On the Padé table for e x and the simple continued fractions for e and e L/M

A recently observed connection between some Padé approximants for the exponential series and the convergents of the simple continued fraction for e is established, leading to an alternative proof of the latter. Similar results for the simple continued fraction e ²,e ^1/M and e ^2/M , when M is a natural number greater than one, are derived.      

Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences

The semi-infinite Toda lattice is the system of differential equations d _n (t)/dt = _n (t)(b _n+1(t) – b _n (t)), db _n (t)/dt = 2( _n ²(t) – _n–1 ²(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences _n (t), b _n (t) which satisfy the conditions _n (0) = _n ,, b _n (0) = b _n , where _n > 0 and b _n are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences _n and b _n are bounded. When at least one of the known sequences _n and b _n is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences _n and b _n such that the system has a unique solution. The results are illustrated with a typical example where the sequences _i (t), b _i (t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d²/dt ² log h _n = h _n+1 + h _n–1 – 2h _n , n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation.     

The distribution of special subsets of the Farey sequence

We will examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are not divisible by a fixed prime p. In particular, we will provide an asymptotic result on the distribution of H-tuples of consecutive fractions in , as Q→∞.     

Units in some families of algebraic number fields

Multi-dimensional continued fractions associated with are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).

     

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of

     

Nuclear Physics in Norway, 1933–1955

In the late 1940s and the 1950s, Norwegian nuclear scientists, engineers, and administrators were deeply split over their nation’s goals, organization, politics, and tools for research in nuclear physics. One faction was determined to build a nuclear reactor in Norway, while another fiercely opposed the reactor plans and focused on particle accelerators. The first faction comprised scientific entrepreneurs and research technologists, the second academic scientists, most of whom began their research careers in nuclear physics in the 1930s. To understand this conflict, I trace the development of nuclear research in Norway from the early 1930s to the mid-1950s, placing it within an international context. Roland Wittje is working on his habilitation thesis in the History of Science Unit at the University of Regensburg, Germany.     

Molecular properties of hemicelluloses located in the surface and inner layers of hardwood and softwood pulps

The molecular properties of hemicelluloses located in the surface and inner layers of fibers present in hardwood and softwood pulps, together with the effects of different bleaching processes on these properties, have been investigated in this study. In order to separate the hemicelluloses located in these two layers, fibers were subjected to mechanical peeling and then separated by filtration into surface (filtrate) and inner layer materials. The materials thus obtained were characterized with respect to their polysaccharide compositions and uronic acid contents. The molar mass parameters of the hemicelluloses (extracted by alkali) were determined by employing size-exclusion chromatography in combination with off-line MALDI mass spectrometry. For all of the pulps examined, the relative content of xylan was found to be greater in the surface layer of the fiber than in the corresponding inner layer. The xylan polymers of the surface layer exhibited higher molar masses and lower frequencies of uronic acid side groups than did the xylans in the inner fiber layer. In connection with ozone treatment, hexenuronic acid residues in the surface layer xylan were removed to a greater extent than in the case of the inner layers, indicating a gradient for the reaction with ozone across the fiber wall. The xylan polymer remaining on the surface of the softwood pulps after completion of the chlorine dioxide bleaching process was predominantly uncharged.     

Proper Tuning of Comprehensive Two‐Dimensional Gas Chromatography (GC×GC) to Optimize the Separation of Complex Oil Fractions

Comprehensive two‐dimensional gas chromatography (GC×GC) is an utterly suitable separation technique for the analysis of complex samples, such as oil fractions. Once the two columns and the operating conditions are properly tuned, the technique is able to provide a detailed characterization of such materials. Some considerations applying to the tuning of a GC×GC system for a specific separation are presented and discussed. The authors present a number of different column sets and conditions which allow the separation of a non‐aromatic hydrocarbon solvent, a kerosene, the light end of a crude oil, and an olefinic fraction, respectively. The highly structured GC×GC chromatograms, together with chemical knowledge about the samples, provide a much more comprehensive characterization of the samples than hitherto possible.     

Inhibition of thermal induced protein denaturation of extract/fractions of Withania somnifera and isolated withanolides

This study describes the in vitro inhibition of protein denaturation of extract/fractions of Withania somnifera and isolated withanolides including 20β hydroxy-1-oxo(22R)-witha-2,5,24 trienolide (1), (20R,22R-14α,20α)-dihydroxy-1-oxowitha-2,5,16,24 tetraenolide (2). The results showed that the extract/fractions of the plant evoked profound inhibitory effect on thermal-induced protein denaturation. The chloroform fraction caused the most dominant attenuation of 68% at 500 μg/mL. The bioactivity-guided isolation from chloroform fraction led to the isolation of compounds 1 and 2 that showed profound protein inhibition with 78.05% and 80.43% effect at 500 μg/mL and thus strongly complimented the activity of extract/fractions. In conclusion, extract/fractions of W. somnifera possessed strong inhibition of protein denaturation that can be attributed to these isolated withanolides.     

Thermodynamics of fusion and sublimation for a homologous series of eleven alkane-α,ω-diols HO-(CH2)n-OH: Structure-related odd–even effect

Temperatures and enthalpies of fusion and solid-to-solid transition of a homologous series of linear alkane-α,ω-diols, HO-(CH₂)_n-OH where n = (6 to 16), were measured by differential scanning calorimetry (d.s.c.). The enthalpies and temperatures of fusion displayed a marked odd–even effect as a function of the number of methylene groups in the alkyl chains, with even terms showing higher values than odd terms. Thermodynamic parameters of fusion were compared with those of isoelectronic linear alkanes and earlier measured alkane-α,ω-diamines, alkane-α,ω-diamides and alkane-α,ω-dinitriles. Results were discussed with reference to the effects of chain length on crystal structures and packing patterns raised on hydrogen bonding and hydrophobic interaction interplay. The enthalpies of sublimation at T = 298.15 K were obtained from the enthalpies of fusion and the literature enthalpies of vaporisation, both adjusted to 298.15 K. A smoother odd–even pattern was observed for the enthalpies of sublimation.