Envelopes of splines in the projective plane

In this paper a family of curves—Riemannian cubics—inthe unit sphere and the real projective plane is investigated.Riemannian cubics naturally arise as solutions to variationalproblems in Riemannian spaces. It is remarkable to find thatan envelope of lines generated by a Riemannian cubic in onespace is (nearly) a Riemannian cubic in another space.     

Concepts of lower semicontinuity and continuous selections for convex valued multifunctions

We discuss several concepts of continuity, weaker than lower semicontinuity, but still implying the existence of a continuous selection for a closed convex valued multifunction from a paracompact Hausdorff topological space into a Banach space. In this way, an extension of Michael's celebrated selection theorem is given. The behavior of -envelop approximations, as well as the localization of continuous selections, is also discussed.     

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

How random is your heart beat?

We measure the content of random uncorrelated noise in heart rate variability using a general method of noise level estimation using a coarse-grained entropy. We show that usually, except for atrial fibrillation, the level of such noise is within 5-15% of the variance of the data and that the variability due to the linearly correlated processes is dominant in all cases analyzed but atrial fibrillation. The nonlinear deterministic content of heart rate variability remains significant and may not be ignored.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

Toward Transition-Metal-Templated Construction of Arylated B4 Chains by Dihydroborane Dehydrocoupling

The reactivity of a diruthenium tetrahydride complex towards three selected dihydroboranes was investigated. The use of [DurBH₂] (Dur=2,3,5,6-Me₄C₆H) and [(Me₃Si)₂NBH₂] led to the formation of bridging borylene complexes of the form [(Cp*RuH)₂BR] (Cp*=C₅Me₅; 1 a : R=Dur; 1 b : R=N(SiMe₃)₂) through oxidative addition of the B−H bonds with concomitant hydrogen liberation. Employing the more electron-deficient dihydroborane [3,5-(CF₃)₂-C₆H₃BH₂] led to the formation of an anionic complex bearing a tetraarylated chain of four boron atoms, namely Li(THF)₄[(Cp*Ru)₂B₄H₅(3,5-(CF₃)₂C₆H₃)₄] ( 4 ), through an unusual, incomplete threefold dehydrocoupling process. A comparative theoretical investigation of the bonding in a simplified model of 4 and the analogous complex nido-[1,2(Cp*Ru)₂(μ-H)B₄H₉] ( I ) indicates that there appear to be no classical σ-bonds between the boron atoms in complex I , whereas in the case of 4 the B₄ chain better resembles a network of three B−B σ bonds, the central bond being significantly weaker than the other two.     

Injectable poloxamer/graphene oxide hydrogels with well‐controlled mechanical and rheological properties

Although significant progress has been made in the design and application of injectable hydrogels for biomedical applications, concurrent control of rheological and mechanical properties of injectable hydrogels has remained as an open challenge to the researchers. In this work, we introduce and put into practice a photo‐curable poloxamer (also known as Pluronic)/graphene oxide (Plu/GO) injectable hydrogel with well‐controlled rheological and mechanical properties. Acrylate group was anchored to hydrogel structure to endow photo‐crosslinking ability through decelerating degradation rate of poloxamer hydrogels after injection. It was found that the modified Plu remains stable in biological media for a long‐term period without significant weight loss. Rheological properties of hydrogels were also carried out as essential prerequisite for an ideal injectability via frequency sweep, flow curve, recovery, and yield stress before and after modification, signifying shear‐thinning behavior of Plu/GO hydrogels with high recoverability. The viscosity of shear‐thinning‐like hydrogels dropped at higher shear stress, which facilitated injection process. Moreover, mechanical behavior of Plu was optimized by manipulating the content of Plu, degree of modification with reactive precursor, curing, and particularly incorporation of GO without deteriorating effects on rheological behavior of Plu.     

Conjugated Schiff bases. VII—Decarbonylation of some 1,3-hetrodienes under electron impact

The mass spectra of some 1,3- heterodienes containing the 1-oxa-4-aza-butadiene system have been determined and the characteristic fragmentation pathways are discussed. The pathways suggested are supported by the appropriate metastables, the spectra of model compounds and by deuterium labelling. The 1, 3-heterodienes substituted by the thioarylamide group from ions involving participation of the sulphur atom.     

Cationic polymerization of 1,4,6-trioxaspiro [4,4]-nonane

Cationic polymerization of 1,4,6-trioxaspiro [4,4]-nonane ( 1 ) with (CH₃)₃O⁺SbF₆^?(2) and CH₃OSO₂CF₃(3) initiators has been investigated. Although the observed rates of initiation and propagation are relatively slow, they consist of rapid reversible elementary reactions. In ¹H-NMR spectra, a broadening of the monomer signal was observed, indicating a fast exchange between “free” monomer and monomer engaged in the active species. The variety of orthoester bonds were observed in the polymer formed at the early stages of monomer conversion. The final polymer has, however, structure of a linear poly(ester–ether) including two subsequent ester or ether linkages. To account for these new facts, the mechanism of polymerization was proposed, consisting of a rapid reversible opening of one of the rings in the monomer molecule involved in the growing species, followed by the slower opening of the second ring with formation of the ester linkages. It appears that the rings originally present in the chains rearrange into the linear units intramolecularly.