CoMFA and docking studies of 2-phenylindole derivatives with anticancer activity.

Three-dimensional (3D) quantitative structure-activity relationship (QSAR) and docking studies of 43 tubulin inhibitors, 2-phenylindole derivatives with anticancer activity against human breast cancer cell line MDA-MB 231, have been carried out. The established 3D-QSAR model from the comparative molecular field analysis (CoMFA) in training set shows not only significant statistical quality, but also satisfying predictive ability, with high correlation coefficient value (R(2)=0.910) and cross-validation coefficient value (q(2)=0.705). Moreover, the predictive ability of the CoMFA model was further confirmed by a test set, giving the predictive correlation coefficient (R(2)(pred)) of 0.688. Based on the CoMFA contour maps and docking analyses, some key structural factors responsible for anticancer activity of this series of compounds were revealed as follows: the substituent R(1) should have higher electronegativity; the substituent R(2) should be linear alkyl with four or five carbon atoms in length; and the substituent R(3) should be selected to OCH(3)-kind group whereas should not be selected to CF(3)-kind group. Meanwhile, the interaction information between target and ligand was presented in detail. Such results can offer some useful theoretical references for understanding the action mechanism, designing more potent inhibitors and predicting their activities prior to synthesis.

Explicit state vector for Torres-Vega-Frederick phase space representation and its statistical behavior.

We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.

Mother wavelets for complex wavelet transform derived by Einstein-Podolsky-Rosen entangled state representation.

The Einstein-Podolsky-Rosen entangled state representation is applied to studying the admissibility condition of mother wavelets for complex wavelet transforms, which leads to a family of new mother wavelets. Mother wavelets thus are classified as the Hermite-Gaussian type for real wavelet transforms and the Laguerre-Gaussian type for the complex case.

Symplectic wavelet transformation.

Usually a wavelet transform is based on dilated-translated wavelets. We propose a symplectic-transformed-translated wavelet family psi(*)(r,s)(z-kappa) (r,s are the symplectic transform parameters, |s|(2)-|r|(2)=1, kappa is a translation parameter) generated from the mother wavelet psi and the corresponding wavelet transformation W(psi)f(r,s;kappa)=integral(infinity)(-infinity)(d(2)z/pi)f(z)psi(*)(r,s)(z-kappa). This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.

Collins diffraction formula studied in quantum optics.

We find that the Collins diffraction formula in cylindrical coordinates is just the transformation matrix element of a three-parameter two-mode squeezing operator in the deduced entangled state representation. This is a new tie connecting the unitary transform in quantum optics to the generalized Hankel transform in Fourier optics. The group multiplication rule of the squeezing operators maps to the Collins formula related to two successive Hankel transforms.

General formula for finding mother wavelets by virtue of Dirac's representation theory and the coherent state.

The admissibility condition of a mother wavelet is explored in the context of quantum optics theory. By virtue of Dirac's representation theory and the coherent state property we derive a general formula for finding qualified mother wavelets. A comparison between a wavelet transform computed with the newly found mother wavelet and one computed with a Mexican hat wavelet is presented.

Eigenmodes of fractional Hankel transform derived by the entangled-state method.

Using the entangled-state method in quantum mechanics, we find that the eigenmodes of the fractional Hankel transform are two-variable Hermite-Gaussian functions that can be rewritten in a clearer form as Laguerre polynominals.