On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.     

A liquid chromatography – tandem mass spectrometry method to measure a selected panel of uremic retention solutes derived from endogenous and colonic microbial metabolism

Chronic kidney disease (CKD) is associated with an increased risk of mortality and cardiovascular disease, which is, at least partly, mediated by the accumulation of so-called uremic retention solutes. Although there has been an increasing interest in the behavior of these solutes, derived from both the endogenous and colonic microbial metabolism, methods to simultaneously and accurately measure a broad panel of relevant uremic retention solutes remain scarce.     

Template-directed self-assembly of alkanethiol monolayers: selective growth on preexisting monolayer edges

Self-assembled monolayers were investigated for their suitability as two-dimensional scaffolds for the selective growth of alkanethiol edge structures. Heterostructures with chemical contrast could be grown, whose dimensions were governed by both the initial pattern sizes and the process time. n-Octadecanethiol (ODT) was made to expand from the edges of 16-mercaptohexadecanoic acid (MHDA) monolayer patterns. Likewise, 11-mercaptoundecanol (MUD) was grown on MHDA and on ODT monolayer edges. The results of these experiments are in accordance with the moving boundary model for monolayer spreading. In addition to such surface-bound spreading, a vapor-phase contribution to lateral monolayer growth was identified. MUD was observed to be an excellent ink for creating chemical contrast by means of regioselective deposition from a vapor phase. As a proof of principle, ribbons of 11-mercaptoundecanol with submicrometer widths were grown on pentaerythritol-tetrakis(3-mercaptopropionate) edges, and submicrometer wide gold lines were produced by subsequent wet-chemical etching.     

Efficient evaluation of the algebrants of VB wave functions using the successive expansion method. I. Spin S = 0, ½

An efficient expansion method for the evaluation of VB matrix elements is introduced. The overlaps of VB wave functions of N electrons can be treated as algebrants, i.e., generalized determinants, of N × N matrices. An algebrant can be expanded with subalgebrants of lower orders in a successive way. By choosing Rumer spin bases and appropriately arranging the expansion, it is found that the number of unique subalgebrants involved in the expansion increases in a quite moderate way with N. In contrast to the traditional symmetric group approach, which explicitly utilizes all N! representation matrices, the new strategy incorporates the group theoretical factors in a simple way in the successive expansion. As only the unique subalgebrants are further expanded, the computational effort required by the new strategy scales in a very acceptable manner with the increasing number of electrons. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 245–259, 1997     

Synthesis of polystyrene-polysiloxane side-chain liquid crystalline block copolymers

The synthesis of a new liquid crystalline block copolymer consisting of a polystyrene block and a side-chain liquid crystalline siloxane block is reported. The synthetic approach described is based on the anionic polymerization of styrene and cyclic trimethyltrivinyltrisiloxane monomers, followed by functionalization of the siloxane block with side chain mesogens. The siloxane block has a T_g well below 25°C and is designed to exhibit a chiral smectic C^* phase at room temperature. These block copolymers are the first side-chain liquid crystalline block copolymers which contain both a high T_g glassy block and a low T_g liquid crystalline block.     

Linear stability analysis of the natural convection in inclined rotating parallel plates

The linear stability analysis of the natural convection in a rectangular tilted infinite cavity filled with a Boussinesq fluid subject to Coriolis force is presented. The bottom and top surfaces have fixed temperatures. Both unstable and stable thermal conditions are studied (heated from below and heated from above respectively). The rotation axis passes through the center and it is orthogonal to the hot and cold surfaces. The stability equations were solved using the Tau–Chebyshev spectral method. The critical Rayleigh number and critical wave number were obtained for several rotation rates and different orientation of convective oblique rolls in a range of inclination of the cavity from 0 to 120 degrees. The stability analysis show that rotation rate affects the basic velocity profile, onset of convection, wave number and critical orientation of convective rolls.