The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szeg? weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ? 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006).     

On the convergence of <Emphasis Type="Italic">s</Emphasis>-dependent GFR conjugate gradient method for unconstrained optimization

In this paper, the authors present an s-dependent conjugate gradient method for unconstrained optimization problem and make two different kinds of estimations of upper bounds of β _k with respect to \(\beta_{k}^{\mathrm{FR}}\) which are called dependent ratio. The global convergence of s-dependent GFR conjugate gradient method using several step-size rules is obtained.     

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

We provide a partial result on Taylor’s modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analog for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence n-permutability for a fixed n, and satisfying a non-trivial congruence identity.     

Large-scale synthesis and growth habit of 3-D flower-like crystal of PbTe

In this paper, 3-D flower-like crystal of PbTe was successfully synthesized using Pb(CH₃COO)₂·3H₂O and Na₂TeO₃ as precursors under hydrothermal conditions, and characterized by means of scanning electron microscopy (SEM), transmission electron microscopy (TEM) and X-ray diffraction pattern (XRD). The reaction parameters that influenced the evolution of PbTe synthesis and morphology were investigated. It was shown that the flower-like crystal of PbTe was composed of a nucleus with eight pods. A possible growth mechanism was proposed based on the calculation of the surface energies of PbTe and the SEM observation. Furthermore, the temperature-dependent transport properties of 3-D flower-like crystal of PbTe specimen have been evaluated with an average thermoelectric power of 120 S cm^?1 and electrical conductivity of 220 μV K^?1 at 740 K.     

A New Gradient Method with an Optimal Stepsize Property

The gradient method for the symmetric positive definite linear system is as follows

Inelastic partial cross sections for scattering of HF by neon

An analytic expression of the potential energy surface (PES) of the ground state of the Ne-HF complex is obtained by utilizing nonlinear least square method to fit the intermolecular interaction energies [Zhang Y. Guizhou Science, 2003, 21(3): 9–13 (in Chinese)], which have been computed using the augmented correlation-consistent polarized quadruple zeta basis set aug-cc-pVQZ at the theoretical level of CCSD (T). On the basis of the PES, the partial cross sections (PCSs) at the incident energies of 60, 75, 100 and 150 meV for collisions between Ne atoms and HF molecules are calculated using the quantum close coupling approach. The effects of the long-range attractive and the short-range anisotropic interactions on the inelastic PCSs are discussed in detail. The results show: (1) The long-range attractive well of the EPS makes the significant contribution to the lower excitation PCSs, especially the tail maximum for j = 0→j′ = 1 transitions, whereas no contribution is to the j′⩾3 inelastic transitions. (2) The short-range (the repulsive and attractive) interaction makes the significant contribution to the lower excitation PCSs, especially the main peak for j = 0→j′ = 1, 2. As for the transitions of j′⩾3, the short-range interaction plays a key role in the inelastic excitation. (3) Although the positions of the maximums and minimums of the inelastic PCSs are different at the collision energies, they correspond to almost the same impact parameter. Supported by the National Natural Science Foundation of China (Grant Nos. 10676025 and 10574096)     

Preparation of nanosized Bi<Subscript>2</Subscript>WO<Subscript>6</Subscript> using different content of starting materials and their photoactivities

A series of nanosized Bi₂WO₆ catalysts was synthesized using various starting materials, and they were characterized by X-ray diffractometry, transmission electron microscopy, and diffuse reflectance spectroscopy. Rhodamine-B (RhB) photodegradation in aqueous medium was employed as a probe reaction to test the photoactivity of the as-prepared samples. Dependence of the photocatalytic activities on different contents of the starting materials was examined under visible irradiation (λ > 400 nm). The sample prepared in the following conditions: reaction time 24 h, the pH of the solution 7, the Bi³⁺ amount in the start precipitates 5 mmol — exhibited the highest photochemical activity when the hydrothermal temperature was settled at 180°C. Published in Zhurnal Prikladnoi Spektroskopii, Vol. 76, No. 2, pp. 243–249, Martch–April, 2009.     

The eigenvalue complementarity problem

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.     

Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type

We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal Skyrmions.