 11 '11 at 19:13
.....and the design of the software, and to some extent the culture, of MO, tacitly encourages such abuses. –  Michael Hardy Jul 11 '11 at 19:13
A few months ago, I ran across a preprint whose title consisted two simple figures separated by an equals sign. Of course, now I've forgotten the authors. –  JeffE Mar 31 '12 at 18:21

$SL_2(\mathbf{R})$ (link)

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@David: Great example! "$SL_2(R)$ gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example--$SL_2(R)$." –  Joseph O'Rourke Jul 11 '11 at 0:54
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Ha! Has their conjecture stood fast these past dozen years? (Abstract: "We conjecture that 7,373,170,279,850 is the largest integer which cannot be expressed as the sum of four nonnegative integral cubes.") –  Joseph O'Rourke Jul 11 '11 at 0:53
I guess "the largest number not expressible as" offers several opportunities... –  Joseph O'Rourke Jul 11 '11 at 1:07
Let me explain why this precise problems are considered: By a classical result of Hilbert (solving Waring's problem) every nonnegative integer is a sum of a fixed number of k-th nonnegative integral powers. (The fixed depends of course on the k). One can now ask what is the best 'fixed' for a given k. It turns out that small numbers cause most problems and one gets by (for given k) with a smaller number of k-th powers, if one just wants all sufficiently large integers as a sum. Now, this raises the question, what is the 'sufficiently large'. See en.wikipedia.org/wiki/Waring's_problem –  quid Jul 11 '11 at 1:25
This gets my third vote. Gerhard "Email Me About System Design" Paseman, 2011.07.11 –  Gerhard Paseman Jul 11 '11 at 17:48
@Joseph: I think so, see oeis.org/A022566 . –  Charles Dec 20 '11 at 1:20

!

(Title of a talk about the factorial function by Manjul Bhargava at the Clay conference in Paris in the year 2000.)

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Along this line, Doron Zeilberger gave a talk for an REU program about nothing‌​, and his title was, well, nothing at all. (Not "nothing". But nothing.) Unfortunately, I am not sure if for this exercise of Joseph's, whether "consisting of mathematical symbols" requires the subset to be non-trivial. –  Willie Wong Jul 11 '11 at 11:47

Professor Luca and his co-authors are surely fond of this kind of titles:

• F. Luca & B. de Weger, $\sigma_k(F_m)=F_n$. New Zealand J. Math. 40 (2010), 1–13.

• F. Luca & F. Nicolae, $\phi(F_n)=F_m$. Integers 9 (2009), A30, 375–400.

• F. Luca & M. Mignotte, $\phi(F_{11})=88$. Divulg. Mat. 14 (2006), no. 2, 101–106.

• F. Luca & P. Stănică, $F_1F_2F_3F_4F_5F_6F_8F_{10}F_{12}=11!$. Port. Math. (N.S.) 63 (2006), no. 3, 251–260.

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"Two concepts which are often used in the theory of partial differential equations and the calculus of variations are the so-called $H$ spaces and $W$ spaces." –  Joseph O'Rourke Jul 11 '11 at 10:55

McCarthy, Charles A. $c_p.$ Israel J. Math. 5 1967 249–271.

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MathSciNet link: ams.org/mathscinet-getitem?mr=MR0225140 –  Joseph O'Rourke Jul 11 '11 at 10:57

Would IP=PSPACE count?

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$R(4,5)=25$

B. D. McKay and S. P. Radziszowski, J. Graph Theory, 19 (1995) 309-322.

The title is also the main theorem. $R(4,5)$ is a classical Ramsey number (the one most recently determined exactly).

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210=14*15=5*6*7

I may have the title wrong. It is about the simultaneous solution of some Pell-like equations. I will provide more detail as my memory permits.

Gerhard "Email Me About System Design" Paseman, 2011.07.10

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That's a nice one. I heard of this once, but completely forgot. Nice to be reminded. –  quid Jul 11 '11 at 1:33
Pinter and deWeger, "$210 = 14 \times 15 = 5 \times 6 \times 7 = \binom{21}{4} = \binom{10}{4}$". Publ. Math. Debrecen. 51(1-2) 175-189 (1997). "It is given all the solutions to the Diophantine equations $(y−1)y(y+1)=\binom{n}{4}$ and $x (x+1) = \binom{n}{4}$." –  Joseph O'Rourke Jul 11 '11 at 1:34
–  Goldstern Jul 11 '11 at 9:54

$\int_x^{hx}(g^*\alpha-\alpha)$ (by Kedra and Gal)

http://arxiv.org/abs/1105.0825

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$H_g^1(K,V)=H_{st}^1(K,V)$

An unpublished manuscript by Osamu Hyodo (who passed away untimely).

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$H_8$, by Jacques Martinet.

$GL_n$, by William Casselman.

Both these articles appear in the a book edited by Albrecht Fröhlich: Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 525–538. Academic Press, London, 1977.

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$\Delta=b^2-4ac$, by Jean-Pierre Serre (Math. Medley, Singapore Math. Soc. 13, 1985, 1-10).

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J.-P. Serre also has a book entitled $SL_2$'. No wait, come to think of it the title also mentions Arbres, amalgames', whatever they are. –  shane.orourke Sep 28 '12 at 8:14

$$\left(1+\frac{d}{dz}\right)^{-1}$$

only a preprint, though: http://arxiv.org/abs/1203.3045

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"Pi" (I keep "A source book" in parentheses to hide the non-mathematical part), L. B. Berggren, J. M. Borwein, P. B. Borwein (Eds.).

"Z=60", Conference in Honor of Doron Zeilberger's 60th Birthday (this, of course, is influenced by one of ma favorite titles "$A=B$").

Removed (following the healthy criticism): "2012", a 2009 American science fiction disaster movie.

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+1 for Pi and Zeilberger. And, if works of fiction count, I think one should add 2001 (and 2010, 2061, 3001); let's ignore the odyssey add-ons. –  quid Jul 11 '11 at 13:47
Point taken! Thanks –  Wadim Zudilin Jul 11 '11 at 13:51

Here is $H_\infty\not= E_\infty$, wherein Justin Noel gives an example of an $H_\infty$-structure on a ring spectrum which does not descend from an $E_\infty$-structure.

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&

Yes, this is the title. Just "&". :-)

From Mulvey's homepage: "This paper, presented at the Topology Meeting in Taormina, Sicily in April, 1984, introduced the concept of quantale, outlining the programme of work in the spectral theory of C*-algebras and the constructive foundations of quantum mechanics to which it was expected to contribute. The paper is a slight development of that which appeared in the Tagungsbericht of the Category Meeting at Oberwolfach in September, 1983. It is included here since, although often quoted, it is more difficult to obtain in its published form in the Rendiconti del Circulo Matematico di Palermo. "

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I apologize for a bit of vanity, which, worse yet, is not even a proper example: I nearly published a paper entitled $T^0_2(MSP)=PV_1$, but a referee made me rename it in the final version.

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Ha! I wonder how many other pithy paper titles were quashed by referees... –  Joseph O'Rourke Oct 20 '11 at 19:03
I wanted to publish a paper called "B-pairs and (φ,Γ)-modules" but the editors made me change it on the ground that they did not want too many math symbols in a title. –  Laurent Berger Dec 16 '11 at 11:52

$\Gamma_4=0$

is the subtitle of Jean Cerf's famous lecture notes: Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$. (French) Lecture Notes in Mathematics, No. 53 Springer-Verlag, Berlin-New York 1968 xii+133 pp.

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Thomas Forster's Phd thesis is called N.F.'' On his website he claims that this is the shortest title for a Cambrige maths PhD on record. The abstract is also pretty short.

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$K_1(A,B,I)$

S. Geller, C. Weibel, J. Reine Angew. Math. 342 (1983), 12–34.

$K(A,B,I)$: II

S. Geller, C. Weibel, K-Theory 2 (1989), no. 6, 753-760.

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$K_{i}^{loc}(\mathbb{C})$, $i = 0, 1$, by Nicolae Teleman (link).

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