Tensor products of contexts and complete lattices

Although the categoryCLC of complete lattices and complete homomorphisms does not possess arbitrary coproducts, we show that the tensor product introduced by Wille has the universal property of coproducts for so-called distributing families of morphisms (and only for these). As every family of morphisms into a completely distributive lattice is distributing, this includes the known fact that in the category of completely distributive lattices, arbitrary coproducts exist and coincide with the tensor products. Since the definition of tensor products is based on the notion of contexts and their concept lattices, many results on tensor products extend from complete lattices to contexts. Thus we introduce two kinds of tensor products for arbitrary families of contexts, a partial and a complete one, and establish universal properties of these tensor products.Presented by B. Jonsson.     

TrigonometricR matrices related to ‘dilute’ Birman-Wenzl-Murakami algebra

Explicit expressions for three series ofR matrices which are related to a dilute generalisation of the Birman-Wenzl-Murakami algebra are presented. Of those, one series is equivalent to the quantumR matrices of theD _n+1 ⁽²⁾ generalised Toda systems, whereas the remaining two series appear to be new.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

Posets isomorphic to their extensions

A standard extension for a poset P is a system Q of lower ends (descending subsets) of P containing all principal ideals of P. An isomorphism between P and Q is called recycling if [Y]Q for all YQ. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.     

The regioselectivity of the formation of 2-pyrazolylthiazoles and their precursors from the reaction of 2-hydrazinothiazoles with 4,4,4-trifluoro-1-hetaryl-1,3-butanediones

Reaction of 2-hydrazinothiazoles 1 with 1-thienyl- and 1-furyl-1,3-butanediones 2a,b in methanol in the presence of hydrochloric acid mainly leads to a mixture of pyrazoles 3 and pyrazolines 4 or pyrazoles 3 and 5 in strong acidic conditions. Isomeric hydrazones 6 and pyrazolines 4 were formed and isolated in these reactions in the absence of hydrochloric acid. It has been shown that the regioselectivity in the reaction of diketones 2 with hydrazine 1 is governed by both the concentration of acid and the nature of substituents in the 1,3-diketones 2. Cyclization of hydrazones 6 is shown to occur under milder conditions than dehydration for pyrazolines 4. The new heterocyclic compounds were prepared and fully characterized by NMR spectra and by X-ray analysis for 3c.     

Freon (CHF<Subscript>3</Subscript>)-assisted atomization for the determination of titanium using ultrasonic slurry sampling–graphite furnace atomic absorption spectrometry (USS–GFAAS): a simple and advantageous method for solid samples

A simple and advantageous method for the determination of titanium using graphite furnace atomic absorption spectrometry with slurry sampling has been developed. Titanium is one of the refractory elements that form thermally stable carbides in the graphite tube, which leads to severe memory effects. Trifluoromethane (Freon-23) was applied in the purge gas during the atomization step or alternatively just prior to the atomization to successfully eliminate the problems of carbide formation and increase the lifetime of the furnace tube which could be used for more than 600 heating cycles. A flow rate of 40 mL min^–1 (5% of Freon in argon) was used to obtain symmetrical peaks with no tailing. However, when the gas flow rate was too high (250 mL min^–1) the peak-tailing and memory effects reappeared. Ti was determined in various materials covering a wide range of concentrations, from 2.8 g g^–1 to 12% (m/m) Ti. The accuracy of the method was confirmed by analyzing certified reference materials (CRMs) or by comparing the results with those obtained using inductively coupled plasma–atomic emission spectrometry (ICP–AES) after decomposition of the samples. The materials analyzed were soil, plant, human hair, coal, urban particulate matters, toothpaste, and powdered paint.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Comments on the electronic properties of cyclopropane,cyclobutane, and cyclohexane

Spectral quantities of cyclopropane, cyclobutane, cyclohexane, and of several derivatives, have been calculated by a semiempirical all-valence electron SCF-CI MO method. In cyclopropane, HOMO is practically localized in the carbon-frame, and LVMO is purely so. In cyclobutane, these two MO's are based on C-H bonds, while cyclohexane holds an intermediate position. Despite the overall similarity-experimental and computed-of the spectra of these molecules, assignments are non-parallel. Like cyclopropane, cyclobutane can extend conjugation, but to a diminished degree; cyclohexane behaves in this respect like an acyclic alkane. An interpretation of this gradation, in terms of the nature of high-lying MO's, is proposed.     

Statistical Model for Organic Chromatographic Trace Analysis of Complex Samples. A Case Study: Plant Extracts

Abstract

The use of pooled plant extracts is described in the estimation of matrix interference in HPLC (UV and EC) determinations of organic compounds in plant extracts. An extract from freeze dried leaves of 134 different plant species was used for this purpose. It was split in different subgroups with solid extraction clean-up procedures. UV, EC and chromatographic data of the subgroups were used in the calculation of minimum concentrations of organic compounds which are still accurately determinable in plant samples with HPLC methods. The UV and/or EC characteristics of the compound must be known. The contribution of the solid phase extraction procedures and of the analytical system to the selectivity of the method can be estimated. Information is also supplied which allows rapid comparison of the selectivity of the UV and EC (single, or dual parallel) detectors for the determination of a specified compound.