On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.     

On the periodic points of a two-parameter family of maps of the plane

We study the dynamics of a two-parameter family of noninvertible maps of the plane, derived from a model in population dynamics. We prove that, as one parameter varies with the other held fixed, the nonwandering set changes from the empty set to an unstable Cantor set on which the map is topologically equivalent to the shift endomorphism on two symbols. With the help of some numerical work, we trace the genealogies of the periodic points of the family of period 5, and describe their stability types and bifurcations. Among our results we find that the family has a fixed point which undergoes fold, flip and Hopf bifurcations, and that certain families of period five points are interconnected through a codimension-two cusp bifurcation.     

Organic phase conversion of bulk (wurtzite) ZnO to nanophase (wurtzite and zinc blende) ZnO

We describe the all-organic phase conversion of bulk commercial ZnO in the wurtzite modification to sub-30 nm ZnO that we find to be partially in the zinc blende [, a=4.568(3) Å] modification. The conversion involves refluxing ZnO in 2,4-pentanedione (acetylacetone) at 413 K to form the zinc 2,4-pentanedionate, which is decomposed by heating at 573 K in an appropriate high-temperature solvent such as dibenzylether to form nanophase ZnO. This nanophase, partially zinc blende ZnO can also be obtained in a single step by heating commercial zinc 2,4-pentanedionate in refluxing dibenzylether. Thermodiffractometry suggests that the conversion of zinc blende ZnO to wurtzite ZnO commences near 650 K.     

Physico-Chemistry of the Limestone Sulphation Process

Aspects of the mechanism of the overall reaction between CaCO₃/CaO and SO₂/SO₃ under oxidizing conditions are discussed. The limestone and lime sulphation processes were carried out in a thermobalance under conditions relevant to atmospheric fluidized bed combustion. Sulphated samples, prepared in the form of cross-section particles, were examined in a scanning electron microscope by energy-dispersive X-ray and back-scattered electron imaging. Photomicrographs are presented. The reaction proceeded from the outer surface of the particles and along the pores. Surface textural changes during the reaction were considered. The layer of products was identified as controlling both the rate and extent of limestone/lime sulphation. In the products, two sulphur-bearing solids (CaSO₄ and CaS) were identified. The presence of CaS, which may cause difficulties in practice, is attributed to CaSO₃ disproportionation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Structure analytical methods for quantitative reference applications

The analytical methods mass spectrometry, UV/Vis, IR, Raman, Fluorometry, XRD, Mössbauer, and NMR used to elucidate chemical structure are evaluated regarding their capabilities to be used as primary analytical techniques in quantitative measurements, considering the criteria in the CCQM definition of primary methods. This includes a review of the respective measurement equations, the evaluation of the measurement uncertainty, and a discussion of evidence for the “highest metrological level”, as obtained from intercomparisons in contest with other methods. It is shown that only few methods fulfill the CCQM criteria. Quantitative NMR spectroscopy is one of them and may be considered as a potential primary method as recommended by CCQM because of being free of empirical factors in the uncertainty budget.     

Hydrothermal synthesis and crystal structure of a layered vanadium oxide with an interlayer metal coordination complex: [Co(phen)3][V10O26]·H2O

A novel compound, [Co(phen)₃][V₁₀O₂₆]·H₂O, was hydrothermally synthesized and characterized by single-crystal X-ray diffraction. This compound crystallizes in the orthorhombic space group Ccca with a=13.447(3), b=29.936(6), c=23.252(5) Å, V=9360(3) Å³, Z=8 and R=0.0285. Data were collected on a Rigaku R-AXIS RAPID IP diffractometer at 293 K in the range of 1.36<θ<24.99°. The structure of the compound consists of vanadium oxide layers, which are built up from the infinite VO₄ chains by corners and edges sharing. The [Co(phen)₃]²⁺ complexes occupy the interlayer space and contact each other via π−π stacking interactions of the phen groups to form infinite one-dimensional chains.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

Incomplete idempotent Schröder quasigroups and related packing designs

Summary In this paper it is proved that, for any positive integern 2, 3 (mod 4),n 7, there exists an incomplete idempotent Schröder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern 0, 1 (mod 4), there exists an idempotent Schröder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.Research supported by NSERC grant A-5320.Research supported by NSFC grant 19231060-2.     

A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.