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This month's topics:
The "rising star hypothesis" tested in math
"The role of mentorship in protégé performance," by the Northwestern University team of R. Dean Malmgren, Julio M. Ottino, and Luís A. Nunes Amaral, appears in the June 3 2010 Nature. From the Abstract: "... we investigate one aspect of mentor emulation by studying mentorship fecundity-- the number of protégés a mentor trains-- using data from the Mathematics Genealogy Project, which tracks the mentorship record of thousands of mathematicians over several centuries." In particular they wished to test the "rising star" hypothesis: "that mentors select up-and-coming protégés on the basic of their perceived ability and potential and past performance, ... resulting in a 'perpetual cycle' of rising-star protégés that emulate their mentors by seeking other rising stars as their protégés." The Mathematics Genealogy Project ("aggregates  the graduation date, mentor and protégés of 114,666 mathematicians from as early as 1637") is a database "unique in its scope and coverage." They used it to construct "a network in which links are formed from a mentor to each of his k protégés, where k denotes mentorship fecundity," focussing on the 7,259 mathematicians who graduated between 1900 and 1960. An initial observation (from comparing mentorship data to publication records and membership in the National Academy): "although fecundity is not a typical measure of academic performance, it is closely related to other measures of academic success." A more detailed look at the statistics reveals some deviations from a uniform pattern:
The authors speculate about explanations for these divergences and go on to conclude: "Our findings therefore reveal interesting nuances to the rising-star hypothesis."
"Street-fighting Math" on Boing Boing
The relentlessly with-it tech, culture and news blog Boing Boing showcased a math text on July 26, 2010: "Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem-Solving" by Sanjoy Mahajan of MIT. (MIT Press sells bound copies, and also offers the book as a free-download PDF file, through Creative Commons). Boing Boing editor Cory Doctorow characterizes it as a "down and dirty guide to approximation and problem-solving" and goes so far as to say that it "looks like a fun read."
A theorem in Biochemistry
"A biological system shows absolute concentration robustness (ACR) for an active molecular species if the concentration of that species is identical in every positive steady state the system might admit." This is from the beginning of "Structural Sources of Robustness in Biochemical Reaction Networks" by Guy Shinar (Weitzmann Institute) and Martin Feinberg (OSU) in Science, March 12, 2010. Mathematically speaking, a "biological system" is a system of first-order differential equations linking the concentrations cA, cB, ... of a set A, B, ... of "molecular species," i.e. (biological) substances identifiable by their molecular structure, participating in a chemical reaction or set of reactions, each of which is governed by a rate constant. The authors state and prove (in the supplementary material) a stability theorem giving sufficient conditions for a system to show ACR (as above) for a molecular species S among those participating. This means that the steady-state value of cS is invariant under fluctuations in the initial concentrations; a criterion of obvious biological importance.
The theorem is stated in terms of structural properties of the system.
A biological system involving 6 molecular species A ... F. The first line, for example, corresponds to the reactions 2A --> B, B --> 2A, B --> C (reaction rates are not shown). Each node (yellow box) corresponds to the head or the tail of an arrow.
Here the linkage classes (solid boxes) contain nodes linkable by arrows or chains of arrows; the strong linkage classes (dotted boxes) contain nodes linkable forwards and back by chains of arrows. A strong linkage class is terminal (pink nodes) if no arrows lead away from it; otherwise it is non-terminal (blue). Images adapted from Science 327 1389-1391.
Furthermore a rank is assigned to the system by thinking of the species A, B, ... as independent vectors, assigning to a reaction X --> Y the vector Y-X, and calculating the dimension of the space spanned by all the reaction vectors. In the example illustrated, the seven vectors B-2A, C-B, D-B-C, 2B-D, A+E-2B, F-A-E, 2B-F satisfy (A+E-2B) + (F-A-E) + (2B-F) = 0, and also (D-B-C) + (2B-D) + (C-B) = 0; otherwise they are independent, so the rank is 5. The authors then define the deficiency of the system as #(nodes) - #(linkage classes) - rank; in our example this number is 8 - 2 - 5 = 1.
Theorem: If in a biological system (defined as above) of deficiency one, there are two nonterminal nodes whose vector difference is a multiple of the species vector S, then the system has absolute concentration robustness in S. (In our example the blue nodes B+C and B have difference C, so this system has ACR for species C ).
Shinar and Feinberg give applications of their theorem to real-life systems. Their work is explained and situated (references to Waddington and to René Thom) by Jeremy Gunawardena of Harvard in a "Perspectives" piece (same journal, April 30).
P and NP, blogs and wiki
On August 6, 2010 Vinay Deolalikar of HP Research Labs, Palo Alto, posted a paper entitled "P ≠ NP" and setting forth a (negative) resolution of the problem, whether those two classes of problems were distinct or not. Ten days later, the New York Times ran a piece by John Markoff: "Step 1: Post Elusive Proof. Step 2: Watch Fireworks." Markoff sketches some of the background, but then gets to his main point: "In this case, however, the significant breakthrough may not be in the science, but rather in the way science is practiced ... the pace of discussion and analysis, carried out in real time on blogs and a wiki that had been quickly set up for the purpose of collectively analyzing the paper." In fact the wiki Deolalikar P vs NP paper, hosted by the polymath project wiki, links to 27 blog postings (up to August 19) and 52 entries or publications (up to August 23, including the item under review here). Many of these are news items but many are also technical analyses of parts of the proof. Markoff: "Several of the researchers said that until now such proofs had been hashed out in colloquiums that required participants to be physically present at an appointed time. Now, with the emergence of Web-connected software programs it is possible for such collaborative undertakings to harness the brainpower of the world's best thinkers on a continuous basis."
On August 20, the New Scientist took up the same story, with a somber twist: "Flawed proof ushers in era of wikimtath.".
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