Synthesis and properties of 2-amino-5-(3,5,6-trichloro-1,4-benzoquinon-2-yl)thiazoles

The reaction of thiourea of N-substituted thioureas with 2-(2-N,N-diethylaminoethenyl)-3,5,6-trichloro-1,4-benzoquinone leads, through the formation of 2,5-dihydroxy-3,4,6,7-tetrachlorocoumaran, to 2-amino-5-(2,5-dihydroxy-3,4,6-trichlorophenyl)thiazoles (which exist in the form of betaines), the oxidation of which leads to 2-amino-5-(3,5,6-trichloro-1,4-benzoquinon-2-yl)thiazoles.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 692–698, May, 1988.     

Fixed point limit behavior ofN-mode truncated Navier-Stokes equations asN increases

The fixed point behavior ofN-mode truncations of the Navier-Stokes equations on a two-dimensional torus is investigated asN increases. FromN=44 on the behavior does not undergo any qualitative change. Furthermore, the bifurcations occur at critical parameter values which clearly tend to stabilize asN approaches 100.     

Ring — chain tautomerism and reactions of 4-aryl-4-hydroxy-2,3-dimethyl-3,4-dihydrophthalazinones

It has been established by IR spectroscopy that the N,N-dimethylhydrazides of 2-aroylbenzoic and benzil-o-carboxylic acids, with the exception of 2-mesitoylbenzoic acid, have the ring structure of 4-aryl- or 4-benzoyl-4-hydroxy-2,3-dimethyl-3,4-dihydrophthalazinones. The action of electrophilic agents on 4-hydroxy-2,3-dimethyl-4-phenyl-3,4-dihydrophthal-azinone under mild conditions leads to N₍₃₎-demethylation with the formation of 2-methyl-4-phenylphthalazinone.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 8, pp. 1124–1126, August, 1973.     

Symmetry breaking on a model of five-mode truncated Navier-Stokes equations

A symmetryless model of nonlinear first-order differential equations, obtained by perturbing a known model of five-mode truncated Navier-Stokes equations, is studied. Some interesting phenomena, such as the existence of an infinite sequence of bifurcations in a very narrow range of the parameter and the simultaneous presence of a strange attractor either with two fixed attracting points or with a periodic attracting orbit, are shown. Furthermore, two new sequences of period doubling bifurcations are found in the unperturbed model.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

Keto-enol tautomerism of 4-hydroxy-3-phenyl-2-(2-chloroacetamidophenyl)-1-isoquinolone

Three crystalline modifications of the N-chloroacetyl derivative were obtained in the chloroacetylation of 2-(2-aminophenyl)-4-hydroxy-3-phenyl-1-isoquinolone. It was established by IR spectroscopy that they differ with respect to isomeric composition and the character of the intermolecular hydrogen bonds. For the first time, both keto-enol tautomeric forms were isolated in individual form in the 4-hydroxy-1-isoquinolone series.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 542–543, April, 1985.     

Beta relaxation in the shear mechanics of viscous liquids: Phenomenology and network modeling of the alpha-beta merging region

The phenomenology of the beta relaxation process in the shear-mechanical response of glass-forming liquids is summarized and compared to that of the dielectric beta process. Furthermore, we discuss how to model the observations by means of standard viscoelastic modeling elements. Necessary physical requirements to such a model are outlined, and it is argued that physically relevant models must be additive in the shear compliance of the alpha and beta parts. A model based on these considerations is proposed and fitted to data for Polyisobutylene 680.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.