Invariant subspaces and representations of certain von Neumann algebras

Let be a covariant system and let be a covariant representation of on a Hilbert space . In this note, we investigate the representation of the covariance algebra and the -weakly closed subalgebra generated by and in the case of or when there exists a pure, full, -invariant subspace of .

     

Characterization of intermediate values of the triangle inequality II

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C _n that satisfy 0 ≤ C _n ≤ Σ _j=1 ⁿ ‖x _j ‖ ? ‖Σ _j=1 ⁿ x _j ‖, x ₁,...,x _n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.     

Oligomers of β-amino acid bearing non-planar amides form ordered structures

In this report, we explore the feasibility of using bicyclic chiral β-amino acids, (1R,2R,4S)- and (1S,2S,4R)-7-azabicyclo[2.2.1]heptane-2-carboxylic acid (R-Ah2c and S-Ah2c, respectively), to prepare novel peptides with unique properties. Facile cis-trans isomerization of the non-planar amide bonds of these β-amino acids should result in great flexibility of the backbone structure of β-peptides containing them. Indeed, oligomers of these amino acids showed thermostability and characteristic CD absorptions, which were not concentration-dependent, suggesting that the oligomers remained monomeric. The results indicated the formation of self-organized monomeric structures with chain-length-dependent stabilization. Energy calculations suggested that the peptides can take helical structures in which the energy barriers to cis-trans isomerization are greater for the central amide bonds than for the terminal amides.     

Environment-sensitive fluorophore emitting in protic environments

The unusual fluorescence properties of 8-methoxy-4-methyl-2H-benzo[g]chromen-2-one (1) are described. The fluorophore 1 is almost nonfluorescent in aprotic solvent (e.g., fluorescence quantum yield Phi(f) < 0.0003 in n-hexane), whereas it strongly fluoresces at long wavelengths (>450 nm) in protic solvent (e.g., Phi(f) = 0.21 in methanol). The fluorophore 1 also shows good applicability in developing a new fluorogenic (fluorescent "off-on") sensor. [structure: see text]     

Electrons-in-a-box model for conjugation in linear carbanions

Solving the constrained Hartree–Fock equation and using the method of the energy component analysis, we have determined the fundamental factor which causes the electrons at the negative center of conjugated carbanion to delocalize over the whole system. Constraint on the π-electron flow between the anion center and polyethylenic system produces the electronic state that the lone-pair electrons are localized at the anionic center. By doing so, a prominent increase in the kinetic energy of π electrons was observed. Such an increase was found to quantitatively fit in with that by the model that electrons are packed in a box. Thus, we could clearly show that delocalization of the carbanion in the polyethylenic system is brought forth by the relaxation of the kinetic energy pressure. We also examined conjugated carbodianion systems. Dianion involves repulsive interaction between anions causing less conjugation energy. Namely, conjugation itself is given by the release in the kinetic energy pressure, but, it is restrained by the electrostatic repulsion. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 65–72, 1998     

Protonation enthalpies in fluorosulfonic acid using ab initio self-consistent reaction field theory

Electrostatic solvation free energies were computed for several small neutral bases and their conjugate acids using a continuum solvation model called the self-consistent isodensity polarizable continuum model (SCIPCM). The solvation energies were computed at the restricted Hartree–Fock (RHF) and second-order Møller–Plesset (MP2) levels of theory, as well as with the Becke3–Lee–Yang–Parr (B3LYP) density functional theory, using the standard 6–31G** Gaussian basis set. The RHF solvation energies are similar to those computed at the correlated MP2 and B3LYP theoretical levels. A model for computing protonation enthalpies for neutral bases in fluorosulfonic acid solvent leads to the equation ΔH(B)=−PA(B)+ΔE_t(BH⁺)−ΔE_t(B)+β, where PA(B) is the gas phase proton affinity for base B, ΔE_t(BH⁺) is the SCIPCM solvation energy for the conjugate acid, and ΔE_t(B) is the solvation energy for the base. A fit to experimental values of ΔH(B) for 10 neutral bases (H₂O, MeOH, Me₂O, H₂S, MeSH, Me₂S, NH₃, MeNH₂, Me₂NH, and PH₃) gives β=238.4±2.9 kcal/mol when ΔΔE_t is computed using the 0.0004 e⋅bohr⁻³ isodensity surface for defining the solute cavity at the RHF/6–31G** level. The model predicts that for carbon monoxide ΔH(CO)=10 kcal/mol. Thus, protonation of CO is endothermic, and the conjugate acid HCO⁺ (formyl cation) behaves as a strong acid in fluorosulfonic acid. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 250–257, 1998