Maximal Equilateral Sets

A subset of a normed space $X$ X is called equilateral if the distance between any two points is the same. Let $m(X)$ m ( X ) be the smallest possible size of an equilateral subset of $X$ X maximal with respect to inclusion. We first observe that Petty’s construction of a $d$ d - $X$ X of any finite dimension $d\ge 4$ d ≥ 4 with $m(X)=4$ m ( X ) = 4 can be generalised to give $m(X\oplus _1\mathbb R )=4$ m ( X ⊕ 1 R ) = 4 for any $X$ X of dimension at least $2$ 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $\Gamma $ Γ , $m(\ell _p(\Gamma ))$ m ( ? p ( Γ ) ) is finite and bounded above by a function of $p$ p , for all $1\le p<2$ 1 ≤ p < 2 . Also, for all $p\in [1,\infty )$ p ∈ [ 1 , ∞ ) and $d\in \mathbb N $ d ∈ N there exists $c=c(p,d)>1$ c = c ( p , d ) > 1 such that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $c$ c from $\ell _p^d$ ? p d . Using Brouwer’s fixed-point theorem we show that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $3/2$ 3 / 2 from $\ell _\infty ^d$ ? ∞ d . A graph-theoretical argument furthermore shows that $m(\ell _\infty ^d)=d+1$ m ( ? ∞ d ) = d + 1 . The above results lead us to conjecture that $m(X)\le 1+\dim X$ m ( X ) ≤ 1 + dim X for all finite-normed spaces $X$ X .     

Nuclear Magnetic Resonance Spectroscopy Study of a Disaccharidic Crown Ether.

Abstract

The disaccharidic anhydro derivative 6-O-(5,6-anhydro-3-deoxy-1,2-O-isopropylidene α-D-glucofuranos-3-yl)-1,2-O-isopropylidene-3-O-n-dodecyl-α-D-glucofuranose (1) led to the disaccharidic crown ether 2 in 40% yield when treated at low concentration with 2.5 eq. of KOH in toluene-Me₂SO. Compound 2 structure was proved through a detailed NMR analysis (¹H, ¹³C, ¹H-¹H and ¹³C-¹H 2D correlations). This structural elucidation indicated that compound 2 resulted from the intramolecular attack of the C-5-O^?alkoxide group, generated in the basic medium, on the C-6′ carbon of the 5′,6′-anhydro group.     

Centralised reallocation of emission permits using DEA

In this paper a data envelopment analysis (DEA) approach to the problem of emission permits reallocation is presented. It can be used with conventional command and control as well as with an allowance market. It uses a centralized point of view, which represents the common good. In the model it is assumed that firms produce two types of outputs: desirable outputs (i.e. good outputs with positive value for consumers) and undesirable outputs (i.e. bad outputs with negative value for consumers, such as emissions of pollutants). The proposed approach has three phases, which correspond to three objectives that are pursued lexicographically. The three objectives are maximizing aggregated desirable production, minimizing undesirable total emissions and minimizing the consumption of input resources. The relative priority of these objectives is defined by the regulator. The whole approach is units-invariant and does not require information on input and output prices. The approach is applied on a dataset from the Swedish pulp and paper industry.     

Effects of tributyltin(IV) chloride on fertilization of Styela plicata (Ascidiacea: Tunicata): II. Scanning and transmission electron microscopy studies

The morphological aspects of Styela plicata fertilization after treatment with tributyltin(IV) chloride are described by means of scanning and transmission electron microscopy investigations. Alterations have been shown both on female and male gametes; spermatozoa, all the egg envelopes and the mitochondria of the egg cortical cytoplasm are modified in relation to incubation time. As a consequence, the damage to gametes blocks sperm–egg interaction and fertilization does not occur. Copyright © 2003 John Wiley & Sons, Ltd.     

Essential oil composition of the different parts of Eryngium bourgatii Gouan from Spain

The essential oil extracted from the different parts of Eryngium bourgatii Gouan: stems + leaves (E.b.SL), inflorescences (E.b.I) and roots (E.b.R), have been extracted by steam distillation and analysed by gas chromatography (GC) and gas chromatography coupled to mass spectrometry (GC-MS). Quantitative but not qualitative differences have been found between the analysed parts. The principal compounds from the inflorescences oil were found to be phyllocladene (37.6%) and bicyclogermacrene (15.1%), while the oil from stems and leaves showed phyllocladene (20.4%), gamma-muurolene (11.8%) and (E)-caryophyllene (10.1%) as main ones. The oil from the roots presented gamma-muurolene (15.4%) and phyllocladene (15.0%) as major constituents. It is worth mentioning the presence of a diterpene, phyllocladene, as main compound of the essential oil. This is the first report on the essential oil of this species.