Failure of the Denjoy theorem for quasiregular maps in dimension

In 1929 L. V. Ahlfors proved the Denjoy conjecture which states that the order of an entire holomorphic function of the plane must be at least if the map has at least finite asymptotic values. In this paper, we prove that the Denjoy theorem has no counterpart in the classical form for quasiregular maps in dimensions . We construct a quasiregular map of with a bounded order but with infinitely many asymptotic limits. Our method also gives a new construction for a counterexample of Lindelöf's theorem for quasiregular maps of .

     

A study of 1,5-hydrogen shift and cyclization reactions of an alkali isomerized methyl linolenoate

Heating a mixture formed by alkali isomerization of methyl linolenoate (1) produces a complex mixture with the bicyclic hexahydroindenoic esters 4β-(7-methoxycarbonylheptyl)-5α-methyl-2,3,3aα,4,5,7aαhexahydroindene (CL5) and 4β-ethyl-5α-(6-methoxycarbonylhexyl)-2,3,3aα,4,5,7aα-hexahydroindene (CL6) as main components. Similar isomerization reactions of three synthetic model compounds, methyl 9Z,13E,15Z-octadecatrienoate (2), 9Z,14E,16E-octadecatrienoate (4) and 9Z,11E,15Z-octadecatrienoate (5) corroborated the results obtained with alkali isomerized methyl linolenoate.     

Computations on the growth of the first factor for prime cyclotomic fields

Denote byh(p) the first factor of the class number of the prime cyclotomic fieldk(exp (2i/p)). The theorem:h(p ₂)>h(p ₁) if 641 p ₂>p ₁ 19 is proved by straightforward computation.     

High-performance liquid chromatography/electrospray ionization tandem mass spectrometry for characterization of enzymatic degradation of 2,2'-bis(2-oxazoline)-linked poly-epsilon-caprolactone

This paper describes a straightforward and rapid on-line characterization using high-performance liquid chromatography/electrospray ionization tandem mass spectrometry (HPLC/ESI-MS(n)) of the enzymatic degradation products of 2,2'-bis(2-oxazoline)-linked poly-epsilon-caprolactone (PCL-O). These new PCL-O polymers are expected to be used in a variety of pharmaceutical and biomedical applications since they are degraded enzymatically by surface erosion. PCL-O was polymerized in a three-step reaction and characterized by (1)H-NMR and size-exclusion chromatography (SEC). Solvent cast polymer films were exposed to enzymatic degradation in phosphate buffer (pH 7.5, 1% pancreatin). The enzymatic degradation of the polymer produced a wide variety of water-soluble oligomers which were separated and identified by HPLC/ESI-MS(n). Optimization of the gradient HPLC method resulted in effective separation of the oligomers. Furthermore, specific structures of the oligomers were clearly identified by tandem mass spectrometry. According to these results, ester bonds seem to be most sensitive to enzymatic degradation and, correspondingly, pancreatic lipase seems to be mainly responsible for the enzymatic erosion of the PCL-O films. This novel mass spectrometric method provides important knowledge about the enzymatic degradation process and structure of the polymer which is difficult to ascertain by other conventional methods.     

Feasibility of density functional methods to predict dielectric properties of polymers

Feasibility of density functional theory (DFT) to predict dielectric properties such as polarizability of saturated polymers is investigated. Small saturated molecules, methane and propane, which is a monomer of polypropylene chain, are used in testing the methods. Results for polarizabilities based on several density functionals together with different basis sets are compared and contrasted with each other, with results by Hartree-Fock and second-order Moller-Plesset perturbation theory, as well as experimental data. The generalized gradient approximation PW91 method together with the 6-311++G(**) basis set is found to be the most suitable method, in terms of sufficient accuracy and computational efficiency, to calculate polarizabilities for large oligomers of polypropylene. The dielectric constant is then determined using the calculated polarizabilities and the Clausius-Mossotti equation. The molecular DFT methods at the PW916-311++G(**) level together with the Clausius-Mossotti equation give dielectric constants for saturated polymers such as polypropylene in good accordance with the experimental values.     

On the nonnegativity of operator products

Summary A bounded, not necessarily everywhere defined, nonnegative operator A in a Hilbert space <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathfrak{H}$ is assumed to intertwine in a certain sense two bounded everywhere defined operators B and C. If the range of A is provided with a natural inner product then the operators B and C induce two new operators on the completion space. This construction is used to show the existence of selfadjoint and nonnegative extensions of B*A and C*A.     

Measurement uncertainty in microbiological cultivation methods

Microbiological analyses are carried out on clinical, food, feed and environmental samples. The aims of the analyses are diagnostic or estimation of the safety or the quality of the sample. Important decisions are made on the basis of microbiological analyses. Little attention, however, is paid to the uncertainty of measurement of microbiological analyses. In microbiological cultivation techniques the result is obtained by counting individual objects. The normally low number of counted objects strongly affects the result of the analysis and its uncertainty. Because of the importance of the particle statistical variation to the uncertainty, the approaches developed for chemical analyses are not directly applicable to microbiology. This paper discusses microbial analyses and describes a novel guidance document for the estimation of measurement uncertainty in culturing methods [1]. Received: 7 October 2000 Accepted: 8 March 2001     

Serendipitous fragment-based drug discovery: ketogenic diet metabolites and statins effectively inhibit several carbonic anhydrases

Acetoacetic acid and R-3-hydroxy-butyric acid (BHB) are "ketone bodies", metabolites produced during the ketogenic diet. We discovered that they inhibit in the submicromolar-micromolar range several carbonic anhydrase (CA, EC 4.2.1.1) isoforms involved in relevant physiologic processes such as lipogenesis and tumorigenesis. The BHB fragment is also present in the molecules of most statins, widely used drugs for inhibiting cholesterol biosynthesis through the 3-hydroxy-3-methyl-glutaryl-CoA reductase pathway. Three such statins, atorvastatin, fluvastatin and rosuvastatin, showed submicromolar-low nanomolar inhibition of the fifteen human isoforms hCA I-XIV. Our data point out that in addition to their cholesterol lowering properties, these drugs may exert a therapeutic effect by inhibiting lipogenesis through mitochondrial CA inhibition. The statins are also low nanomolar inhibitors of the tumor-associated isoforms CA IX and XII. Based on the BHB/statin scaffolds, antiepileptic, antiobesity and antitumor compounds with higher affinity for the various CA isoforms involved in epileptogenesis (CA VA, VB, VII), lipogenesis (CA III, CA VA, CA VB) and tumorigenesis (CA IX and CA XII) may be designed.     

A Simple Preparation of 2-Pyrone Structural Unit

The synthesis of 4-substituted 2(5H)-furanones via organosilicon intermediates has been reported from this laboratory.¹ This method gave possibilities to extend the synthesis to include 2-pyrone species (e. g. 1), which are structure units of some natural products.