Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker

Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c

Model Data Fusion: developing Bayesian inversion to constrain equilibrium and mode structure

Recently, a new probabilistic "data fusion" framework based on Bayesian principles has been developed on JET and W7-AS. The Bayesian analysis framework folds in uncertainties and inter-dependencies in the diagnostic data and signal forward-models, together with prior knowledge of the state of the plasma, to yield predictions of internal magnetic structure. A feature of the framework, known as MINERVA (J. Svensson, A. Werner, Plasma Physics and Controlled Fusion 50, 085022, 2008), is the inference of magnetic flux surfaces without the use of a force balance model. We discuss results from a new project to develop Bayesian inversion tools that aim to (1) distinguish between competing equilibrium theories, which capture different physics, using the MAST spherical tokamak; and (2) test the predictions of MHD theory, particularly mode structure, using the H-1 Heliac.Comment: submitted to Journal of Plasma Fusion Research 10/11/200

An implementation of Deflate in Coq

The widely-used compression format "Deflate" is defined in RFC 1951 and is based on prefix-free codings and backreferences. There are unclear points about the way these codings are specified, and several sources for confusion in the standard. We tried to fix this problem by giving a rigorous mathematical specification, which we formalized in Coq. We produced a verified implementation in Coq which achieves competitive performance on inputs of several megabytes. In this paper we present the several parts of our implementation: a fully verified implementation of canonical prefix-free codings, which can be used in other compression formats as well, and an elegant formalism for specifying sophisticated formats, which we used to implement both a compression and decompression algorithm in Coq which we formally prove inverse to each other -- the first time this has been achieved to our knowledge. The compatibility to other Deflate implementations can be shown empirically. We furthermore discuss some of the difficulties, specifically regarding memory and runtime requirements, and our approaches to overcome them

Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online

Hyperfine interaction and electronic spin fluctuation study on Sr2−x_{2-x}Lax_xFeCoO6_6 (x = 0, 1, 2) by high-resolution back-scattering neutron spectroscopy

The study of hyperfine interaction by high-resolution inelastic neutron scattering is not very well known compared to the other competing techniques viz. NMR, M\"ossbauer, PACS etc. Also the study is limited mostly to magnetically ordered systems. Here we report such study on Sr$_{2-x}$La$_x$FeCoO$_6$ (x = 0, 1, 2) of which first (Sr$_2$FeCoO$_6$ with x = 0) has a canonical spin spin glass, the second (SrLaFeCoO$_6$ with x = 1) has a so-called magnetic glass and the third (La$_2$FeCoO$_6$ with x = 2) has a magnetically ordered ground state. Our present study revealed clear inelastic signal for SrLaFeCoO$_6$, possibly also inelastic signal for Sr$_2$FeCoO$_6$ below the spin freezing temperatures $T_{sf}$ but no inelastic signal at all for for the magnetically ordered La$_2$FeCoO$_6$ in the neutron scattering spectra. The broadened inelastic signals observed suggest hyperfine field distribution in the two disordered magnetic glassy systems and no signal for the third compound suggests no or very small hyperfine field at the Co nucleus due to Co electronic moment. For the two magnetic glassy system apart from the hyperfine signal due only to Co, we also observed electronic spin fluctuations probably from both Fe and Co electronic moments. \end{abstract

Effect of spin orbit scattering on the magnetic and superconducting properties of nearly ferromagnetic metals: application to granular Pt

We calculate the effect of scattering on the static, exchange enhanced, spin susceptibility and show that in particular spin orbit scattering leads to a reduction of the giant moments and spin glass freezing temperature due to dilute magnetic impurities. The harmful spin fluctuation contribution to the intra-grain pairing interaction is strongly reduced opening the way for BCS superconductivity. We are thus able to explain the superconducting and magnetic properties recently observed in granular Pt as due to scattering effects in single small grains.Comment: 9 pages 3 figures, accepted for publication in Phys. Rev. Letter

The Diffusion of Humans and Cultures in the Course of the Spread of Farming

The most profound change in the relationship between humans and their environment was the introduction of agriculture and pastoralism. [....] For an understanding of the expansion process, it appears appropriate to apply a diffusive model. Broadly, these numerical modeling approaches can be catego- rized in correlative, continuous and discrete. Common to all approaches is the comparison to collections of radiocarbon data that show the apparent wave of advance of the transition to farming. However, these data sets differ in entry density and data quality. Often they disregard local and regional specifics and research gaps, or dating uncertainties. Thus, most of these data bases may only be used on a very general, broad scale. One of the pitfalls of using irregularly spaced or irregularly documented radiocarbon data becomes evident from the map generated by Fort (this volume, Chapter 16): while the general east-west and south-north trends become evident, some areas appear as having undergone anomalously early transitions to farming. This may be due to faulty entries into the data base or regional problems with radiocarbon dating, if not unnoticed or undocumented laboratory mistakes.Comment: 20 pages, 5 figures, submitted to Diffusive Spreading in Nature, Technology and Society, edited by Armin Bunde, J\"urgen Caro, J\"org K\"arger, Gero Vogl, Chapter 1

The use of Bayesian inversion to resolve plasma equilibrium

Recently, bayesian probability theory has been used at a number of experiments to fold uncertainties and interdependencies in the diagnostic data and forward models, together with prior knowledge of the state of the plasma, to increase accuracy of inferred physics variables. A new probabilistic framework, MINERVA, based on bayesian graphical models, has been used at JET and W7-AS to yield predictions of internal magnetic structure. A feature of the framework is the bayesian inversion for poloidal magnetic flux without the need for an explicit equilibrium assumption. Building on this, we discuss results from a new project to develop bayesian inversion tools that aim to (1) distinguish between competing equilibrium theories, which capture different physics, using the MAST spherical tokamak, and (2) test the predictions of MHD theory, particularly mode structure, using the H-1 Heliac. Specifically, we report on correction of the motional Stark effect, pickup coils, flux-loop constrained bayesian inferred equilibrium for varying toroidal flux.This work was jointly funded by the Australian Government through International Science Linkages Grant No. CG130047, the Australian National University, the United Kingdom Engineering and Physical Sciences Research Council, and by the European Communities under the contract of Association between EURATOM and CCFE

About one long-range contribution to K+ -> pi+ l+ l- decays

We investigate the mechanism of K+ -> pi+ l+ l- (l= e, mu) decays in which a virtual photon is emitted either from the incoming K+ or the outgoing pi+. We point out some inconsistencies with and between two previous calculations, discuss the possible experimental inputs, and estimate the branching fractions. This mechanism alone fails to explain the existing experimental data by more than one order-of-magnitude. But it may show itself by its interference with the leading long-range mechanism dominated by the a_1^+ and rho^0 mesons.Comment: 12 pages, RevTeX, epsf.sty, 2 embedded figure