Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

The Asymptotics of Spectral Gaps of a 1D Dirac Operator with Cosine Potential

Letters in Mathematical Physics -     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

Tau Function Solutions to a q-Deformation of the KP Hierarchy

We construct tau-function solutions to the q-KP hierarchy as deformations of classical tau functions.     

Enhanced specificity of bacterial spore identification by oxidation and mass spectrometry

Addition of an oxidizing agent (e.g., hydrogen peroxide) to intact spores selectively and completely oxidizes Met-containing biomarker proteins by formation of Met sulfoxides. This reaction increases the masses of the biomarker proteins observed in matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS) of Bacillus spores by Deltam = (16 x n) Da, where n is the number of Met residues in the sequence of each individual protein. The procedure is very rapid, and can be performed in situ (i.e., on the MALDI target). It confirms the identity of individual biomarkers by comparing the number of Met amino acids from the experimentally determined mass shifts with predictions for n from the tentative amino acid sequence for each protein. In turn, accurate determination of n for several biomarkers allows rapid validation of the initial spore identification by MALDI-MS.     

Dynamic surface tension of micellar solutions studied by the maximum bubble pressure method. 1. Experiment

The effect of the micelles on the dynamic surface tension of micellar surfactant solutions is studied experimentally by means of the maximum bubble pressure method. Different frequencies of bubbling ranging approximately between 1 and 30 s^–1 are applied. The time dependence of the surface tension is calculated using a dead time correction. Water solutions of two types of surfactants with different concentrations are investigated: sodium dodecyl sulfate and nonylphenol polyglycol ether. The surface tension relaxes more quickly in the presence of micelles. The characteristic times of relaxation of the surface tension seem to be in the millisecond range. The time constants observed experimentally are explained in terms of the theory of surfactant diffusion affected by micellization kinetics.     

An Unexpected Formation of a 14‐Membered Cyclodepsipeptide

The treatment of diluted solutions of the hydroxy diamides 6a and 6b in toluene with HCl gas at 100° gave the dimeric, 14‐membered cyclodepsipeptide 10 in up to 72% yield (Scheme 3). The same product was formed from the linear dimer of 6b , the depsipeptide 11 , under the same conditions (cf. Scheme 4). All attempts to prepare the cyclic seven‐membered monomer 9 , starting with different precursors and using different lactonization methods failed, and 10 was the only product which was isolated (cf. Scheme 6). For example, the reaction of the ester 20 with NaH in toluene at 80° led exclusively to the cyclodimer 10 . On the other hand, the base‐catalyzed cyclization of the hydroxy diester 22 , which is the ‘O‐analogue' of 20 , yielded neither the seven‐membered dilactone, nor the 14‐membered tetralactone, but only the known trimer 23 and tetramer 24 of 2,2‐dimethylpropano‐3‐lactone (cf. Scheme 7).     

cis‐(6RS,13RS)‐3,3,10,10‐Tetra­methyl‐6,13‐diphenyl‐1,8‐dioxa‐4,11‐diaza­cyclo­tetra­decane‐2,5,9,12‐tetra­one and two of its precursors

The title macrocycle, C₂₆H₃₀N₂O₆, (VI), was obtained by `direct amide cyclization' from the linear precursor 3‐hydr­oxy‐N‐[1‐methyl‐1‐(N‐methyl‐N‐phenyl­carbamoyl)ethyl]‐2‐phenylpropanamide, the N‐methyl­anilide of rac‐2‐methyl‐2‐[(3‐hydroxy‐2‐phenyl­propanoyl)­amino]­propanoic acid, C₁₃H₁₇NO₄, (IV). The reaction proceeds via the inter­mediate rac‐2‐(2‐hydroxy‐1‐phenyl­ethyl)‐4,4‐dimethyl‐1,3‐oxazol‐5(4H)‐one, C₁₃H₁₅NO₃, (V), which was synthesized independently and whose structure was also established. Unlike all previously described analogues, the title macrocycle has the cis‐diphenyl configuration. The 14‐membered ring has a distorted rect­angular diamond‐based [3434] configuration and inter­molecular N—H⋯O hydrogen bonds link the mol­ecules into a three‐dimensional framework. The propanoic acid precursor forms a complex series of inter­molecular hydrogen bonds, each of which involves pairwise association of mol­ecules and which together result in the formation of extended two‐dimensional sheets. The oxazole inter­mediate forms centrosymmetric hydrogen‐bonded dimers in the solid state.     

Reaction of γ‐Hydroxy‐N‐[1‐(dimethylcarbamoyl)ethyl]butanamides under the ‘Direct Amide Cyclization’ Conditions

The preparation of the title compounds was achieved via the ‘azirine/oxazolone method’ starting from the corresponding γ‐hydroxy acids. Upon subjecting the γ‐hydroxy‐N‐[1‐(dimethylcarbamoyl)ethyl]butanamides 4 to the so‐called ‘direct amide cyclization’ (DAC) conditions, chlorinated acids 11 or imino lactones 12 were obtained as the sole products instead of the expected cyclodepsipeptides A or their cyclodimers (Scheme 4). Variation of the substituents in 4 did not affect the outcome of the reaction and a mechanism for the formation of both products from the intermediate oxazolone 13 has been proposed. Under the acidic conditions of the DAC, the imino lactones are formed as their HCl salts 12 , which, in polar solvents or on silica gel, reacted further to give the chlorinated acids 11 . Stabilization of the imino lactones was achieved by increasing the substitution in the five‐membered ring, and their structure, in the form of the hydrochlorides, was established independently by X‐ray crystallography (Fig. 4). A derivative 15 of the imino lactone 12a was prepared by the reaction with the 2H‐azirin‐3‐amine 10a ; its structure was also established by an X‐ray crystal‐structure determination (Fig. 3). Furthermore, the structures of the ω‐chloro acids 11a and 11b were determined by X‐ray crystallography (Fig. 2).