Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson (Ann. of Math. (2) 100 (1974) 433) is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a “coordinate” approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of Arveson. We solve some problems posed in Arveson (1974).     

Deciding uniqueness in norm maximization

NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytopeP, and whose question is whether, onP, the global maximum of the Euclidean norm is attained at more than one vertex ofP. The NP-hardness persists even for the restricted problem in whichP is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.Research of the first author was supported in part by the Deutsche Forschungsgemeinschaft. Research of the second author was supported in part by the National Science Foundation.     

Searle freed of every flaw

Strong Al presupposes (1) that Super-Searle (henceforth ‘Searle’) comes to know that the symbols he manipulates are meaningful, and (2) that there cannot be two or more semantical interpretations for the system of symbols that Searle manipulates such that the set of rules constitutes a language comprehension program for each interpretation. In this paper, I show that Strong Al is false and that presupposition #1 is false, on the assumption that presupposition #2 is true. The main argument of the paper constructs a second program, isomorphic to Searle’s, to show that if someone, say Dan, runs this isomorphic program, he cannot possibly come to know what its mentioned symbols mean because they do not mean anything to anybody. Since Dan and Searle do exactly the same thing, except that the symbols they manipulate are different, neither Dan nor Searle can possibly know whether the symbols they manipulate are meaningful (let alone what they mean, if they are meaningful). The remainder of the paper responds to an anticipated Strong Al rejoinder, which, I believe, is a necessary extension of Strong Al.     

Improved predictability of two-dimensional turbulent flows using wavelet packet compression

We propose to use new orthonormal wavelet packet bases, more efficient than the Fourier basis, to compress two-dimensional turbulent flows. We define the “best basis” of wavelet packets as the one which, for a given enstrophy density, condenses the L² norm into a minimum number of non-negligible wavelet packet coefficients. Coefficients below a threshold are discarded, reducing the number of degrees of freedom. We then compare the predictability of the original flow evolution with several such reductions, varying the number of retained coefficients, either from a Fourier basis, or from the best-basis of wavelet packets. We show that for a compression ratio of 1/2, we still have a deterministic predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. Likewise, for compression ratios of 1/20 and 1/200 we still have statistical predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. In fact, the significant wavelet packet coefficients in the best-basis appear to correspond to coherent structures. The weak coefficients correspond to vorticity filaments, which are only passively advected by the coherent structures. In conclusion, the wavelet packet best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components. It gives us some hope of drastically reducing the number of degrees of freedom necessary to the computation of two-dimensional turbulent flows.     

Spectrum Determination of Terahertz Sources Using Fabry-Perot Interferometer and Bolometer Detector

We describe a method to determine the radiation spectrum of terahertz sources using a transmitted Fabry-Perot interferometer and a bolometer detector. Our novel Fabry-Perot spectrometer can measure the spectrum of an unknown broadband sub-terahertz and terahertz source. The spectrometer does not need to be pre-tuned. We develop a new algorithm to support the measurements using this spectrometer. Our technique allows us to measure the spectrum over more than an octave, and our numerical algorithm is very stable and robust, providing for an accurate spectrum determination. Two Gunn oscillation sources with the main frequencies at 200 GHz and 600 GHz are used to test the proposed method. The spectrum extracted from the measured data shows that this method is accurate and reliable.     

Interpretation of the Lavrentiev phenomenon by relaxation

We consider functionals of the calculus of variations of the form F(u)= ∝₀¹ f(x, u, u′) dx defined for u ε W^1,∞(0, 1), and we show that the relaxed functional with respect to weak W^1,1(0, 1) convergence can be written as

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, where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F.     

A Water Rich Crystalline Hydrate Formed by 1,4,7,10-Tetraazacyclododecane, Siliconhydrofluoric Acid and Water: Structure and Properties

Novel complementary associations have been found in a crystalline ternarycomplex of the macrocyclic tetramine, [12]aneN₄ (e.g. =1,4,7,10-tetraazacyclododecane, cyclen, L), hexafluorosilicate ions andwater (I). The final compound belongs to the host—guest type with themacrocycle as the host (H), the inorganic entity as a guest (G) and watermolecules. It was characterized by X-ray techniques, IR spectroscopy andthermogravimetric study. X-ray crystal analysis revealed that the structureis built of the charged entities, with a 3D-network uniting thequadriprotonated form of the macrocycle, hexafluorosilicate as counterionsand lattice water molecules viaN—H⁺...F^-,N—H⁺...O(w),OH...F^- and O—-H(w)...O(w)hydrogen bonds. The overall complex stoichiometry comprises fourSiF₆ ^2- anions and seven water molecules per two[LH₄]⁴⁺ cations. Crystals of (I) are monoclinic,space group P2₁/n with a = 14.585(3), b = 16.384(2), c =16.500(3) Å, = 92.84(3)°, V = 3938(1)ų and Z = 8 forC₈H₃₁F₁₂N₄O_3.5Si₂.     

[1,2,5]Oxadiazolo[3,4-d]pyridazine 1,5,6-trioxides: efficient synthesis via the reaction of 3,4-bis(hydroxyimino)methyl)-1,2,5-oxadiazole 2-oxides with a mixture of concentrated nitric and trifluoroacetic acids and structural characterization

An efficient synthesis of [1,2,5]oxadiazolo[3,4-d]pyridazine 1,5,6-trioxides (1) from 3,4-bis(hydroxyimino)methyl)-1,2,5-oxadiazole 2-oxides using a mixture of concentrated nitric and trifluoroacetic acids has been developed. The scope of the unconventional reaction was established. The 4,7-dinitro[1,2,5]oxadiazolo[3,4-d]pyridazine 1,5,6-trioxide 1f represents a new high energy compound, unfortunately with low thermal stability. The parent [1,2,5]oxadiazolo[3,4-d]pyridazine 1,5,6-trioxide 1c was studied by single-crystal X-ray diffraction analysis which revealed a planar molecule with an unusually long intracyclic NN bond of 1.668(5)?Å and unexpected exo-cyclic bond angles at the nitroxyl nitrogen atoms. In the crystal, the molecules of 1c are bound to each other by strong π-π stacking and CH?O hydrogen bonding interactions into a three-dimensional framework that results in a high crystal density of 1.833?gcm^?3.     

First platinum(ii)–alkaline-earth acetate-bridged complexes Ptii(m-OAc)4Mii(AcOH)4 (M = Ca,Sr, Ba)

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