Reliability Estimation of Classification Algorithms. I. Introduction to the Problem. Point Frequency Estimates

The study deals with estimating the reliability of classification algorithms. The present article (the first of three) examines the statistical methods used for point estimation of the reliability of classifiers in the frequency approach. The estimation results are applicable to the small-sample case, which is of considerable importance in practice.     

Simulation of Convective Flow of an Electrically Conducting Fluid in a Cavity

A new quasi-hydrodynamic algorithm is proposed for numerical analysis of convective flows in the presence of a homogeneous external magnetic field. The Marangoni convection problem in a square cross-section cavity is solved.     

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non‐conforming FEM systems

Preconditioners based on various multilevel extensions of two‐level finite element methods (FEM) lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Such methods were first presented in Axelsson and Padiy (SIAM. J. Sci. Stat. Comp. 1990; 20 :1807) and Axelsson and Vassilevski (Numer. Math. 1989; 56 :157), and are based on (recursive) two‐level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant γ in the so‐called Cauchy–Bunyakowski–Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. It turns out that only existence of uniform estimates for this constant is not enough but accurate quantitative bounds for γ have to be found as well. More precisely, the value of the upper bound for γ∈(0,1) is part of the construction of various multilevel extensions of the related two‐level methods. In this paper, an algebraic two‐level preconditioning algorithm for second‐order elliptic boundary value problems is constructed, where the discretization is done using Crouzeix–Raviart non‐conforming linear finite elements on triangles. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper two‐level basis is considered, which enables us to fit the general framework for the construction of two‐level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The major contribution of this paper is the derived estimates of the related constant γ in the strengthened CBS inequality. These estimates are uniform with respect to both coefficient and mesh anisotropy. To our knowledge, the results presented in the paper are the first such estimates for non‐conforming FEM systems. Copyright © 2004 John Wiley & Sons, Ltd.     

An Integral Formula for the Complex Intersection Number of Real Cycles in a Real Algebraic Variety with Topologically Rational Singularities

A formula of type indicated in the title is presented and discussed. Bibliography: 7 titles.     

Differential Operator Rings for Which Homogeneous Functions are Linear

Let ε be the collection of those rings S such that, for every S-module V and every homogeneous function ?. V → V, ?(sv) = s?(v), s ∈ S, v ∈ V, ? is linear on V. In this paper we characterize those Stanley-Reisner rings, R, such that D(R) is in ε.     

Solubility parameter determination of cationic surfactants by inverse GC

Summary The solubility parameters of cationic surfactants were obtained using the inverse gas chromatographic technique. The surfactants didodecyl dimethyl ammonium bromide, dioctadecyl dimethyl ammonium bromide and dodecyl pyridinium chloride were used as stationary phase and retention data of different probe solutes were measured at different temperatures. The results were analysed by the combination of Flory-Huggins and Hildebrand theories, and the solubility parameters of the surfactants were obtained in a range of temperatures between 80–120°C.     

Viscoelastic properties of nanostructured natural rubber/polystyrene interpenetrating polymer networks

The effects of the blend ratio and initiating system on the viscoelastic properties of nanostructured natural rubber/polystyrene‐based interpenetrating polymer networks (IPNs) were investigated in the temperature range of ?80 to 150 °C. The studies were carried out at different frequencies (100, 50, 10, 1, and 0.1 Hz), and their effects on the damping and storage and loss moduli were analyzed. In all cases, tan δ and the storage and loss moduli showed two distinct transitions corresponding to natural rubber and polystyrene phases, which indicated that the system was not miscible on the molecular level. However, a slight inward shift was observed in the IPNs, with respect to the glass‐transition temperatures (T_g's) of the virgin polymers, showing a certain degree of miscibility or intermixing between the two phases. When the frequency increased from 0.1 to 100 Hz, the T_g values showed a positive shift in all cases. In a comparison of the three initiating systems (dicumyl peroxide, benzoyl peroxide, and azobisisobutyronitrile), the dicumyl peroxide system showed the highest modulus. The morphology of the IPNs was analyzed with transmission electron microscopy. The micrographs indicated that the system was nanostructured. An attempt was made to relate the viscoelastic behavior to the morphology of the IPNs. Various models, such as the series, parallel, Halpin–Tsai, Kerner, Coran, Takayanagi, and Davies models, were used to model the viscoelastic data. The area under the linear loss modulus curve was larger than that obtained by group contribution analysis; this showed that the damping was influenced by the phase morphology, dual‐phase continuity, and crosslinking of the phases. Finally, the homogeneity of the system was further evaluated with Cole–Cole analysis. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1680–1696, 2003