The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

The Monster Lie Algebra is a rank 2 Borcherds-Kac-Moody Lie algebra, and this implies it has a self-centralizing subalgebra of dimension 2, and decomposes under the adjoint action into a collection of eigenspaces, called root spaces. Here is the decomposition as a -graded vector space:

A brief examination reveals a bit of a mess in the upper left and lower right. Ignoring the mess for the moment, we have a in the middle, and two copies of the complex numbers in the upper left and lower right. This four dimensional vector space is a Lie subalgebra isomorphic to , i.e., we can view it as the Lie algebra of 2-by-2 matrices under conjugation, where the diagonal matrices form the 2-dimensional space in the middle.

The vector spaces in the mess have a symmetry between the degree piece and the degree piece, imposed by the action of the subalgebra . However, we can say more: the root multiplicities are determined by the coefficients of the normalized modular j-invariant . More precisely, when either m or n is nonzero, the degree subspace has dimension , the coefficient of the -coefficient of .

The simple roots of the Monster Lie Algebra span the degree spaces, i.e., those in the column, and they generate the “positive” subalgebra . That is, one has one simple root of degree , 198664 simple roots of degree , 21493760 simple roots of degree , etc. In the diagram below, the red line divides the Lie algebra into positive and negative subalgebras, and the semi-infinite red box contains the simple roots.

The degree simple root is called “real”, while the rest are “imaginary”, because the root space has an inner product where the vector has norm . To generate the full Monster Lie algebra from the simple roots, we follow the standard recipe for Borcherds-Kac-Moody Lie algebras, using quadratic relations, together with Serre’s relations for real roots. This makes the positive subalgebra into a Lie algebra “freely generated over ” by the imaginary simple roots.

The first miraculous fact about the Monster Lie algebra is that we started with simple roots whose multiplicities are coefficients of , and ended up with all roots having multiplicities given by coefficients of . This fact implies an infinite collection of identities relating the coefficients of . For example, an examination of the degree part reveals that , so .

We can do a more systematic version of this examination using the Weyl (-Kac-Borcherds) denominator formula:

which arises from the Chevalley-Eilenberg resolution of the trivial representation of the positive subalgebra. Here, is the Weyl vector , and is an alternating sum of over finite orthogonal subsets of simple imaginary roots that add up to . The -th homology of the resulting complex is the subspace of the -th exterior power of supported in degree satisfying . For the Monster Lie algebra, the Weyl group has order 2 (as it is isomorphic to ), so the contributions to homology are easy to enumerate. If we set to be a basis vector in degree , we have , , , and for . We decategorify by evaluating the Hilbert-Poincaré series, and obtain the Koike-Norton-Zagier identity:

When considering characters, this identity is naturally an identity of formal power series, although we may turn this into a complex analytic identity near infinity by setting and . The product actually converges in the region where the product of the imaginary parts of and is greater than , since that is where is nonvanishing.

This identity has one quite remarkable property, namely the power series on the left is a sum of power series that are pure in and , while the power series on the right appears to be full of mixed terms containing both and . The vanishing of mixed terms on the left is what yields the identities between coefficients of that I mentioned before. In particular, the vanishing of the coefficient on the right is equivalent to .

The Koike-Norton-Zagier identity was proved independently during the 1980s by the three people named, but it seems that none of them bothered to write up a proof. An elementary argument can be given by multiplying both sides by and taking logs – the right side becomes a sum of where is the -th Hecke operator, while the left side is the sum of , where is the unique polynomial in of the form .

There is a higher-level argument using the theory of Borcherds products. Basically, the Borcherds-Harvey-Moore multiplicative theta lift sends to a function on that is invariant under , with zeroes of multiplicity one along the divisors for , and a cusp expansion as an infinite product whose exponents are coefficients of . It therefore suffices to examine the polar part at infinity to identify with , and the product formula yields the term .

I’d like to recapitulate what I said in the beginning about monster actions on the Monster Lie Algebra. Any linear action of a group on the simple root spaces extends naturally to an action by homogeneous Lie algebra automorphisms, so the bare fact that the monster acts is not so special.

However, Borcherds gave an alternative construction of the Monster Lie Algebra that produced a very well-behaved action. Instead of using generators and relations as above, he used a stringy quantization functor, which also goes by the name . This functor takes in a representation of the Virasoro algebra at central charge 26, and produces a vector space. If the representation has a product structure, in particular from a vertex algebra, then the output has a Lie algebra structure. The “cancellation of oscillators” theorem asserts that if the input has the form , where is a unitarizable representation of Virasoro with central charge 24, and is a Fock space for 2 free bosons with momentum , then the output is the weight part of (when ). This was first conjectured by Lovelace in 1971, and proved by Goddard and Thorn in 1972. We typically attach the name “no-ghost theorem” to this result, although the name refers to a somewhat different aspect of their theorem. In particular, the fact that the output space has no negative-norm states (known as ghosts) was a big deal in the early development of string theory.

Borcherds chose to input the tensor product of the Monster Vertex Algebra with the Lorentzian lattice vertex algebra . The lattice vertex algebra is -graded, and each graded piece is a rank 2 free boson. By cancellation of oscillators, we get an identification between graded pieces of the Lie algebra in degree away from and graded pieces of as monster modules. If we write , then the degree piece of the Monster Lie Algebra is identified with .

Here then is the distinguishing property of the monster action on the Monster Lie Algebra: for any , the root space of degree is a monster module whose isomorphism type depends only on the product . That is, the monster representation is constant along hyperbolas .

This property has the following impact: The identities we found relating the coefficients of , or equivalently the dimensions of root spaces, are promoted to relations between monster representations. For example, the identity of coefficients is promoted to a monster module isomorphism . This requires the identification of the monster action on the degree vector space with the action on the degree space, which does not hold for general monster actions. This additional information means that in addition to the ordinary Weyl denominator identity, we have a twisted denominator identity for each element of the monster, and one can use equivariant Hecke operators to organize the terms.

The application to moonshine is the following: if we want to understand the character of an element in the monster acting on , we may use the twisted denominator identities to obtain recursion relations between the traces on graded pieces. For example, the previous monster module isomorphism yields for any element in the monster simple group. The general form of these recursion relations is known as complete replicability, and Koike showed in unpublished work that the candidate moonshine functions listed by Conway and Norton are completely replicable. Once these recursions were in place for characters of the monster action on , Borcherds proved the Monstrous Moonshine conjecture by comparing the graded pieces of the monster vertex algebra to Conway and Norton’s candidate representations for .

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I don’t yet have any of the juicy numbers revealing what libraries are paying for their Elsevier subscriptions (as Timothy Gowers has been doing in the UK; if you haven’t read his post do that first!). Nevertheless there are some interesting details.

Essentially all the Australian universities, excepting a few tiny private institutes, subscribe to the Freedom collection (this is the same bundle that nearly everyone is forced into subscribing to). The contracts are negotiated by CAUL (the Council of Australian University Librarians).

My librarian was very frank about Article Processing Charges (APCs) constituting double-dipping, whatever it is that Elsevier and the other publishers say. The pricing of journal bundles is so opaque, and to the extent we understand it primarily based on the historical contingencies of print subscription levels more than a decade ago, that in practice the fraction of articles in a subscription bundle for which APCs have been paid has no meaningful effect on the prices libraries pay for their bundles.

I think this point needs wider dissemination amongst mathematicians — whatever our complaints about APCs inhibiting access to journals for mathematicians without substantial funding, we are just plain and simple being ripped off. **Gold open access hybrid journals are a scam.**

Now, on to some details about contracts. First, my librarian confirmed the impression from Gowers’ investigations in the UK — bundle pricing is based largely on historical spending on print subscriptions, with annual price increases. Adding some interesting context on the numbers we’re now seeing out of the UK, she told me that the UK is widely perceived as having received a (relatively) great deal from Elsevier, in terms of annual price increases. If the UK numbers scared you, be aware that here in Australia we may well have it worse. A curious anecdote about historical pricing of subscriptions is that one division of CSIRO happened to have cancelled most of their print journals the year before they took out an electronic subscription with a commercial publisher, and as a result got an excellent deal. The Australian universities have apparently mostly signed confidentiality agreements regarding their journal subscription costs (as we expect, by now), but my understanding of the conversation was that the ANU in particular had not.

Finally, my librarian pointed out that doing what I hope to do next, namely use the FOI act to obtain detailed information on Elsevier subscription costs, may be counterproductive, as the most likely result of unusual discrepancies in pricing being revealed is some libraries simply having budgets cut, rather than actually giving the negotiators any more power in the future. I got the impression she’d talked to other Australian librarians about this, and there was some amount of nervousness.

I’ve been told I should go talk to Andrew Wells, the librarian at UNSW, and after posting this I’m going to get in touch with him!

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In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.

Let be some parameter space such as the space of pairs of two conics. Let be some space of solutions, such as the space of triples where is a point on . Let be a map, such as projection onto the components. We want theorems which will discuss the size of the fibers of , in terms of some global degree of the map .

We work over some field . For simplicity of presentation, we’ll assume that is affine, meaning that it is a subset of defined by polynomial equations

We’ll write for the ring .

It would be silly to ask for any such results if were disconnected. A very basic observation of algebraic geometry is that is connected if and only if has no nontrivial idempotents. In fact, we will ask for something stronger: That is an integral domain. The terminology for this is that is **irreducible**. From now on, we will make:

**Assumption** is irreducible. ( is an integral domain.)

If is also affine, with corresponding ring , then is an module. We define the **degree** of in this case to be the dimension of as a vector space. Degree can be defined in much greater generality; we will feel free to refer to it in greater generality without giving the definition. We will denote the degree of by . Roughly, we want theorems which say that the fibers of have size .

Here is our first result.

**Theorem** (Shafarevich, II.6.3, Theorem 4) If has characteristic zero and is algebraically closed then for almost all in . More precisely, there is some polynomial , not identically zero on , so that implies .

**Warning** This isn’t true if is not algebraically closed: Consider the map from .

**Warning** This isn’t true in characteristic : Consider .

We now want results which let us say something, not just about almost all , but about all .

We will at first focus on counting the size of in a naive sense: We think of as sitting in (or in ) and we literally count points of the fiber. We can’t hope for the fibers to always be of full size because even the nicest map, , has fiber of size , not , over the point . So, using the naive size, we can only hope for upper bounds.

There are two additional problems. The first one is if we have something like projecting onto the coordinate. In this case, the degree is but the fiber over has size . When is affine, with corresponding ring , we can fix this by requiring that is torsion free as an -module. In general, the right condition is that no irreducible component of maps to a proper subvariety of .

More subtly, suppose that is a nodal curve, such as , and is its desingularization. (In this case, the line with as the map .) Then the degree of the map is , but the fiber over is , of size . The hypothesis to rule this out is that is integrally closed in its fraction field. By definition, this is the same as saying that is **normal**.

Once we rule out these possibilities, we have

**Theorem** (Shafarevich, II.6.3, Theorem 3) If is normal, and no irreducible component of maps to a proper subvariety of , then every fiber of has naive size .

I can’t resist mentioning a result which far harder than these:

**Theorem** (A consequence of Zariski’s Main Theorem) Let be normal and let have degree . Assume that no irreducible component of maps to a proper subvariety of . For any in , the number of connected components of is at most $d$.

We now consider counting size in a less naive way. Again, for simplicity, suppose that is affine, with corresponding ring . Let be a point of , so there is a map of rings by . Consider the ring , where acts on by the above map. The maps from this ring to are the point in . Thus, is an upper bound for the number of points of above . We will call this dimension the **scheme theoretic size** of the fiber. Once again, it can be defined when is not affine as well.

We have the following cautionary example: Let mapping onto the coordinate. Then the degree is , but the fiber above has size , either scheme theoretically or naively. To rule this out, we impose that is **finite** over . By definition, this means that is affine, and is a finitely generated module.

You might worry about how we could ever prove that is affine if it is not given to us as a closed subset of . Fortunately, we have:

**Theorem** (Hartshorne, Exercise III.11.2) If is projective with finite fibers, then it is a finite map. Here projective means that is a closed subset of , projecting onto . (This is not the morally right definition of a projective map, but if you are ready for the right definition, then you should be working with “proper” rather than “projective” anyway.)

We then have

**Theorem** (Hartshorne, Exercise II.5.8) If is finite over , and no irreducible component of maps to a proper subvariety of , then every fiber of has scheme theoretic size .

**Theorem** Let be a finite map. Then all fibers have scheme theoretic size if and only if is **flat** over .

Unfortunately, flat is a rather technical condition. The first thing to understand is that some nice looking maps can fail to be flat:

**Warning** Let be , let and let the map be . This is a finite map. (We can alternately describe as .) This map is degree , but the fiber over has scheme theoretic size (and naive size ).

If your eye is well enough trained that this doesn’t look nice to you, try the examples here.

There are two good conditions that imply flatness:

**Theorem** (Hartshorne III.9.7) If is normal and one dimensional, and no irreducible component of maps to a proper subvariety of , then is flat over .

**Theorem** (The miracle flatness theorem) If is Cohen-Macaulay, is smooth of the same dimension as , and is finite, then is flat.

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We still have room for a number more applicants, so we would like to encourage more of you to apply. Please note that the application deadline of March 1 is firm.

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Summary:A number of mathematics journals (e.g. Quantum Topology, Forum of Mathematics Sigma and Pi, and probably many others), are not listed on the new official journal list in Australia. Please, help identify missing journals, and submit feedback via http://jacci.arc.gov.au/.

Every few years the Australian Research Council updates their “official list of journals”. One might wonder why it’s necessary to have such a list, but nevertheless it is there, and it is important that it is accurate because the research outputs of Australian mathematicians are essentially filtered by this list for various purposes.

There is a new draft list out, and the purpose of this post is to coordinate finding missing journals, and to ensure that interested mathematicians submit feedback before the deadline of March 15. Please note that while in the past this list included dubious rankings of journals, the current list is just meant to track all peer reviewed journals in each subject. Having a journal missing entirely means that some published papers will not be counted in measures of a department’s or university’s research output.

You can access the full list here, just journals marked as mathematics here, and just the journals marked a pure mathematics here. These are not the “official” lists, which you have to create an account (follow the instructions at http://www.arc.gov.au/era/current_consult.htm) to view, and even then only an Excel version is available. I hope that by making these mathematics specific lists available in a standard format, more mathematicians will take the time to look over the list.

Please look through the lists. If you see something missing, please comment here so we all know about it. In any case, please submit feedback via http://jacci.arc.gov.au/ (you’ll have to create an account first) recommending inclusion of the journals identified so far. Submitting a missing journal requires identifying an article published in it by an Australia author; feel free to add this information here as well if appropriate. (Thanks to Anthony Henderson for pointing out this detail!)

It is also possible to submit additional “FoR” (field of research) codes for journals on the list, and this may be of interest to people publishing cross-disciplinary research. Feel free to make suggestions along these line here too: the AustMS has been advised that “multiple responses, rather than a single AustMS one, will carry more weight on this aspect”.

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We intend that these will be 2 year positions, with minimal teaching requirements.

There is an informal description of the jobs at http://tqft.net/web/postdoc, including some information about the grants funding these positions. The official ad is online at http://jobs.anu.edu.au/PositionDetail.aspx?p=3736, and you can find it on MathJobs at http://www.mathjobs.org/jobs/jobs/5678.

Please contact us if you have questions, and please encourage good Ph.D. students (especially with interests in subfactors, fusion categories, categorification, or related subjects) to apply!

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(The colour coded bars show the fractions of papers available on the arXiv, available on authors’ webpages, and not freely accessible at all; these now appear all over the wiki, but unfortunately don’t update automatically. Over at the wiki you can hover over these bars to get the numerical totals, too.)

Thanks everyone for your contributions so far! If you’ve just arrived, check out the tutorial I made on editing the wiki. Now, it’s time to do a little planning.

Here’s one we can start to answer right away.

What fraction of recent papers are available on the arXiv or on authors webpages?For good generalist journals (e.g. Adv. Math. and Annals), almost everything! For subject area journals, there is wide variation (probably mostly depending on traditions in subfields): AGT is almost completely freely accessible, while Discrete Math. is at most half.

I hope we’ll soon be able to say this for many other journals, too.

Here’s the question I really want to have answers for:

Does being freely accessible correlate well with quality?It’s certainly tempting to think so, seeing how accessible Advances and Annals are. I think to really answer this question we’re going to have to classify all the articles in slightly older issues (2010?) and then start looking at the citation counts for articles in the two pools. If we get coverage of more journals, we can also look for the correlation between, say, impact factor and the ratio of freely accessible content.

I don’t want to just list every journal on the wiki; it’s best if editors (and the helpful bots working in the background) can focus attention and enjoy the pleasures of finishing off issues and journals. Suggestions for journals to add next welcome in the comments. I’ve already included the tables of contents for the Journal of Number Theory, and the Journal of Functional Analysis. (It will be nice to be able to make comparisons between JFA and GAFA, I think.)

I’ve been working with some people on automating the entry of data in the wiki (mainly by using arXiv metadata; there are actually way more articles there with journal references and DOIs than I’d expected). Hopefully this will make the wiki editing experience more fun, as a lot of the work will have already been done, and humans just get to handle the hard and interesting cases.

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(For the impatient, go visit http://tqft.net/mlp, or for the really impatient http://tqft.net/mlp/wiki/Adv._Math./232_(2013).)

It would be nice to know how much of the mathematical literature is freely accessible. Here by ‘freely accessible’ I mean “there is a URL which, in any browser anywhere in the world, resolves to the contents of the article”. (And my intention throughout is that this article is legitimately hosted, either on the arxiv, on an institutional repository, or on an author’s webpage, but I don’t care how the article is actually licensed.) I think it’s going to be okay to not worry too much about discrepancies between the published version and a freely accessible version — we’re all grown ups and understand that these things happen. Perhaps a short comment field, containing for example “minor differences from the published version” could be provided when necessary.

This post outlines an idea to achieve this, via a human editable database containing the tables of contents of journals, and links, where available, to a freely accessible copy of the articles.

It’s important to realize that the goal is *not* to laboriously create a bad search engine. Google Scholar already does a very good job of identifying freely accessible copies of particular mathematics articles. The goal is to be able to definitively answer questions such as “which journals are primarily, or even entirely, freely accessible?”, to track progress towards making the mathematical literature more accessible, and finally to draw attention to, and focus enthusiasm for, such progress.

I think it’s essential, although this is not obvious, that at first the database is primarily created “by hand”. Certainly there is scope for computer programs to help a lot! (For example, by populating tables of contents, or querying google scholar or other sources to find freely accessible versions.) Nevertheless curation at the per-article level will certainly be necessary, and so whichever route one takes it must be possible for humans to edit the database. I think that starting off with the goal of primarily human contributions achieved two purposes: one, it provides an immediate means to recruit and organize interested participants, and two, hopefully it allows much more flexibility in the design and organization of the collected data — hopefully many eyes will reveal bad decisions early, while they’re easy to fix.

That said, we better remember that eventually computers may be very helpful, and avoid design decisions that make computer interaction with the database difficult.

What should this database look like? I’m imagining a website containing a list of journals (at first perhaps just one), and for each journal a list of issues, and for each issue a table of contents.

The table of contents might be very simple, having as few as four columns: the title, the authors, the link to the publishers webpage, and a freely accessible link, if known. All these lists and table of contents entries must be editable by a user — if, for example no freely accessible link is known, this fact should be displayed along with a prominent link or button which allows a reader to contribute one.

At this point I think it’s time to consider what software might drive this website. One option is to build something specifically tailored to the purpose. Another is to use an essentially off-the-shelf wiki, for example tiddlywiki as Tim Gowers used when analyzing an issue of Discrete Math.

Custom software is of course great, but it takes programming experience and resources. (That said, perhaps not much — I’m confident I could make something usable myself, and I know people who could do it in a more reasonable timespan!) I want to essentially ignore this possibility, and instead use mediawiki (the wiki software driving wikipedia) to build a very simple database that is readable and editable by both humans and computers. If you’re impatient, jump to http://tqft.net/mlp and start editing! I’ve previously used it to develop the Knot Atlas at http://katlas.org/ with Dror Bar-Natan (and subsequently many wiki editors). There we solved a very similar set of problems, achieving human readable and editable pages, with “under the hood” a very simple database maintained directly in the wiki.

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One of the central problems in fusion categories is to determine to what extent fusion categories can be classified in terms of finite groups and quantum groups (perhaps combined in strange ways) or whether there are exceptional fusion categories which cannot be so classified. My money is on the latter, and in particular I think extended Haagerup gives an exotic fusion category. However, there are a number of examples which seem to involve finite groups, but where we don’t know how to classify them in terms of group theoretic data. For example, the Haagerup fusion category has a 3-fold symmetry and may be built from or (as suggested by Evans-Gannon). The simplest examples of these kind of “close to group” categories, are called “near-group categories” which have only one non-invertible object and have the fusion rules

for some group of invertible objects . A result of Evans-Gannon (independently proved by Izumi in slightly more generality), says that outside of a reasonably well understood case (where and the category is described by group theoretic data), we have that must be a multiple of . There are the Tambara-Yamagami categories where , and many examples (E6, examples of Izumi, many examples of Evans-Gannon) where

Here’s the question: Are there examples where n is larger than ?

It turns out the answer is yes! In fact the answer is given by the -graded part of the quantum subgroup of quantum from Ocneanu’s tables here. I’ll explain why below.

The category of representations of a group has a restriction functor to the category of representations of any subgroup. This suggests a generalization of the notion of “subgroup” to an arbitrary tensor category. If C is a tensor category, then a “quantum subgroup” (of type I) is a tensor category D with a tensor functor which is dominant (every object in D is a summand of an object in the image of F). In particular, this makes D into a module category. (A simple module category which doesn’t come from a tensor functor is called a subgroup of type II, Ocneanu’s list includes both types.)

Ocneanu’s notation here is as follows. The quantum subgroups of type I are the ones with a starred vertex. The vertices of the graph are the simple objects in D, and the starred vertex is the trivial object. The category of representations of quantum SU(3) is -graded. Sometimes this grading descends to a grading on D, if it does then Ocneanu denotes the grading by coloring the vertices white, grey, and black. Note that the zero graded vertices (the white ones) form a tensor subcategory.

The edges of the graph give the fusion rules for tensoring with the fundamental representations of SU(3). That is the number of edges from A to B is the dimension of the hom spaces between where is a fundamental representation. In fact,there are two fundamental representations, and , and it is possible to distinguish which edge is which by looking at the coloring of vertices since adds 1 to the grading and subtracts 1.

(The graph is also decorated with some additional information, that won’t be needed here. For example, the vertices are circled if the object is “dyslectic” and the subcategory of dyslectic objects is braided.)

For E9 there are four objects which are 0-graded. Three of dimension 1 (which we call 1, , and ), and one of dimension which we will call . We can compute that . Thus we can work out the rules for tensoring with by counting paths of length 2 which go white-black-white, but subtracting 1 from the total count of paths from a vertex to itself. Using this we can see that , where this 6 appears as . Of course, .

If you know what a conformal inclusion is (I only sort of do), this example comes from the conformal inclusion of SU(3) at level 9 including into the exceptional group E6 at level 1.

Also, it turns out that for the special case when G is , a result of (then high school student) Hannah Larson, shows that n can’t be any larger than 6. This example shows that her result is sharp.

I think as the amount of mathematics grows, it will be increasingly important to find better ways to arrange old information in ways that make searching easier. In a perfect world, there should be a searchable database of fusion categories where one could just ask for all known examples of rank 4 fusion categories with 3 invertible objects and have this example returned. (In this case, the example would definitely be in the database because the paper itself is well-known, it just has a huge list of examples.)

(Finally, I’d like to note that this example will eventually be mentioned in the formal literature in a paper of Zhengwei’s.)

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