Characterization of supercommuting matrices

For a given n-by-n matrix A, we consider the set of matrices which commute with A and all of whose principal submatrices commute with the corresponding principal submatrices of A. The properties of this set are examined, with particular attention to its dimension.     

The completely positive and doubly nonnegative completion problems

We prove the following. Let G be an undirected graph. Every partially specified symmetric matrix, the graph of whose specified entries is G and each of whose fully specified submatrices is completely positive (equal to BB^T for some entrywise nonnegative matrix B), may be completed to a completely positive matrix if and only if G is a block-clique graph (a chordal graph in which distinct maximal cliques overlap in at most one vertex). The same result holds for matrices that are doubly nonnegative (entrywise nonnegative and positive semidefinite).     

Orientation and relaxation of polymer–clay solutions studied by rheology and small-angle neutron scattering

The influence of shear on viscoelastic solutions of poly(ethylene oxide) (PEO) and clay [montmorillonite, i.e., Cloisite NA+ (CNA)] was investigated with rheology and small-angle neutron scattering (SANS). The steady-state viscosity and SANS were used to measure the shear-induced orientation and relaxation of the polymer and clay platelets. Anisotropic scattering patterns developed at much lower shear rates than in pure clay solutions. The scattering anisotropy saturated at low shear rates, and the CNA clay platelets aligned with the flow, with the surface normal parallel to the gradient direction. The cessation of shear led to partial and slow randomization of the CNA platelets, whereas extremely fast relaxation was observed for laponite (LRD) platelets. These PEO–CNA networklike solutions were compared with previously reported PEO–LRD networks, and the differences and similarities, with respect to the shear orientation, relaxation, and polymer–clay interactions, were examined. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3102–3112, 2004     

Thermal conductivity, dislocation density and GaN device design

The performance of high power transistor devices is intimately connected to the substrate thermal conductivity. In this study, the relationship between thermal conductivity and dislocation density is examined using the 3 omega technique and free standing HVPE GaN substrates. Dislocation density is measured using imaging cathodoluminescence. In a low dislocation density regime below 10⁵ cm⁻², the thermal conductivity appears to plateau out near 230 W/K m and can be altered by the presence of isotopic defects and point defects. For high dislocation densities the thermal conductivity is severely degraded due to phonon scattering from dislocations. These results are applied to the design of homoepitaxially and heteroepitaxially grown HEMT devices and the efficiency of heat extraction and the influence of lateral heat spreading on device performance are compared.     

Deuterium exchange of 4-pyrimidones and 4-pyrimidthiones

The hydrogen-deuterium exchange of H-2 in 4-pyrimidone, a number of 1- and 3-alkyl-4-pyrimidones and several of their thione analogs in deuterium oxide at moderate temperatures is reported. This reaction is apparently not susceptible to acid or base catalysis. Cations of the 4-pyrimidones were found not to exchange H-2 at all in acid media. Catalytic amounts of bases do not alter the exchange rates whereas more concentrated alkali decompose the pyrimidone. However, 1,4(3,4)-dihydro-1,3-dimethyl-4-oxopyrimidinium iodide (and its 6-methyl analog) exchange H-2 quite rapidly in neutral deuterium oxide, almost instantly when triethylamine is added as a catalyst, and very slowly in dilute solutions of deuterium chloride in deuterium oxide. Mechanisms are proposed to account for these phenomena.     

Synthesis and reactivity of dimeric Ar'TlTlAr' and trimeric (Ar"T1)3 (Ar', Ar" = bulky terphenyl group) thallium(I) derivatives: Tl(I)-Tl(I) bonding in species ligated by monodentate ligands

The synthesis and characterization of three new organothallium(I) compounds are reported. Reaction of (Ar'Li)(2) (Ar' = C(6)H(3)-2,6-(C(6)H(3)-2,6-Pr(i)(2))(2)) and Ar"Li (Ar" = C(6)H(3)-2,6-(C(6)H(3)-2,6-Me(2))(2)) with TlCl in Et(2)O afforded (Ar'Tl)(2) (1) and (Ar' 'Tl)(3) (2). The "dithallene" 1 is the heaviest group 13 dimetallene and features a planar, trans-bent structure with Ar'Tl-Tl = 119.74(14) degrees and Tl-Tl = 3.0936(8) A. Compound 2 is the first structurally characterized neutral, three-membered ring species of formula c-(MR)(3) (M = Al-Tl; R = organo group). The Tl(3) ring has Tl-Tl distances in the range ca. 3.21-3.37 A as well as pyramidal Tl geometries. The Tl-Tl bonds in 1 and 2 are outside the range (2.88-2.97 A) of Tl-Tl single bonds in R(2)TlTlR(2) compounds. The weak Tl-Tl bonding in 1 and 2 leads to their dissociation into Ar'Tl and Ar' 'Tl monomers in hexane. The Ar'Tl monomer behaves as a Lewis base and readily forms a 1:1 donor-acceptor complex with B(C(6)F(5))(3) to give Ar'TlB(C(6)F(5))(3), 3. Adduct 3 features an almost linear thallium C(ipso)-Tl-B angle of 174.358(7) degrees and a Tl-B distance of 2.311(2) A, which indicates strong association. Treatment of 1 with a variety of reagents resulted in no reactions. The lower reactivity of 1 is in accord with the reluctance of Tl(I) to undergo oxidation to Tl(III) due to the unreactive character of the 6s(2) electrons.     

Rectangles inscribed in symmetric continua

LetQ be any rectangle and letK ^d (d 2) be a continuum which is either symmetric across a hyperplane or symmetric through a pointz K. We show thatK contains the vertices of a rectangle similar toQ which exhibits the same symmetry as doesK.