A simple, efficient way to solve Ax = b where A is a graph Laplacian!

For an overview see:

http://web.mit.edu/newsoffice/2013/short-algorithm-long-range-consequences-0301.html

#spnetwork #recommend   #graphTheory #graphLaplacian #networks   

arXiv:1301.6628

David Spivak describes an operad for sticking together graphs with specified vertices called 'terminals'... like electrical circuits.  This is an interesting alternative to the approach I'm developing, where we create a symmetric monoidal bicategory of graphs with input and output terminals. It would also be interesting to compare the modular operad and maybe some other famous operads.

#spnetwork #mustread  #graph_rewriting #networks #operads #circuits #cs arXiv:1305.0297 

This looks like a really great book on the interface between control theory and systems biology.  In particular, it talks about stability for chemical reaction networks... and how organisms ensure stability in their biochemistry.  It also talks about how the 'identification' problem: how to take empirical data and guess what chemical reaction network fits it.

#spnetwork #control_theory #systems_biology #reaction_networks   #recommend isbn:9781615832347

This paper describes the Graph Grammar Library - software that implements a large class of graph rewrite systems using a "generic double push out" approach.  Very roughly, a graph rewrite rule is a pair of graphs L and R with a morphism L -> R.  The idea is that given any graph G with a copy of L showing up as a subgraph, we can rewrite G by replacing that copy of L by a copy of R. 

I would like to understand this category-theoretically, and the term "double pushout" suggests it can be done, but the category theory is not explained in this paper - I'll need to go to a reference.  Clearly given a monomorphism L -> G and a morphism L -> R we can form the pushout of these, but it's not clear that's right, or why the term "double pushout" is used.  Any help understanding this would be appreciated!

#spnetwork #graphRewriting #reactionNetworks   arXiv:1304.1356

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Does anyone know about Agostino Prastaro, professor at the University of Rome, who has papers on the arXiv claiming to prove the Goldbach Conjecture, the Twin Primes Conjecture and the Riemann Hypothesis?  My source tells me that he is currently a reviewer for MR and Zentralblatt fuer Mathematik and he is the author of many papers reviewed there (but mainly published in conference proceedings, with only the abstract as review).  He apparently lists his fields of research as:
 
Geometry of PDE’s (Differential Geometry, Algebraic Geometry and Algebraic Topology); (Co)bordism in PDE’s and quantum PDE’s; Geometry of PDE’s in Continuum Mechanics; Geometry of PDE’s in Quantum Field Theory and Quantum Supergravity; Geometry of PDE’s in Mathematical Physics.

A rebuttal of his claimed proof of the Goldbach Conjecture is here:

http://mixedmath.wordpress.com/2012/08/24/reviewing-goldbach/

Given that his proof method shows a much stronger result - all odd numbers are prime - I'm not really asking if his other 'proofs' are correct.  I'm more curious about what he's like, what they think about him at the University of Rome, etcetera.

#spnetwork  #disagree #numberTheory arXiv:1208.2473  arXiv:1305.6845

I had dinner with Gregory Benford last weekend, and he raised a provocative point.  So far, radio searches for extraterrestrial life have only seen puzzling brief signals - not long transmissions.  But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…”  Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, Gray, 1994; 199 Tarter, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993).  Another striking example is the “WOW” signal seen at the Ohio SETI site...

That's a quote from this paper:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons, #spnetwork #mustread #SETI arXiv:0810.3966.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky.  So, for anyone receiving this signal, there would be a lot of time between pulses.  That might explain some of the above mysteries, or this:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons, #spnetwork #mustread #SETI arXiv:0810.3964.

and you can see a summary in this talk by James Benford, who works on high-power microwave technologies.

This article by Perelman outlines a proof of the Poincare conjecture!

#spnetwork #mustread #geometry #poincareConjecture
arXiv:math/0211159

This paper by Aleks Kissinger uses graph rewriting to study the diagrammatic approach to monoidal categories.  Morphisms in monoidal categories can be described using 'string graphs', and these form a 'partial adhesive category', so we can describe rewriting of string graphs using the 'double pushout approach' to graph rewriting.  

An adhesive category is one that has pullbacks, has pushouts along monos, and such that a pushout of two monos has a special nice property: it's a so-called 'van Kampen square'.  This is the usual context for graph rewriting, which Kissinger generalizes a bit.   But luckily, he explains it before generalizing it!

I am still looking for good basic papers on graph rewriting and adhesive categories.  However, Kissinger's work uses these concepts in the way I want: to study rewriting of string diagrams for morphisms in monoidal categories.  There should clearly a monoidal bicategory where these rewrites are 2-morphisms.

#spnetwork #recommend #graph_rewriting #categories #adhesive_categories   arXiv:1203.0202

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Samson Abramsky and Jonathan Zvesper analyze the so-called 'Brandenbruger-Keisler paradox', not really a paradox, which says it's impossible for Ann to believe that Bob's assumption that Ann's belief about Bob's assumptions is wrong.  Does anyone here understand this so-called paradox?

  #spnetwork   #gameTheory #logic     arXiv:1006.0092

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