How to compare power towers?

Lithuanian Mathematical Journal - We provide various tools that give an answer to the following question: given two distinct finite tower powers with entries at least 1 and (possibly) different...     

Epidemics spread in heterogeneous populations

Individuals building populations are subject to variability. This variability affects progress of epidemic outbreaks, because individuals tend to be more or less resistant. Individuals also differ with respect to their recovery rate. Here, properties of the SIR model in inhomogeneous populations are studied. It is shown that a small change in model’s parameters, e.g. recovery or infection rate, can substantially change properties of final states which is especially well-visible in distributions of the epidemic size. In addition to the epidemic size and radii distributions, the paper explores first passage time properties of epidemic outbreaks.     

An analysis of nonlinear normal contact microvibrations excited by a harmonic force

This paper presents the results of numerical investigations of nonlinear normal contact microvibrations excited by a harmonic force in a system of two bodies in planar contact that models the slideway connections of machine tools. The aim of the computation analysis was to determine the plots of the forced vibrations as a function of frequency and amplitude and to study the resonances taking place in the system. The studies showed that beside the main resonance, at exciting frequencies lower than the natural frequency (f _w < f _o ), a number of interesting phenomena can occur in the investigated system such as: high amplitude vibration with complex motion, resonance peaks, amplitude jumps, bifurcations, and chaotic vibrations.     

Tip‐enhanced Raman spectroscopy – an interlaboratory reproducibility and comparison study

Since its first experimental realization, tip‐enhanced Raman spectroscopy (TERS) has emerged as a potentially powerful nanochemical analysis tool. However, questions about the comparability and reproducibility of TERS data have emerged. This interlaboratory comparison study addresses these issues by bringing together different TERS groups to perform TERS measurements on nominally identical samples. Based on the spectra obtained, the absolute and relative peak positions, number of bands, peak intensity ratios, and comparability to reference Raman and surface‐enhanced Raman spectroscopy (SERS) data are discussed. Our general findings are that all research groups obtained similar spectral patterns, irrespective of the setup or tip that was used. The TERS (and SERS) spectra consistently showed fewer bands than the conventional Raman spectrum. When comparing these three methods, the spectral pattern match and substance identification is readily possible. Absolute and relative peak positions of the three major signals of thiophenol scattered by 19 and 9 cm⁻¹, respectively, which can probably be attributed to different spectrometer calibrations. However, within the same group (but between different tips), the signals only scattered by 3 cm⁻¹ on average. This study demonstrated the suitability of TERS as an analytical tool and brings TERS a big step forward to becoming a routine technique. Copyright © 2014 John Wiley & Sons, Ltd.     

A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy

In this paper we develop a new efficient and stable Lagrangian numerical method for computing the evolution of 3D curves driven in the normal plane by a driving force and curvature. This new method contains asymptotically uniform tangential redistribution of grid points designed originally for 3D curve evolution in this paper which makes our computations stable and is crucial for the presented application. Together with the design of new tangentially stabilized algorithm for 3D evolving curves, we develop a new method for the fully automatic finding of the optimal trajectory of the camera in the virtual colonoscopy. The proposed method consists of three steps: 3D segmentation of the colon from CT images, finding an initial trajectory guess inside the segmented 3D subvolumes, and driving the initial 3D curve to its optimal position. To that goal, a suitable intrinsic advection-diffusion partial differential equation with a driving force is designed and solved numerically in fast and robust way in order to find a smooth uniformly discretized 3D curve representing the ideal path of the camera in the virtual colonoscopy.     

General Synthesis of Phosphatidylinositol Phosphates

Abstract

A general method for synthesis of 3-phosphorylated phosphatidylinositols is described.

Phosphatidylinositol (PI) and phosphatidylinositol phosphates (PIP_n), such as PI-4-P, PI-4,5-P₂ undergo receptor-mediated cleavage by phosphatidylinositol-specific phospholipase C, and are precursors of second messengers important in diverse cellular signaling pathways. In contrast, the 3-phosphorylated phosphatidylinositols, such as PI-3-P (1), PI-3,4-P₂ (2) and PI-3,4,5-P₃ (3), which are formed in response to signals of growth factors, are resistant to hydrolysis by this enzyme, and their cellular function is only beginning to emerge [1,2]. These compounds are formed in minute quantities, and therefore have to be synthesized for many practical applications.     

Testing the 1,1,3,3-tetramethyldisiloxane linker in olefin metathesis

Compounds 12-15, possessing two styrenes connected by a silicon linker [1,1,3,3 tetramethyl-di-siloxane], were synthesized, characterized and used as model compounds for the ring-closing metathesis (RCM) catalyzed by commercially available ruthenium catalysts 1, 2 and 3. The RCM reactions of 12 and 15 in the presence of catalysts 1 or 2 resulted exclusively in the formation of (E)-stilbenes. The RCM reactions of 13 and 14, compounds possessing alkoxide substituents in the ortho position to styrene functionality, were not observed in the presence of 2, presumably due to the formation of inactive Hoveyda type ruthenium complexes. The RCM of mixture of 12 and 15, with 2, was used for the detailed examination of the mechanism of metathesis reactions investigated in this work. They revealed that both inter- and intramolecular metathesis is possible, in this case, despite the use of siloxane linker.     

Synthesis and isomer distribution of 2-alkyltropinones and 2-alkylgranatanones

Tropinone (8-methyl-8-azabicyclo[3.2.1]octan-3-one) metal (Li, Na, K, Mg) enolates were used to achieve α-alkylation. The reactions, regardless of the metal or conditions used, were low-yielding. N,N-Dimethylhydrazones of tropinone and granatanone (pseudopelletierine, 9-methyl-9-azabicyclo[3.3.1]nonan-3-one) were prepared and α-alkylated using n-butyllithium as the lithiating reagent. Lithium amides, including a polymer-supported lithium amide, were less effective. The reactions were modelled using DFT calculations at the B3LYP 6-31G(d) level and the CPCM solvent model, revealing that the face-selective alkylations of tropinone and granatanone hydrazones favoured the exo-isomers. Granatanone and α-isopropyl tropinone derivatives resisted typical mild hydrolytic hydrazone cleavage (aqueous trifluoroacetic acid) and required more forceful conditions (p-TsOH, boiling dioxane). Using the hydrazone alkylation strategy, 16 α-alkyl derivatives (benzyl, methyl, propyl, isopropyl, allyl, pentyl, heptyl, p-methoxybenzyl) were prepared in 52–90% yields overall. For the α-alkylated tropinones and granatanones (10 examples), the DFT calculations and experimental thermodynamic distributions in base catalysed equilibrations showed that except for the α-isopropyl derivatives, the endo-isomers were more stable than the exo-isomers and were the major products. For 2-isopropyltropinone, the bulky substituent favoured the exo (axial) position in the bicyclic skeleton. The thermodynamic distribution for the α,α′-dibenzyl tropinone isomers was also evaluated.