20 December 2017:
Here is this week's progress, as far as I know:
1) I told you last time how Kenny and Daniel's paper was rejected from TAC
based on a mistaken counterexample to their main theorem. They explained this to the editor and referee, and now their paper has been accepted!
The new referee's report is so nice I'm going to quote it:
This is a nice, carefully written paper which presents a new and interesting
result. I recommend that it be accepted for TAC.
The authors continue the study of the bicategory of cospans in a topos
introduced by the first author in "Spans of cospans" to appear in TAC. Their motivation lies in the theory of complex networks and in order to accommodate
the various constructions that they wish to perform, they must take as 2-cells,
not the usual morphisms of cospans, but monic spans of cospans.
Their main result is that this gives a symmetric compact closed bicategory.
Now, this bicategory is relatively complicated, and proving that any bicategory
is symmetric monoidal, let alone compact, is no mean feat. To accomplish this,
the authors use a theorem of Shulman's which constructs a symmetric monoidal
bicategory from an isobrant symmetric monoidal double category. This last
structure sounds more complicated but in fact it is much easier to use because
the coherences are isomorphisms rather than equivalences. So they enlarge their
bicategory to a double category in which the vertical arrows are isomorphisms
rather than the identities which are the hallmark of a bicategory. This seemingly
minor change allows the coproduct to be a monoidal structure on the objects and
then Shulman's theorem applies. Once this double category has done its job it
is summarily discarded. Shulman's theorem cannot be used to get compactness,
so the authors resort to a theorem of Pstragowski for this.
Lately I've been feeling we should not "summarily discard" the double categories that Kenny has been using to build bicategories - that's a nice observation by the referee, with a nice touch of humor.
There probably is a Shulman-esque theorem that says "given a .... double category you get a compact closed symmetric monoidal bicategory", and I bet Kenny and Daniel could easily extract this theorem from the argument they used in this special case. Maybe Kenny should do that in his thesis!
2) Now I have a paper that seems to have been erroneously rejected despite a very positive referee's report.
This is not really progress; I'm mentioning it just because it outraged me. Here is the letter I received:
Dear Prof. Baez,
We have received the reports from our advisors on your manuscript NACO-D-14-00036R1 "Quantum Techniques for Reaction Networks".
With regret, I must inform you that, based on the advice received, the Editors have decided that your manuscript cannot be accepted for publication in Natural Computing.
Below, please find the comments for your perusal.
I would like to thank you very much for forwarding your manuscript to us for consideration and wish you every success in finding an alternative place of publication.
With kind regards,
Journals Editorial Office
COMMENTS TO THE AUTHOR:
Reviewer #4: The manuscript entitled "Quantum Techniques for Reaction Networks" investigates reaction networks utilizing quantum mechanical tools. Given a set of species and transitions which takes complexes to each other, the author derived the equations that governs concentration of the species. From a continuous-deterministic point of view, he derived rate equation for expected number of population of each species and from a discrete-stochastic point of view (by leveraging the probabilistic interpretation of amplitudes in quantum theory and using operators), he was able to obtain the underlying master equation. Then he made a connection between the state vector of the rate equation (classical state) and state vector of the master equation (mixed state) by averaging over probability distribution of the second one(amplitudes). Finally the author shows the required condition where both equations and formalism can be equivalent, which is large number limit and coherent
I am completely satisfied with this paper. Its a coherent and well-written paper with a new approach to reaction networks. This view may be of interest not only for researchers in this field but also for people in entire network society. I strongly recommend this paper for publication in this journal. Besides, I have two questions for the author and I suggest him to rephrase these questions and add them as future works in the last part of the paper (Its up to the author to do this or not).
1-Relation between rate equation and master equation has been discussed in the regime of the mass-action law. Does it work in other regimes?
2-The author has derived the exact relation between rate equation formalism and master equation formalism when the initial state is a coherent state. Is it possible to have a non-coherent state which satisfy the equivalence condition (stated under the theorem 8)?
Aargh! Emphasis mine: I've never gotten such a positive referee's report together with a rejection. I've asked some flunky at Springer to look into it, and I'll see what they have to say.