Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Biosynthesis of iron and silver nanoparticles at room temperature using aqueous sorghum bran extracts

Iron and silver nanoparticles were synthesized using a rapid, single step, and completely green biosynthetic method employing aqueous sorghum extracts as both the reducing and capping agent. Silver ions were rapidly reduced by the aqueous sorghum bran extracts, leading to the formation of highly crystalline silver nanoparticles with an average diameter of 10 nm. The diffraction peaks were indexed to the face-centered cubic (fcc) phase of silver. The absorption spectra of colloidal silver nanoparticles showed a surface plasmon resonance (SPR) peak centered at a wavelength of 390 nm. Amorphous iron nanoparticles with an average diameter of 50 nm were formed instantaneously under ambient conditions. The reactivity of iron nanoparticles was tested by the H(2)O(2)-catalyzed degradation of bromothymol blue as a model organic contaminant.     

In situ measurement of the thermal and photo-assisted MOVPE of ZnTe using laser reflectometry

In situ laser reflectometry has been used to study the growth kinetics of ZnTe under pyrolytic and photo-assisted conditions from diisopropyltellurium (DIPTe) and dimethylzinc.triethylamine (DMZn.TEN). The growth rate of ZnTe was monitored as a function of VI:II ratio, temperature and laser power density. From the results a model for the growth involving the surface decomposition of DIPTe via a bimolecular reaction with methyl radicals homolytically released from the decomposition of DMZn.TEN is proposed. The activation energy under pyrolytic conditions in the low temperature regime for a 1:1 VI:II ratio was found to be 27.6 kcal/mol and under photo-assisted conditions this activation energy was seen to be lowered to 21.9 kcal/mol. The growth kinetics are explained in terms of two competitive processes. The decomposition of DMZn on the surface will be enhanced with increasing substrate temperature; however, the desorption rate of DIPTe will also be enhanced. The competitive nature of these two processes is seen to be particularly pronounced under laser illumination at high power densities, where a growth rate of 13 a.u./s was observed at 360°C, laser power density 67 mW/cm², whereas at 380°C for a similar power density the growth rate was seen to be reduced to 10.5 a.u./s.     

The Will of the People

The paper discusses the cybernetic mechanisms whereby our institutions fail to translate the will of the people into effective policies, and those by which the will of the people is an attenuated version of human potentiality in the first place. A systemic model is developed to account for the observed phenomena in terms of a cybernetic theory of the management process, and this is then exemplified from current dilemmas facing humankind. The model is subsequently extended to encompass the theory of viable systems, the principle of self-reference, and a model of self-hood which promotes new concepts that close the model into its starting point of human potential. The total approach bears on the capability of individuals, groups, institutions, societies and nations to realize themselves, and to thwart the dangers in which our civilization is plunged.     

Serum lithium determinations using flameless atomic absorption spectroscopy

The use of lithium salts and lithium carbonate in particular, in the control and treatment of certain types of psychiatric patients requires a means of monitoring lithium levels in serum. The currently accepted theraputic concentrations cover the range 0.5–1.5 milliequiv-alents of lithium per liter (mEq/l.) of serum with toxic symptoms reported above 1.5–2.0 mEq/l. Atomic absorption spectroscopy has been the method of choice for this analysis for several years and with the development of flameless methods the smaller samples may be used and multiple determinations made in about the same time required for a single determination using an air-acetylene flame and aspiration technique. The standard deviation of the flameless method is approximately 5% relative over the range of interest, 0.5–2.0 mEq/l. A comparison of results is made using a Perkin-Elmer Model 303 AA unit and a Varian Model AA5 with Model 63 Carbon Rod Atomizer.     

Rational Cherednik algebras and Hilbert schemes

Let H_c be the rational Cherednik algebra of type A_n-1 with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators, with associated graded ring where W is the nth symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod;• the associated graded Z-algebra has grB-lqgr?coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.This can be regarded as saying that U_c simultaneously gives a non-commutative deformation of h⊕h^*/W and of its resolution of singularities Hilb(n)→h⊕h^*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of H_c and its relationship to Hilb(n).