Refinement of Vahlen's Theorem for Minkowski Bases of Three-Dimensional Lattices

In the paper, the analog of Vahlen's theorem for Minkowski bases of three-dimensional lattices is sharpened.     

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth-grade students’ multiplicative structures and their construction of improper fractions. Students’ multiplicative structures are the units coordinations that they can take as given prior to activity—i.e., the units coordinations that they have interiorized. This research indicates that the construction of improper fractions requires having interiorized three levels of units. Students who have interiorized only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. These findings call for a revision in Steffe's hypothesis (Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20, 267-307) that upon the construction of the splitting operation, students’ fractional schemes can be regarded as essentially including improper fractions. While the splitting operation seems crucial in the construction of improper fractions, it is not necessarily accompanied by the interiorization of three levels of units.     

The explicit formulas and evaluations of Ramanujan's theta-function ψ

We define two quotients of theta-function ψ depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function, theta-function φ, and the Ramanujan-Weber class invariants. Explicit formulas for determining values of the theta-function ψ are derived, and several examples will be given. In addition, we give some applications of these parameters for the famous Rogers-Ramanujan continued fraction R(q), Ramanujan's cubic continued fraction G(q), and the modular j-invariant.     

‘Classical’ convergence theorems for generalized continued fractions

In this paper the classical convergence theorems by Śleszyński-Pringsheim, Worpitzky and Van Vleck for ordinary continued fractions will be generalized to continued fractions generalizations (along the lines of the Jacobi–Perron algorithm) with four-term recurrence relations.      

ON THE MINIMAL PERIOD FOR PERIODIC SOLUTION PROBLEM OF NONLINEAR HAMILTONIAN SYSTEMS

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ALGORITHMS FOR LACUNARY VECTOR VALUED RATIONAL INTERPOLANTS

Efficient algorithms are established for the computation of bivariate lacunary vector valued rational interpolants based on the branched continued fractions and a numerical example is given to show how the algorithms are implemented,     

Quality test of a mechanical switching valve for two-dimensional open tubular gas chromatography

A commercially available mechanical micro switching valve has been tested for inertness. The test mixtures used were a modified Grob test and a solution consisting of eight typical aroma compounds, both at the 2-ng level. Cryogenic focusing of the sample after chromatography including one passage through the valve revealed that irreversible interaction of some test substances with valve components was rather small. A composite main peak of a chromatogram from a fruit juice aroma exhibiting a “mixed odor” was selectively transferred (heart-cutting) to a second column and resolved into three peaks. GC-GC/sniffing assigned a minor peak to a substance formerly hidden under the composite peak. Separate heating of the valve during temperature programming of the column was found to be unnecessary.     

3‐Designs of PSL(2, 2n) With block sizes 4 and 5

We determine the distribution of 3‐designs among the orbits of 4‐ and 5‐element subsets under the action of PSL(2,2ⁿ) on the projective line. Thus we give complete information on all Kramer–Mesner matrices for the action of PSL(2,2ⁿ) on 3‐sets versus 4‐ and 5‐sets. As a consequence, all 3‐designs with block sizes 4 and 5 and automorphism group PSL(2,2ⁿ) can immediately be obtained. © 2003 Wiley Periodicals, Inc.     

Use of capillary electrophoresis to monitor wheat maturation

Summary The maturation of wheat varieties with different harvest times has been examined by high-performance capillary electrophoresis. The unique proteins of the albumin, gliadin and glutenin fractions of Hungarian winter wheat cultivars Bánkúti 1201 (early harvest time), Martonvásári 23 (medium harvest time), and Martonvásári 15 (semi-late harvest time) were analysed. An acidic phosphate buffer containing a polymeric additive and organic modifiers was used in capillary zone electrophoresis mode. Formation of albumin followed the same time scale, and the patterns were quite similar, for all three cultivars. For gliadins and glutenins the time scale and patterns were different and some correlation was observed between harvest time and gliadin formation. Presented at Balaton Symposium '01 on High-Performance Separation Methods, Siófok, Hungary, September 2–4, 2001     

GRAFTING OF PEROXIDE-INITIATED MALEIC ANHYDRIDE ON SPHERICAL PE／PP IN-REACTOR BLEND GRANULES

Spherical polyethylene/polypropylene (PE/PP) in-reactor blend granules with various ethylene/propylene molar ratios and high porosity were synthesized using a high yield TiCl4/MgCl2 supported catalyst. A solution of benzoyl peroxide (BPO)/maleic anhydride (MAH)/xylene (interfacial reagent) or BPO/MAH/St (comonomer) was absorbed onto the PE/PP inreactor blend granules, and solid phase gratt polymerization of MAH on PE/PP was conducted. The amount of grafted MAH on PE/PP was measured through chemical titration. The results showed that solid phase graft polymerization of MAH in PE/PP in-reactor blend granules produced graft copolymer with high amount of grafted MAH, and the amount of grafted MAH was raised slightly when St was introduced as comonomer. The graft in-reactor blend was fractionated into five fractions through temperature-gradient extraction fractionation (TGEF), and the fractions were analyzed by FTIR. The results revealed that MAH is mainly grafted on the PE segments, whereas MAH was predominantly grafted on the PP segments when St was present in the graft polymerization system. In addition, the final product is still in the form of regular spherical granules, which is beneficial for industrial processing.