Asymptotic values of clique partition numbers

The problem is considered of decomposing a given graph into the minimum number of complete subgraphs. Asymptotic results are obtained for the case where the graph is the complement of a graph with relatively few unisolated vertices. This research was carried out while the author was visting Queen’s University, Kingston, whose hospitality is gratefully acknowledged.     

Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

Laser cooling of antihydrogen

We present a short review of the essential techniques of cooling free atoms by resonant laser radiation. The different contributions to the light forces are explained and their application to the problem of damping the thermal motion of free atoms is described. Due to quantum mechanical fluctuations of the light force there exists a limit temperature for a given atomic transition. Deceleration of atomic beams by the radiation pressure demands techniques to maintain the resonance condition while the Doppler shift of the decelerated atom is rapidly changing. Radiation forces may serve to compress and deflect slow atomic beams as well as to trap cold atoms. The possible use of pulsed laser radiation is discussed.     

A comparison of commercial sources of epitaxial material for GaN HFETs fabrication

This paper describes a comparison of material and device results obtained from AlGaN/GaN epitaxial HFET wafers from three commercial sources. Although all three sources supplied material to the same nominal specification, X-ray diffraction, Hall effect and C–V profiling revealed significant differences between them. Wafers from two of the suppliers showed poor inter-device isolation characteristics, indicative of a conducting buffer layer. Wafers from the third supplier showed excellent inter-device isolation, but C–V measurements showed that the AlGaN was about twice as thick as specified, resulting in devices with high pinch-off voltages (−16 V). For the wafers with poor buffer isolation, RF measurements on 1.2 μm gate length devices gave values of f_T 5.0 GHz and values of f_max from 8.0 to 11.7 GHz (exact values depending on DC bias conditions), while for the wafer with over-thick AlGaN the corresponding values were 8.0 and 20.0 GHz.     

Crystal structures of three 1-arylpropane-1,2-diol 2-aryl ether derivatives and the acetate of 2,6-dimethoxy-4-(1-(E)-propenyl)phenol

The crystal structures ofthreo-2-[2,6-dimethoxy-4-(1-(Z)-propenyl)phenoxy]-1-(4-hydroxy-3,5-dimethoxyphenyl)-1-propanol (7b),threo-2-(2,6-dimethoxy-4-propylphenoxy)-1-(3,4-5-trimethoxyphenyl)-1-propanol (7c), the diacetate oferythro-2-[2,6-dimethoxy-4-(1-(Z)-propenyl)phenoxy]-1-(4-hydroxy-3,5-dimethoxyphenyl)-1-propanol (8d) and the acetate of 2,6-dimethoxy-4-(1-(E)-propenyl)phenol (9) have been determined by single-crystal X-ray diffraction methods;7b,7c, and8d are structurally closely related to a class of optically-active neolignans of 1-arylpropane-1,2-diol 2-aryl ether type occurring in plant extracts. The reflection intensities were recorded at room temperature for7b and8d, at -140°C for7c and at -120°C for9. Compound7b crystallized as needles in the orthorhombic space groupPna2₁ witha=27.055(3),b=11.850(8),c=6.717(4)Å andZ=4.R became 0.041 for 1792 observed [I>3(I)] reflections used in the refinement of the structure. Compound7c crystallized as prisms in the monoclinic space groupP2₁/a witha=12.565(4),b=13.96(1),c=12.846(7)Å,=93.86(3)° andZ=4. The refinement gaveR=0.051 for 2101 observed [I>3(I)], unique reflections. Compound8d crystallized as prisms in the monoclinic space groupP2₁/n witha=13.182(4),b=8.518(5),c=23.322(5)Å,=106.12(2)° andZ=4. The finalR-valuebecame 0.042 for 2131 observed [I>3(I)], unique reflections. Compound9 crystallized as prisms in the orthorhombic space groupPca2₁ witha=18.216(4),b=9.483(3),c=7.416(2)Å andZ=4. TheR-value became 0.038 for 1502 observed [I>3(I)], unique reflections. The crystals of7b are stabilized by moderately strong hydrogen bonds. All the 1-arylpropane-1,2-diol 2-aryl ethers examined adopted conformations in which the aromatic groups tend to be as remote as possible. Observed double bond lengths in the propenyl groups of the compounds examined (1.30 Å) deviate significantly from the expected value (1.34 Å). The cause of this discrepancy is discussed.     

Selective syntheses of vinylenedithiathiophenes (VDTTs) and dithieno[2,3-b;2′,3′-d]thiophenes (DTTs); building blocks for π-conjugated systems

Selective syntheses of 3,4-vinylenedithiathiophenes (VDTTs) and dithieno[2,3-b;2′,3′-d]thiophenes (DTTs), having Ph, 4-CH₃OC₆H₄, 4-BrC₆H₄, 4-NO₂C₆H₄ and 4-(CH₃)₂NC₆H₄ groups, were achieved through the reaction of 1,8-diketone with phosphorus decasulfide (P₄S₁₀). The reaction could be shifted between VDTT and DTT by adding base (sodium bicarbonate) or acid (para-toluenesulfonic acid), respectively, to the reaction mixture. While the VDTTs were obtained in moderate yields, an important achievement has been made with the syntheses of the DTTs, obtaining them in higher yields compared with the previous report. Polymers of the VDTTs, which are the analogues of ethylenedioxythiophene, EDOT, were prepared using FeCl₃. Unfortunately, all attempts for their electropolymerization failed. Spin density calculations revealed that none of the VDTTs had a significant positive spin density at the ‘α’ carbon atoms of the thiophene ring. Considering their solubility and functional groups, which could be further derivatized, they are useful building blocks for the preparation of new organic materials.