Euler, on a Swiss bank note

In principle, a mathematician can communicate to the others his own work using solely public free access Internet academic sites (either personal or global). A famous example is the work of the Fields Medalist Grisha Perelman on Thurston’s geometrization conjecture published on arXiv.org. Even if arXiv.org is moderated, it does not provide a peer reviewing process. For historical and sociological reasons, most professional mathematicians publish their results in peer reviewed mathematical journals. Many of these journals are handled by capitalistic companies such as Elsevier, Springer, Taylor and Francis, etc. The aim of these companies is to make profit, and some of them are not willing to make knowledge freely accessible. Basically, the mathematical publications rely on three pillars: science, money, and human comedy:

• Science. Mathematicians use peer reviewing in order to improve science quality and to reduce pollution. Peer reviewing is the first step in the long transformation of live mathematics into “dead” mathematics which can be considered as true and certified. This does not mean that peer reviewing is good for everything. Mathematicians need also media and space to discuss ideas. Internet has good tools for such live interactions, such as MathOverflow, a sort of modern version of the venerable sci.math usenet newsgroup. Actually, MathOverflow implements some kind of peer rating, while keeping freedom.
• Money. Everything has a cost. Non free journals are actually paied by readers via their institution. Freely accessible journals or sites have also a cost, but the circuit of money is different. For instance, the cost of arXiv.org is supported mostly by North American and European Universities and Research Centers. Other circuits of money are possible. For instance, some journals are freely accessible but the authors have to pay at publication time (some private companies such as Springer propose this scheme to the authors of non free journals in order to make their papers freely accessible). Many prestigious journals are handled by capitalistic companies, asking for high prices to make profit. On the other hand, being handled by an academic institution does not guarantee a low price.
• Human comedy. Mathematicians are first of all humans, members of an academic community. The academic community needs a way to measure the merits of each individual in order to decide recognition (and thus promotion and funding). In general, recognition is gained by obtaining good or influential results, published in respected journals. Publishing in respected journals can be seen as a way to obtain a label of quality, a sort of delegation of evaluation. The respect or prestige of a journal comes from its history, its editorial board, and its content. The editorial and peer reviewing job of most peer reviewed mathematical journals (free or not) is done for free by mathematicians (they gain social recognition).

Many aspects are not specific to mathematics. Ideally, one can imagine the introduction of editorial labels in arXiv.org, attributed by editorial boards to papers at author request and after peer reviewing (these boards may leave completely capitalistic publishers). One can also imagine to make arXiv.org less static by adding on top of it a dynamic collaborative stack-exchange structure such as MathOverflow, allowing the discussion of each paper. These ideas are floating around for a while. Unfortunately, discussing these matters with many colleagues and with the Editors of certain leading journals reveals that the mathematical community is a bit conservative. However, it is remarkable that some few scientific leaders such as the Fields Medalist Tim Gowers are promoting such a revolution.

Some related reading:

• Elsevier — my part in its downfall by Tim Gowers
• The Cost of Knowledge by Terry Tao
• What’s wrong with electronic journals? by Tim Gowers
• Tim Gowers’ Model of Mathematical Publishing by NuitBlanche
• Science fiction ? (in French) by the author of this blog
• Can Peer Review be better Focused? by Paul Ginsparg

http://www.youtube.com/watch?v=XP9cfQx2OZY

Diff is a standard utility from the Unix world outputting the differences between two files. Even if diff can handle binary files, it is used mostly to compare two versions of a text file. For mathematicians, diff is helpful for the comparison of $\LaTeX$ files written with co-authors, in particular via its graphical user interface tkdiff written in Tcl/Tk. The is also diffpdf, which is very convenient to compare two versions of a PDF file. There is another well known utility called patch, a sort of reversed diff, used millions of times a day for software development, worldwide. On the technical side, diff relies on an algorithm for solving the longest common subsequence problem. By the way, if you are curious about text algorithms, you may take a look at the (free!) books written by Maxime Crochemore, from Marne-la-Vallée.

Yesterday, a colleague of mine asked during a dinner “is there an elementary way to construct martingales which are not Markov chains?” Let us show that the answer is positive, by using a recursive recipe. Let \( {{(f_n)}_{n\geq1}} \) be a sequence of functions where \( {f_n:\mathbb{R}^{n+1}\rightarrow\mathbb{R}} \). Let \( {{(\varepsilon_n)}_{n\geq1}} \) be a sequence of i.i.d. real random variables of zero mean, independent of a real random variable \( {X_0} \). We now define the sequence \( {{(X_n)}_{n\geq0}} \) by setting, for every \( {n\geq0} \),

\[ X_{n+1}:=f_{n+1}(X_0,\ldots,X_n,\varepsilon_{n+1}). \]

We may safely assume that our ingredients \( {X_0} \), \( {{(f_n)}_{n\geq1}} \), and \( {{(\varepsilon_n)}_{n\geq1}} \) are chosen in such a way that \( {X_n} \) is integrable for every \( {n\geq0} \). The stochastic process \( {{(X_n)}_{n\geq0}} \) is a martingale for its natural filtration as soon as for every \( {n\geq0} \) and \( {x_0,\ldots,x_n\in\mathbb{R}} \),

\[ \mathbb{E}(f_{n+1}(x_0,\ldots,x_n,\varepsilon_{n+1}))=x_n. \]

However, if for instance \( {f_{n+1}} \) depends on the first variable \( {x_0} \) for every \( {n\geq0} \) then \( {{(X_n)}_{n\geq0}} \) is not a Markov chain (of any order). This is clearly the case if we take

\[ f_{n+1}(x_0,\ldots,x_n,\varepsilon) =\varepsilon g_{n+1}(x_0,\ldots,x_n)+x_n \]

where \( {g_{n+1}:\mathbb{R}^{n+2}\rightarrow\mathbb{R}} \) depends on its first variable for every \( {n\geq0} \). This leads to the following simple example of a martingale which is not a Markov chain (of any order):

\[ X_{n+1}=\varepsilon_{n+1}X_0+X_n. \]

Another way to construct martingales which are not Markov chains (of any order) consists in perturbing a martingale. Namely, let \( {{(M_n)}_{n\geq0}} \) be a martingale. Let \( {{(\varepsilon_n)}_{n\geq1}} \) be a sequence of i.i.d. random variables of zero mean, independent of \( {{(M_n)}_{n\geq0}} \). Let \( {{(g_n)}_{n\geq1}} \) be a sequence of functions with \( {g_n:\mathbb{R}^n\rightarrow\mathbb{R}} \). Then one may define the sequence \( {{(Y_n)}_{n\geq0}} \) by \( {Y_0=M_0} \), and for every \( {n\geq0} \),

\[ Y_{n+1}=\varepsilon_{n+1}g_{n+1}(Y_0,\ldots,Y_n)+M_{n+1}. \]

The stochastic process \( {{(Y_n)}_{n\geq0}} \) is a martingale (one can easily guess the filtration), but is not a Markov chain (of any order) if \( {g_{n+1}} \) depends on the first variable.

Note: a Markov chain (of any order) is a stochastic recursive sequence of finite order, or equivalently an auto-regressive process of finite order (possibly nonlinear). In contrast, the martingale property does not put constraints on the order of recursion, while imposing a linear projection condition. If \( {{(M_n)}_{n\geq0}} \) and \( {{(M_n’)}_{n\geq0}} \) are two martingales for the same filtration then \( {{(M_n+M_n’)}_{n\geq0}} \) is a martingale. In contrast, the sum of two Markov chains is not a Markov chain in general (delicate counter examples are available).

Converse: the converse is well known. Namely, if \( {{(M_n)}_{n\geq0}} \) is a Markov chain with state space \( {E} \) and kernel \( {P} \), and if \( {f:E\rightarrow\mathbb{R}} \) is such that \( {f(M_n)} \) is integrable for every \( {n\geq0} \), then \( {{(f(M_n))}_{n\geq0}} \) is a martingale as soon as \( {f} \) is harmonic, i.e. \( {Pf=f} \). In particular, if \( {E=\mathbb{R}} \) then \( {{(M_n)}_{n\geq0}} \) is a martingale as soon as \( {P(x,\cdot)} \) has mean \( {x} \) for every \( {x} \).