Tag Archives: peer community

Reproducibility vs peer review

Here are my thoughts about replacing peer review by validation. Peer review is the practice where the work of a researcher is commented by peers. The content of the commentaries (reviews) is clearly not important. The social practice is to not make them public, nor to keep a public record about those. The only purpose of peer review is to signal that at least one, two, three or four members of the professional community (peers) declare that they believe that the said work is valid. Validation by reproducibility is much more than this peer review practice. Validation means the following:

  • a researcher makes public (i.e. “publishes”) a body of work, call it W. The work contains text, links, video, databases, experiments, anything. By making it public, the work is claimed to be valid, provided that the external resources used (as other works, for example) are valid. In itself, validation has no meaning.
  • a second part (anybody)  can also publish a validation assessment of the work W. The validation assessment is a body of work as well, and thus is potentially submitted to the same validation practices described here. In particular, by publishing the validation assessment, call it W1, it is also claimed to be valid, provided the external resources (other works used, excepting W) are valid.
  • the validation assessment W1 makes claims of the following kind: provided that external works A,B,C are valid, then this piece D of the work W is valid because it has been reproduced in the work W1. Alternatively, under the same hypothesis about the external work, in the work W1 is claimed that the other piece E of the work D cannot be reproduced in the same.
  • the means for reproducibility have to be provided by each work. They can be proofs, programs, experimental data.

As you can see the validation can be only relative, not absolute. I am sure that scientific results are never amenable to an acyclic graph of validations by reproducibility. Compared to peer review, which is only a social claim that somebody from the guild checked it, validation through reproducibility is much more, even if it does not provide means to absolute truths. What is preferable: to have a social claim that something is true, or to have a body of works where “relative truth” dependencies are exposed? This is moreover technically possible, in principle. However, this is not easy to do, at least because:

  • traditional means of publication and its practices are based on social validation (peer review)
  • there is this illusion that there is somehow an absolute semantical categorification of knowledge, pushed forward by those who are technically able to implement a validation reproducibility scheme at a large scale.

UPDATE: The mentioned illusion is related to outdated parts of the cartesian method. It is therefore a manifestation of the “cartesian disease”.

I use further the post More on the cartesian method and it’s associated disease. In that post the cartesian method is parsed like this:

  • (1a) “never to accept anything for true which I did not clearly know to be such”
  • (1b) “to comprise nothing more in my judgement than what was presented to my mind”
  • (1c) “so clearly and distinctly as to exclude all ground of doubt”
  • (2a) “to divide each of the difficulties under examination into as many parts as possible”
  • (2b) “and as might be necessary for its adequate solution”
  • (3a) “to conduct my thoughts in such order that”
  • (3b) “by commencing with objects the simplest and easiest to know, I might ascend […] to the knowledge of the more complex”
  • (3c) “little and little, and, as it were, step by step”
  • (3d) “assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence”

Let’s take several researchers who produce works, some works related to others, as explained in the validation procedure.

Differently from the time of Descartes, there are plenty of researchers who think in the same time, and moreover the body of works they produce is huge.

Every piece of the cartesian method has to be considered relative to each researcher and this is what causes many problems.

Parts (1a),(1b), (1c) can be seen as part of the validation technique, but with the condition to see “true”and “exclude all grounds of doubt” as relative to the reproducibility of work W1 by a reader who tries to validate it up to external resources.

Parts (2a), (2b) are clearly researcher dependent; in a interconnected world these parts may introduce far more complexity than the original research work W1.

Combined with (1c), this leads to the illusion that the algorithm which embodies the cartesian method, when run in a decentralized and asynchronous world of users, HALTS.

There is no ground for that.

But the most damaging is (3d). First, every researcher embeds a piece of work into a narrative in order to explain the work. There is nothing “objective” about that. In a connected world, with the help of Google and alike, who impose or seek for global coherence, the parts (3d) and (2a), (2b) transform the cartesian method into a global echo chamber. The management of work bloats and spill over the work itself and in the same time the cartesian method always HALT, but for no scientific reason at all.

__________________________________

Advertisements

Gamification of peer review with Ingress

Seems possible to adapt Ingress in order to play the Game of Research and Review.

In the post MMORPGames at the knowledge frontier I propose a gamification of peer review which is, I see now very close to the Ingress game:

“… we could populate this world and play a game of conquest and exploration. A massively multiplayer online game.  Peer-reviews of articles decide which units of places are wild and which ones are tamed. Claim your land (by peer-reviewing articles), it’s up for grabs.  Organize yourselves by interacting with others, delegating peer-reviews for better management of your kingdoms, collaborating for the exploration of new lands.

Instead of getting bonus points, as mathoverflow works, grab some piece of virtual land that you can see! Cultivate it, by linking your articles to it or by peer-reviewing other articles. See the boundaries of your small or big kingdom. Maybe you would like to trespass them, to go into a near place? You are welcome as a trader. You can establish trade with other near kingdoms by throwing bridges between the land, i.e. by writing interdisciplinary articles, with keywords of both lands. Others will follow (or not) and they will populate the new boundary land you created.”

In Ingress (from the wiki source):

“The gameplay consists of establishing “portals” at places of public art, landmarks, cenotaphs, etc., and linking them to create virtual triangular fields over geographic areas. Progress in the game is measured by the number of Mind Units, i.e. people, nominally controlled by each faction (as illustrated on the Intel Map).[7][8] The necessary links between portals may range from meters to kilometers, or to hundreds of kilometers in operations of considerable logistical complexity.[9] International links and fields are not uncommon, as Ingress has attracted an enthusiastic following in cities worldwide[10] amongst both young and old,[11] to the extent that the gameplay is itself a lifestyle for some, including tattoos. ”

 

Instead of public art, Portals could be openaccess articles (from the arXiv, for example, not from the publishers).

 

“A portal with no resonators is unclaimed; to claim a portal for a faction, a player deploys at least one resonator on it.”

Resonators are reviews.

Links between portals are keywords.

 

Something to think about!

____________________________________________________

 

 

Sometimes an anonymous review is “a tale told by an idiot …”

… “full of sound and fury, signifying nothing.” And the editor believes it, even if it is self-contradictory, after sitting on the article for half a year.

There are two problems:

  • the problem of time; you write a long and dense article, which may be hard to review and the referee, instead of declining to review it, it keeps it until the editor presses him to write a review, then he writes some fast, crappy report, much below the quality of the work required.
  • the problem of communication: there is no two way communication with the author. After waiting a considerable amount of time, the author has nothing else to do than to re-submit the article to another journal.

Both problems could be easily solved by open peer-review. See Open peer-review as a service.

The referee can well be anonymous, if he wishes, but a dialogue with the author and, more important, with other participants could only improve the quality of the review (and by way of consequence, the quality of the article).

I reproduce further such a review, with comments. It is about the article “Sub-riemannian geometry from intrinsic viewpoint” arXiv:1206.3093 .  You don’t need to read it, maybe excepting the title, abstract and contents pages, which I reproduce here:

Sub-riemannian geometry from intrinsic viewpoint
Marius Buliga
Institute of Mathematics, Romanian Academy
P.O. BOX 1-764, RO 014700
Bucuresti, Romania
Marius.Buliga@imar.ro
This version: 14.06.2012

Abstract

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Caratheodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character.
In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead.
Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.
MSC2000: 51K10, 53C17, 53C23

1 Introduction       2
2 Metric spaces, groupoids, norms    4
2.1 Normed groups and normed groupoids      5
2.2 Gromov-Hausdorff distance     7
2.3 Length in metric spaces       8
2.4 Metric profiles. Metric tangent space      10
2.5 Curvdimension and curvature     12

3 Groups with dilations      13
3.1 Conical groups     14
3.2 Carnot groups     14
3.3 Contractible groups   15

4 Dilation structures  16
4.1 Normed groupoids with dilations     16
4.2 Dilation structures, definition    18

5 Examples of dilation structures 20
5.1 Snowflakes, nonstandard dilations in the plane    20
5.2 Normed groups with dilations    21
5.3 Riemannian manifolds    22

6 Length dilation structures 22
7 Properties of dilation structures    24
7.1 Metric profiles associated with dilation structures    24
7.2 The tangent bundle of a dilation structure    26
7.3 Differentiability with respect to a pair of dilation structures    29
7.4 Equivalent dilation structures     30
7.5 Distribution of a dilation structure     31

8 Supplementary properties of dilation structures 32
8.1 The Radon-Nikodym property    32
8.2 Radon-Nikodym property, representation of length, distributions     33
8.3 Tempered dilation structures    34
9 Dilation structures on sub-riemannian manifolds   37
9.1 Sub-riemannian manifolds    37
9.2 Sub-riemannian dilation structures associated to normal frames     38

 

10 Coherent projections: a dilation structure looks down on another   41
10.1 Coherent projections     42
10.2 Length functionals associated to coherent projections    44
10.3 Conditions (A) and (B)     45

11 Distributions in sub-riemannian spaces as coherent projections    45
12 An intrinsic description of sub-riemannian geometry    47
12.1 The generalized Chow condition     47
12.2 The candidate tangent space    50
12.3 Coherent projections induce length dilation structures  53

Now the report:

 

Referee report for the paper


 Sub-riemannian geometry from intrinsic viewpoint

Marius Buliga
for

New York Journal of Mathematics (NYJM).

One of the important theorems in sub-riemannian geometry is a result
credited to Mitchell that says that Gromov-Hausdorff metric tangents
to sub-riemannian manifolds are Carnot groups.
For riemannian manifolds, this result is an exercise, while for
sub-riemannian manifolds it is quite complicate. The only known
strategy is to define special coordinates and using them define some
approximate dilations. With this dilations, the rest of the argument
becomes very easy.
Initially, Buliga isolates the properties required for such dilations
and considers
more general settings (groupoids instead of metric spaces).
However, all the theory is discussed for metric spaces, and the
groupoids leave only confusion to the reader.
His claims are that
1) when this dilations are present, then the tangents are Carnot groups,
[Rmk. The dilations are assumed to satisfy 5 very strong conditions,
e.g., A3 says that the tangent exists – A4 says that the tangent has a
multiplication law.]
2) the only such dilation structures (with other extra assumptios) are
the riemannian manifolds.
He misses to discuss the most important part of the theory:
sub-riemannian manifolds admit such dilations (or, equivalently,
normal frames).
His exposition is not educational and is not a simplification of the
paper by Mitchell (nor of the one by Bellaiche).




The paper is a cut-and-past process from previous papers of the
author. The paper does not seem reorganised at all. It is not
consistent, full of typos, English mistakes and incomplete sentences.
The referee (who is not a spellchecker nor a proofread) thinks that
the author himself could spot plenty of things to fix, just by reading
the paper (below there are some important things that needs to be
fixed).


The paper contains 53 definitions – fifty-three!.
There are 15 Theorems (6 of which are already present in other papers
by the author of by other people. In particular 3 of the theorems are
already present in [4].)
The 27 proofs are not clear, incomplete, or totally obvious.

The author consider thm 8.10 as the main result. However, after
unwrapping the definitions, the statement is: a length space that is
locally bi-lipschitz to a commutative Lie group is locally
bi-lipschitz to a Riemannian manifold. (The proof refers to Cor 8.9,
which I was unable to judge, since it seems that the definition of
“tempered” obviously implies “length” and “locally bi-lipschitz to the
tangent”)


The author confuses the reader with long definitions, which seems very
general, but are only satisfied by sub-riemannian manifolds.
The definitions are so complex that the results are tautologies, after
having understood the assumptions. Indeed, the definitions are as long
as the proofs. Just two examples: thm 7.1 is a consequence of def 4.4,
thm 9.9 is a consequence of def 9.7.

Some objects/notions are not defined or are defined many pages after
they are used.



Small remarks for the author:

def 2.21 is a little o or big O?


page 13 line 2. Which your convention, the curvdim of a come in infinite.
page 13 line -2. an N is missing in the norm


page 16 line 2, what is \nu?

prop 4.2 What do you mean with separable norm?

page 18 there are a couple of “dif” which should be fixed.
in the formula before (15), A should be [0,A]

pag 19 A4. there are uncompleted sentences.

Regarding the line before thm 7.1, I don’t agree that the next theorem
is a generalisation of Mitchell’s, since the core of his thm is the
existence of dilation structures.

Prop 7.2 What is a \Gamma -irq

Prop 8.2 what is a geodesic spray?

Beginning of sec 8.3 This is a which -> This is a

Beginning of sec 9 contains a lot of English mistakes.

Beginning of sec 9.1 “we shall suppose that the dimension of the
distribution is globally constant..” is not needed since the manifold
is connected

thm 9.2 rank -> step

In the second sentence of def 9.4, the existence of the orthonormal
frame is automatic.

 

Now, besides some of the typos, the report is simply crap:

  • the referee complains that I’m doing it for groupoids, then says that what I am doing applies only to subriemannian spaces.
  • before, he says that in fact I’m doing it only for riemannian spaces.
  • I never claim that there is a main result in this long article, but somehow the referee mentions one of the theorems as the main result, while I am using it only as an example showing what the theory says in the trivial case, the one of riemannian manifolds.
  • the referee says that I don’t treat the sub-riemannian case. Should decide which is true, among the various claims, but take a look at the contents to get an opinion.
  • I never claim what the referee thinks are my two claims, both being of course wrong,
  • in the claim 1) (of the referee) he does not understand that the problem is not the definition of an operation, but the proof that the operation is a Carnot group one (I pass the whole story that in fact the operation is a conical group one, for regular sub-riemannian manifolds this translates into a Carnot group operation by using Siebert, too subtle for the referee)
  • the claim 2) is self-contradictory just by reading only the report.
  • 53 definitions (it is a very dense course), 15 theorems and 27 proofs, which are with no argument: “ not clear, incomplete, or totally obvious
  • but he goes on hunting the typos, thanks, that’s essential to show that he did read the article.

There is a part of the text which is especially perverse: The paper is a cut-and-past process from previous papers of the
author.

Mind you, this is a course based on several papers, most of them unpublished! Moreover, every contribution from previous papers is mentioned.

Tell me what to do with these papers: being unpublished, can I use them for a paper submitted to publication? Or else, they can be safely ignored because they are not published? Hmm.

This shows to me that the referee knows what I am doing, but he does not like it.

Fortunately, all the papers, published or not, are available on the arXiv with the submission dates and versions.

 

______________________________________

See also previous posts:

________________________________________

 

 

Gamifying peer-review?

Fact is: there are lots of articles on arXiv and only about a third published traditionally (according to their statistics). Contrary to biology and medical science, where researchers are way more advanced in new publishing models (like PLoS and PeerJ, the second being almost green in flavour), in math and physics we don’t have any other option than  arXiv, which is great, the greatest in fact, the oldest, but … but only if it had a functional peer-review system attached. Then it would be perfect!

It is hard though to come with a model of peer-review for the arXiv. Or for any other green OA publication system, I take the arXiv as example only because I am most fond of. It is hard because there has to be a way to motivate the researchers to do the peer-reviews. For free. This is the main type of psychological argument against having green OA with peer-review. It is a true argument, even if peer-review is made for free in the traditional publishing model.  The difference is that the traditional publishing model is working since the 1960’s and it is now ingrained in the people minds, while any new model of peer-review, for the arXiv or any other green OA publication system, has first to win a significant portion of researchers.

Such a new model does not have to be perfect, only better than the traditional one. For me, a peer-review which is technical, open, pre- and post- “publication” would be perfect. PLoS and PeerJ already have (almost) such a peer-review. Meanwhile, us physicists and mathematicians sit on the greatest database of research articles, greener than green and older than the internet and we have still not found a mean to do the damn peer-review, because nobody has found yet a viral enough solution, despite many proposals and despite brilliant minds.

So, why not gamify the peer-review process? Researchers  like to play as much as children do, it’s part of the mindframe requested for being able to do research. Researchers are driven also by vanity, because they’re smart and highly competitive humans which value playful ideas more than money.

I am thinking about Google Scholar  profiles. I am thinking about vanity surfing. How to add peer-review as a game-like rewarding activity? For building peer communities? Otherwise? Any ideas?

UPDATE:  … suppose that instead of earning points for making comments, asking questions, etc, suppose that based on the google scholar record and on the keywords your articles have, you are automatically assigned a part, one or several research areas (or keywords, whatever). Automatically, you “own” those, or a part, like having shares in a company. But in order to continue to own them, you have to do something concerning peer-reviewing other articles in the area (or from other areas if you are an expansionist Bonaparte). Otherwise your shares slowly decay. Of course, if you have a stem article with loads of citations then you own a big domain and probably you are not willing to loose so much time to manage it. Then, you may delegate others to do this. In this way bonds are created, the others may delegate as well, until the peer-review management process is sustainable. Communities may appear. Say also that the domain you own is like a little country and citations you got from other “countries” are like wealth transfer: if the other country (domain) who cites you is more wealthy then the value of the citation increases. As you see, until now, with the exception of “delegation” everything could be done automatically. From time to time, if you want to increase the wealth of your domain, or to gain shares in it, then you have to do a peer-review for an article where you are competent, according to keywords and citations.

MORE: MMORPGames at the knowledge frontier.

Something like this could be tried and it could be even funny.

“Future of peer review” by Maria Kowalczuk

Via the post “Peer pressure: the changing role of peer review” at BioMed Central blog, which I highly recommend as reading for those interested in the problem of peer review. I embed further the presentation “Future of peer review” by Maria Kowalczuk, because I think it exactly applies to publishing in mathematics. There’s a lot to learn form, or to discuss about.

______________

UPDATE: Here is an almost similar presentation, from dec. 2011, by Iain Hrynaszkiewicz: (pdf)

Peer-review, what is it for?

An interesting discussion started at Retraction Watch, in the comments of the post Brian Deer’s modest proposal for post-publication peer review. Let me repeat the part which I find interesting: post-publication peer review.

The previous post “Peer-reviews don’t protect against plagiarism and articles retraction. Why?”  starts with the following question:

After reading one more post from the excellent blog Retraction Watch, this question dawned on me: if the classical peer-review is such a good thing, then why is it rather inefficient when it comes to detecting flaws or plagiarism cases which later are exposed by the net?

and then I claimed that retractions of articles which already passed the traditional peer-review process are the best argument for an open, perpetual peer-review.

Which brings me to the subject of this post, namely what is peer-review for?

Context. Peer-review is one of the pillars of the actual publication of research practice. Or, the whole machine of traditional publication is going to suffer major modifications, most of them triggered by its perceived inadequacy with respect to the needs of researchers in this era of massive, cheap, abundant means of communication and organization. In particular, peer-review is going to suffer transformations of the same magnitude.

We are living interesting times, we are all aware that internet is changing our lives at least as much as the invention of the printing press changed the world in the past. With a difference: only much faster. We have an unique chance to be part of this change for the better, in particular  concerning  the practices of communication of research. In front of such a fast evolution of  behaviours, a traditionalistic attitude is natural to appear, based on the argument that slower we react, a better solution we may find. This is however, in my opinion at least, an attitude better to be left to institutions, to big, inadequate organizations, than to individuals. Big institutions need big reaction times because the information flows slowly through them, due to their principle of pyramidal organization, which is based on the creation of bottlenecks for information/decision, acting as filters. Individuals are different in the sense that for them, for us, the massive, open, not hierarchically organized access to communication is a plus.

The bottleneck hypothesis. Peer-review is one of those bottlenecks, traditionally. It’s purpose is to separate the professional  from the unprofessional.  The hypothesis that peer-review is a bottleneck explains several facts:

  • peer-review gives a stamp of authority to published research. Indeed, those articles which pass the bottleneck are professional, therefore more suitable for using them without questioning their content, or even without reading them in detail,
  • the unpublished research is assumed to be unprofessional, because it has not yet passed the peer-review bottleneck,
  • peer-reviewed publications give a professional status to authors of those. Obviously, if you are the author of a publication which passed the peer-review bottleneck then you are a professional. More professional publications you have, more of a professional you are,
  • it is the fault of the author of the article if it does not pass the peer-review bottleneck. As in many other fields of life, recipes for success and lore appear, concerning means to write a professional article, how to enhance your chances to be accepted in the small community of professionals, as well as feelings of guilt caused by rejection,
  • the peer-review is anonymous by default, as a superior instance which extends gifts of authority or punishments of guilt upon the challengers,
  • once an article passes the bottleneck, it becomes much harder to contest it’s value. In the past it was almost impossible because any professional communication had to pass through the filter. In the past, the infallibility of the bottleneck was a kind of self-fulfilling prophecy, with very few counterexamples, themselves known only to a small community of enlightened professionals.

This hypothesis explains as well the fact that lately peer-review is subjected to critical scrutiny by professionals. Indeed, in particular, the wave of detected plagiarisms in the class of peer-reviewed articles lead to the questioning of the infallibility of the process. This is shattering the trust into the stamp of authority which is traditionally associated with it.  It makes us suppose that the steep rise of retractions is a manifestation of an old problem which is now revealed by the increased visibility of the articles.

From a cooler point of view, if we see the peer-review as designed to be a bottleneck in a traditionally pyramidal organization,  is therefore questionable if the peer-review as a bottleneck will survive.

Social role of peer-review. There are two other uses of peer-review, which are going to survive and moreover, they are going to be the main reasons for it’s existence:

  • as a binder for communities of peers,
  • as a time-saver for the researchers.

I shall take them one-by-one. What is strange about the traditional peer-review is that although any professional is a peer, there is no community of peers. Each researcher does peer-reviewing, but the process is organized in such a manner that we are all alone. To see this, think about the way things work: you receive a demand to review an article, from an editor, based on your publication history, usually, which qualifies you as a peer. You do your job, anonymously, which has the advantage of letting you be openly critical with the work of your peer, the author. All communication flows through the editor, therefore the process is designed to be unfriendly with communications between peers. Hence, no community of peers.

However, most of the researchers who ever lived on Earth are alive today. The main barrier for the spread of ideas is a poor mean of communication. If the peer-review becomes open, it could foster then the appearance of dynamical communities of peers, dedicated to the same research subject. As it is today, the traditional peer-review favours the contrary, namely the fragmentation of the community of researchers which are interested in the same subject into small clubs, which compete on scarce resources, instead of collaborating. (As an example, think about a very specialized research subject which is taken hostage by one, or few, such clubs which peer-reviews favourably only the members of the same club.)

As for the time-saver role of peer-review, it is obvious. From the sea of old and new articles, I cannot read all of them. I have to filter them somehow in order to narrow the quantity of data which I am going to process for doing my research. The traditional way was to rely on the peer-review bottleneck, which is a kind of pre-defined, one size for all solution. With the advent of communities of peers dedicated to narrow subjects, I can choose the filter which serves best my research interests. That is why, again, an open peer-review has obvious advantages. Moreover, such a peer-review should be perpetual, in the sense that, for example, reasons for questioning an article should be made public, even after the “publication” (whatever such a word will mean in the future). Say, another researcher finds that an older article, which passed once the peer-review, is flawed for reasons the researcher presents. I could benefit from this information and use it as a filter, a custom, continually upgrading filter of my own, as a member of one of the communities of peers I am a member of.

Multiple peer-reviews, a story with a happy-end

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“.