I shall tell you the story of this article, from its inception to its publication. I hope it is interesting and funny. It is an old story, not like this one, but nevertheless it might serve to justify my opinion that open peer-review (anonymous or not, this doesn’t matter) is much better than the actual peer-review, in that by being open (i.e. peer-reviews publicly visible and evolving through contributions by the community of peers), it discourages abusive behaviours which are now hidden under the secrecy, motivated by a multitude of reasons, like conflict of interests, protection of it’s own little group against stranger researchers, racism, and so on .
Here is the story.
In 2001, at EPFL I had the chance to have on my desk two items: a recent article by Bernard Dacorogna and Chiara Tanteri concerning quasiconvex hulls of sets of matrices and the book A.W. Marshall, I. Olkin, Inequalities: Theory of Majorisation and it’s Applications, Mathematics in science and engineering, 143, Academic Press, (1979). The book was recommended to me by Tudor Ratiu, who was saying that it should be read as a book of conjectures in symplectic geometry. (Without his suggestion, I would have never decided to read this excellent book.)
At the moment I was interested in variational quasiconvexity (I invented multiplicative quasiconvexity, or quasiconvexity with respect to a group), which is still a fascinating and open subject, one which could benefit (but it does not) from a fresh eye by geometers. On the other hand, geometers which are competent in analysis are a rare species. Bernard Dacorogna, a specialist in analysis with an outstanding and rather visionary good mathematical sense, was onto this subject from some time, for good reasons, see his article with J. Moser, On a partial differential equation involving the Jacobian determinant, Annales de l’Institut Henri Poincaré. Analyse non linéaire 1990, vol. 7, no. 1, pp. 1-26, which is a perfect example of the mixture between differential geometry and analysis.
Therefore, by chance I could notice the formal similarity between one of Dacorogna’s results and a pair (Horn, Thompson) of theorems in linear algebra, expressed with the help of majorization relation. I quickly wrote the article “Majorization with applications to the calculus of variations“, where I show that by using majorization techniques, older than the quasiconvexity subject (therefore a priori available to the specialists in quasiconvexity), several results in analysis have almost trivial proofs, as well as giving several new results.
I submitted the article to many journals, without success. I don’t recall the whole list of journals, among them were Journal of Elasticity, Proceedings of the Royal Society of Edimburgh, Discrete and Continuous Dynamical Systems B.
The reports were basically along the same vein: there is nothing new in the paper, even if eventually I changed the name of the paper to “Four applications of majorization to convexity in the calculus of variations”. Here is an excerpt from such a report:
“Usually, a referee report begins with a description of the goal of the paper. It is not easy here, since Buliga’s article does not have a clear target, as its title suggests. More or less, the text examines and exploits the relationships between symmetry and convexity through the so-called majorization of vectors in Rn , and also with rank-one convexity. It also comes back to works of Ball, Freede and Thompson, Dacorogna & al., Le Dret, giving a few alternate proofs of old results.
This lack of unity is complemented by a lack of accuracy in the notations and the statements. […] All in all, the referee did not feel convinced by this paper. It does not contain a striking statement that could attract the attention. Thus the mathematical interest does not balance the weak form of the article. I do not see a good argument in favor of the publication by DCDS-B.”
At some point I renounced to submit it.
After a while I made one more try and submit it to a journal which was not in the same class as the previous ones, (namely applied mathematics and calculus of variations). So I submitted the article to Linear Algebra and its Applications and it has been accepted. Here is the published version Linear Algebra and its Applications 429, 2008, 1528-1545, and here is an excerpt from the first referee report (from LAA)
“This paper starts with an overview of majorization theory (Sections 1-4), with emphasis on Schur convexity and inequalities for eigenvalues and singular values. Then some new results are established, e.g. characterizations of rank one convexity of functions, and one considers applications in several areas as nonlinear elasticity and the calculus of variation. […] The paper is well motivated. It presents new proofs of known results and some new theorems showing how majorization theory plays a role in nonlinear elasticity and the calculus of variation, e.g. based on the the notion of rank one convexity.
A main result, given in Theorem 5.6, is a new characterization of rank one convexity (a kind of elliptic condition) […] This result involves Schur convexity.
Some modiﬁcations are needed to improve readability and make the paper more self-contained. […] Provided that these changes are done this paper can be recommended for publication.”
PS. The article which, from my experience, took the most time from first submission to publication is this one: first version submitted in 1997, which was submitted as well to many journals and it was eventually published in 2011, after receiving finally an attentive, unbiased peer-review (the final version can be browsed here)The moral of the story is therefore: be optimistic, do what you like most in the best of ways and be patient.
PS2. See also the very interesting post by Mike Taylor “The only winning move is not to play“.