A General Representation of Geometrical Solution Model for Predicting Ternary Thermodynamic Properties

The existing geometrical solution models for predicting ternary thermodynamic properties from relevant binary ones have been analysed,and a general representation was proposed in an integral form on the bases of these models.

EVALUATION OF SINGULAR AND NEARLY SINGULAR INTEGRALS IN THE BEM WITH EXACT GEOMETRICAL REPRESENTATION＊

The geometries of many problems of practical interest are created from circular or ellip- tic arcs. Arc boundary elements can represent these boundaries exactly, and consequently, errors caused by representing such geometries using polynomial shape functions can be removed. To fully utilize the geometry of circular boundary, the non-singular boundary integral equations （BIEs） and a general nonlinear transformation technique available for arc elements are introduced to remove or damp out the singular or nearly singular proper- ties of the integral kernels. Several benchmark 2D elastostatic problems demonstrate that the present algorithm can effectively handle singular and nearly singular integrals occur- ring in the boundary element method （BEM） for boundary layer effect and thin-walled structural problems. Owing to the employment of exact geometrical representation, only a small number of elements need to be divided along the boundary and high accuracy can be achieved without increasing other more computational efforts.

A Geometric Approach to Conditioning and the Search for Minimum Variance Unbiased Estimators

Our purpose is twofold: to present a prototypical example of the conditioning technique to obtain the best estimator of a parameter and to show that this technique resides in the structure of an inner product space. The technique uses conditioning of an unbiased estimator on a sufficient statistic. This procedure is founded upon the conditional variance formula, which leads to an inner product space and a geometric interpretation. The example clearly illustrates the dependence on the sampling methodology. These advantages show the power and centrality of this process.

A Variant of Fermat’s Diophantine Equation

A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

Lagrangian Grassmann manifold （2）

Based on the relationship between symplectic group Sp（2） and （2）, we provide an intuitive explanation （model） of the 3-dimensional Lagrangian Grassmann manifold （2）, the singular cycles of （2）, and the special Lagrangian Grassmann manifold S（2）. Under this model, we give a formula of the rotation paths defined by Arnold.