On Certain Fejér and Tchebychev Type Integrals

Let m and n denote a pair of positive integers. In this paper, we call upon the Hadamard product and computer algebra techniques to evaluate the Fejér integral π ^?1 ∫₀ ^π (sin mθ / sin θ) ²ⁿ dθ. Using symmetry arguments, it is proved that the value of this integral is an odd polynomial in m of degree 2n ? 1. This permits using polynomial curve fitting methods and mathematical software packages to obtain evaluation formulas for n relatively small. Some cases of the above integral with 2n replaced by 2n + 1 are also discussed. A familiar identity shows that these yield evaluations of integrals of powers of certain Tchebychev polynomials.     

Water vapour line intensities and self-broadening coefficients in the 5000-5600 cm spectral region

This paper presents results from an intercomparison of self-broadening coefficients and intensities of approximately 440 of the strongest water vapour lines in the spectral region 5000-5600 cm⁻¹. Line parameters retrieved and reported recently by two scientific groups, both using Fourier transform spectroscopy, are compared with parameters taken from the HITRAN-2008 database and with theoretical linelist BT2. This comparison has revealed marked systematic differences in the self-broadening coefficients (up to 20%) and to lesser degree in the line intensities (up to 6%) between different sources.     

Fitting models to correlated data II (small samples, large correlations, and wrong models)

The estimation of experimental error, data correlation and model error by the analysis of the fit residuals in an ordered sequence of data [J. Mol. Spectrosc. 224 (2004) 73] is improved by a better insight in the calculation of the statistical relevant quantities. The biases previously observed for too small samples of data, very wrong models or highly correlated data, are partially corrected and this leads to better estimates in some extreme cases provided these unfavourable conditions are not combined. However, such results are always obtained with a decrease in accuracy.     

A Cartesian method for fitting the bathymetry and tracking the dynamic position of the shoreline in a three-dimensional, hydrodynamic model

This paper presents a Cartesian method for the simultaneous fitting of the bathymetry and shorelines in a three-dimensional, hydrodynamic model for free-surface flows. The model, named LESS3D (Lake & Estuarine Simulation System in Three Dimensions), solves flux-based finite difference equations in the Cartesian-coordinate system (x,y,z). It uses a bilinear bottom to fit the bottom topography and keeps track the dynamic position of the shoreline. The resulting computational cells are hybrid: interior cells are regular Cartesian grid cells with six rectangular faces, and boundary/bottom cells (at least one face is the water–solid interface) are unstructured cells whose faces are generally not rectangular. With the bilinear interpolation, the shape of a boundary/bottom cell can be determined at each time step. This allows the Cartesian coordinate model to accurately track the dynamic position of the shorelines. The method was tested with a laboratory experiment of a Tsunami runup case on a circular island. It was also tested for an estuary in Florida, USA. Both model applications demonstrated that the Cartesian method is quite robust. Because the present method does not require any coordinate transformation, it can be an attractive alternative to curvilinear grid model.     

方形LED阵列光斑发散特性的幂函数拟合

利用LED照度公式推导方形LED阵列的光斑半径和发散角公式,建立了研究方形LED阵列光斑发散特性的数值计算方法.通过拟合得到方形LED阵列的光斑半径和发散角随目标距离、m值及阵列边长变化的幂函数公式.结果表明,幂函数拟合方法与数值计算结果吻合,平均相对误差小于1%.该方法弥补了数值计算方法不能对方形LED阵列光斑发散特性解析研究的缺陷.     

NON-LINEAR LEAST SQUARE CURVE FITTING OF COPOLYMERIZATION MODELS TO THE ALTERNATING TRIAD FRACTION OF COPOLYMERS OF STYRENE AND CITRACONIC ANHYDRIDE,AND COPOLYMERS OF STYRENE AND MALEIC ANHYDRIDE PREPARED IN N,N-DIMETHYLFORMAMIDE

Copolymers of styrene (ST) and citraconic anhydride (α-methylmaleic anhydride) (CA) were prepared in a very polar solvent, N,N-dimethylformamide (DMF), at 50.0°C with AIBN. The monomer unit triad fractions were determined by ¹³C NMR in acetone-d ₆ solution. Non linear least square (NLLS) curve fitting was performed for the copolymerization models of the terminal model, the penultimate unit effect model, the complex participation model, the complex dissociation model, and the so-called comppen model. The theoretical equations for the ST-centered alternating triad mole fraction were fitted by NLLS minimization routine to the triad fraction data of the ST-CA copolymers and that of the ST-maleic anhydride (MA) copolymers prepared in identical polymerization conditions. It was found that for rigidly alternating copolymers of ST-MA, the difference among the copolymerization models disappeared and all models merged together. The difference among the copolymerization models were somewhat more apparent for less alternating copolymers of ST-CA copolymers. The sum of squares values indicated that the copolymerization models, which involved some complex participation, fit the data better with the comppen model. This was a combination of a complex participation and penultimate unit effects, which performed best.     

On finite groups with some subgroups of prime indices

We establish the solvability of each finite group whose every proper nonmaximal subgroup lies in some subgroup of prime index.     

ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS

This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primary decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.     

On the distributivity of the lattice of solvable totally local fitting classes

It is proved that the lattice of all solvable totally local Fitting classes is algebraic and distributive. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 662–673, May, 2000.     

On the Orbit Sizes of Permutation Groups on the Power Set

We study the action of finite permutation groups of odd order on the power set of a set on which they act naturally, and establish a theorem that guarantees the existence of a lot of distinct orbit sizes in this action. 1991 Mathematics Subject Classification: 20B05