On some peculiar aspects of the constructive theory of point‐free spaces

This paper presents several independence results concerning the topos‐valid and the intuitionistic (generalised) predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Kripke schema in metric topology

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop‐style constructive reverse mathematics.     

图M(P_m)和M(C_m)的点可区别边色数

本文研究了圈Cm和路Pm的Mycielski图的点可区别边染色问题.利用构造法给出了M(Cm)图的点可区别边染色法,得到了它的点可区别边色数,进而从图的结构关系,有效获得了M(Pm)图的相应点可区别边染色法和其边色数.该方法对研究存在结构关系的图染色问题具有重要的借鉴意义.     

Boolean Algebras with Finite Families of Computable Automorphisms

We prove that each computable Boolean algebra has a computable presentation in which for every computable family of automorphisms the set of atoms moved by at least one of its members is finite. This implies that each computable atomic Boolean algebra has a computable presentation in which its every computable family of automorphisms is finite. The priority argument is not used in the proof.     

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

Modified realizability and predicate logic

Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 259–269, February, 1997. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00416 and by the International Science Foundation under grant No. NFQ000. Translated by V. N. Dubrovsky     

Competition between Quantum Interference and Stimulated Electronic Raman Scattering Processes via Hybrid Excitation in a Na2-Na System

An experimental observation of the competition between quantum interference and stimulated electronic Raman scattering (SERS) via equal-frequency hybrid excitation and collision-aided radiative excitation in a Na2-Na system is reported for the first time.It is found that the SERS for 3P1/2,3/2-4P are suppressed and the optically pumped stimulated emission (OPSE) for 5S-4P are enhanced dramatically at a particular excitation wavelength where constructive quantum interference between two transitions 3P1/2-5S and 3P3/2-5S in atomic sodium takes place.A theoretical analysis based on the dressed-atom approach is carried out to explain the experimental results.     

Synthesis,characterization, and photopolymerization of (meth)acrylate-functional polyisobutylene macromers produced by cleavage/alkylation of butyl rubber

Linear, multi-functional polyisobutylene (PIB) macromers bearing pendent and terminal (meth)acrylate moieties were prepared via electrophilic cleavage/alkylation of butyl rubber in the presence of (3-bromopropoxy)benzene, followed by displacement of the resulting bromide moieties with potassium (meth)acrylate. Number average functionality (F_n) ranged from 2.8–7.9; functional equivalent weights ranged from 2.3–4.7 kg/mol. For comparison, a three-arm, end-functional PIB triacrylate with equivalent weight of 3.3 kg/mol was also synthesized via living polymerization and end quenching with 4-phenoxy-1-butyl acrylate. All polymers were photocured using Darocur 1173 photoinitiator, and curing kinetics were monitored by real time Fourier-transform infrared spectroscopy. All systems reached ~100% conversion by 1,800 s, but the linear macromers displayed slower curing rates compared to the PIB triacrylate. The curing rate of linear macromers increased as molecular weight decreased. Cured networks were characterized using dynamic mechanical analysis and tensile testing. Tensile strength varied from 0.15–0.80 MPa. Young's modulus varied from 0.13–1.8 MPa. Strain at break for most networks ranged from 34–54%, but the network derived from the lowest molecular weight PIB reached 113% at failure. Percent extractables, measured using solvent extraction, was about 2% for linear macromers and about 4% for PIB triacrylate.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

Constructive approximation of solution for fourth‐order nonlinear boundary value problems

In this paper, we obtain a sequence of approximate solution converging uniformly to the exact solution of a class of fourth‐order nonlinear boundary value problems. Its exact solution is represented in the form of series in the reproducing kernel space. The n‐term approximation u_n(x) is proved to converge to the exact solution u(x). Moreover, the derivatives of u_n(x) are also convergent to the derivatives of u(x). Copyright © 2008 John Wiley & Sons, Ltd.