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In this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal. [ABSTRACT FROM PUBLISHER]

We establish a general theorem on definite integrals of a large class of rational functions. Several definite integral evaluations are proposed as challenging examples. [ABSTRACT FROM AUTHOR]

We establish a general theorem on definite integrals of a large class of rational functions. Several definite integral evaluations are proposed as challenging examples. [ABSTRACT FROM AUTHOR]

Molecular structural optimization is an invaluable tool in computer-aided molecular design. A new flexible docking optimization model based on induced-fit interaction is proposed. In the model, a concept of residue groups is introduced to describe the protein movement approximately and the movement of ligand is described by the translation, votation and torsion motions. A new iteration scheme in conjunction with the &-means clustering algorithm and genetic algorithm is developed to solve the optimization model for the molecular docking. Multi-population genetic strategy and entropy-based searching technique with narrowing down space are employed in the method, making the efficiency of genetic evolution very high. A new docking program FlexGAsDock considering the protein flexibility has been developed. The docking results show that the method can be used in the drug molecular design efficiently. [ABSTRACT FROM AUTHOR]