Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Activity of microemulsion-based nanoparticles at the human bio-nano interface: concentration-dependent effects on thrombosis and hemolysis in whole blood

Background: Although microemulsion-based nanoparticles (MEs) may be useful for drug delivery or scavenging, these benefits must be balanced against potential nanotoxicological effects in biological tissue (bio-nano interface). We investigated the actions of assembled MEs and their individual components at the bio-nano interface of thrombosis and hemolysis in human blood. Methods: Oil-in-water MEs were synthesized using ethylbutyrate, sodium caprylate, and pluronic F-68 (ME4) or F-127 (ME6) in 0.9% NaCl_w/v. The effects of MEs or components on thrombosis were determined using thrombo-elastography, platelet contractile force, clot elastic modulus, and platelet counting. For hemolysis, ME or components were incubated with erythrocytes, centrifuged, and washed for measurement of free hemoglobin by spectroscopy. Results and conclusions: The mean particle diameters (polydispersity index) for ME6 and ME4 were 23.6 ± 2.5 nm (0.362) and 14.0 ± 1.0 nm (0.008), respectively. MEs (0, 0.03, 0.3, 3 mM) markedly reduced the thromboelastograph maximal amplitude in a concentration-dependent manner (49.0 ± 4.2, 39.0 ± 5.6, 15.0 ± 8.7, 3.8 ± 1.3 mm, respectively), an effect highly correlated (r2 = 0.94) with similar changes caused by pluronic surfactants (48.7 ± 10.9, 30.7 ± 15.8, 20.0 ± 11.3, 2.0 ± 0.5) alone. Neither oil nor sodium caprylate alone affected the thromboelastograph. The clot contractile force was reduced by ME (27.3 ± 11.1–6.7 ± 3.4 kdynes/cm², P = 0.02, n = 5) whereas the platelet population not affected (175 ± 28–182 ± 23 10⁶/ml, P = 0.12, n = 6). This data suggests that MEs reduced platelet activity due to associated pluronic surfactants, but caused minimal changes in protein function necessary for coagulation. Although pharmacological concentrations of sodium caprylate caused hemolysis (EC₅₀ = 213 mM), MEs and pluronic surfactants did not disrupt erythrocytes. Knowledge of nanoparticle activity and potential associated nanotoxicity at this bio-nano interface enables rational ME design for in vivo applications.     

Derivations in Commutative Regular Algebras

Mathematical Notes -     

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid

In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n= 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.     

Thermal conductivity of crystalline chrysotile asbestos

The thermal conductivity of crystalline chrysotile asbestos made up of hollow tubular Mg₃Si₂O₅(OH)₄ filaments is measured in the range 5–300 K. The paper discusses the possibility of using this material in studies of the thermal conductivity of thin filaments of metals and semiconductors incorporated into the channels of crystalline chrysotile asbestos tubes.     

Approximate Solution of the de la Vallée Poussin Problem for a Differential Equation of Neutral Type by the Projection-Iterative Method

We solve the de la Vallée Poussin problem for a functional-differential equation by the projection-iterative method. We construct an algorithm, establish conditions sufficient for the convergence of the method, and present a computational scheme.     

Necessary stationary condition for polarized radiation in scattering media

The stationary condition is derived taking into account the polarization of radiation in the general case of a scattering inhomogeneous medium in an arbitrary-shape emitter. The necessary stationary condition for an emitter in which radiation is emitted and extinguished simultaneously is complete extinction of the entire emitted radiation. Radiation extinction as a result of absorption by the medium and the emergence of radiation from the emitter is analyzed. The stationary condition is an analytical form of writing that extinction of radiation is a sure event whose probability is equal to unity. The passage of radiation through the medium is described on the basis of the linear transport theory with the help of the matrices of the Green functions. The stationary condition includes the characteristics of polarized radiation extinction of which is analyzed, the absorption coefficients of the medium, and the elements of the matrices of the Green functions, which are determined by optical and geometrical parameters of the emitter. The stationary condition obtained is used for deriving the relations between the components of scalar intensity observed in an arbitrary region of the emitter. These relations include, in addition to the absorption coefficients and the matrix elements of the Green functions, the powers of the primary radiation. Possible applications of the stationary condition and the relations between intensity components in computations and experimental studies are considered.     

Beryllium (boron) clustering quest in relativistic multifragmentation (BECQUEREL Project)

A physical program of irradiation of emulsions in beams of relativistic nuclei named the BECQUEREL Project is reviewed. It is destined to study in detail the processes of relativistic fragmentation of light radioactive and stable nuclei. The expected results would make it possible to answer some topical questions concerning the cluster structure of light nuclei. Owing to the best spatial resolution, the nuclear emulsions would enable one to obtain unique and evident results. The most important irradiations will be performed in the secondary beams of He, Be, B, C, and N radioactive nuclei formed on the basis of JINR Nuclotron beams of stable nuclei. We present results on the charged state topology of relativistic fragmentation of the ¹⁰B nucleus at low energy-momentum transfers as the first step of the research.