Rotation three-dimensional rotations

Ion trap analyzer. A mass-resonance analyzer that produces a three-dimensional rotationally symmetric quadrupole field capable of storing ions at selected mass-to-charge (m/z) ratios. 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

Rhee HW, Zhau HE, Pathak S et al (2001) Permanent phenotypic and genotypic changes of prostate cancer cells cultured in a three-dimensional rotating-wall vessel. In Vitro Cell Dev Biol Anim 37 127-140... 

Release of two water molecules from the cavity of a-cyclodextrin (form I) (19) is accompanied not only by the loss of van der Waals interaction (/fj jdw) and hydrogen bonding ( —2 A/fH.bond), but also by the gain of motional freedoms of two water molecules as to translation (2S ans) and three-dimensional rotation (2S ,I(3 D)). At the same time, a change in conformational energy of a-cyclodextrin is involved which is estimated by the use of Allinger s method (49). 

For a particular case of the group of three-dimensional rotations, we have the equation... 

The relative nuclear configuration RNC Xk( ), Zk, Mk) is defined as the set of informations determining a NC up to translations and rotations in 3 3, i.e. invariant with respect to transformations of the inhomogeneous three-dimensional rotation group 10(3). Conveniently the RNC is determined by internal structural parameters, 2.13K-6 which are invariant with respect to (w.r.t.) 10(3). 

Dyer, F. F., L. C. Bate, and J. E. Strain Three-Dimensionally Rotating Sample Holder for 14-Million Electron Volt Neutron Irradiations. Anal. Chem. 39, 1907 (1967). 

The Irreducible Representations of the Three-dimensional Rotation Group... 

In the present paper the angular overlap model is elucidated by discussing it in the fight of the transformation properties of the involved atomic orbitals under the three-dimensional rotation group. 

The functions, here occurring in standard order, are our standard basis functions for the real irreducible representations of the full three-dimensional rotation-reflection group, Rg x I, and for its subgroups, Aoft. and Coo . All functions are normalized to 4nj(2l- -1), where / is the azimuthal quantum number. 

Dynamic fluorescence anisotropy is based on rotational reorientation of the excited dipole of a probe molecule, and its correlation time(s) should depend on local environments around the molecule. For a dye molecule in an isotropic medium, three-dimensional rotational reorientation of the excited dipole takes place freely [10]. At a water/oil interface, on the other hand, the out-of-plane motion of a probe molecule should be frozen when the dye is adsorbed on a sharp water/oil interface (i.e., two-dimensional in respect to the molecular size of a probe), while such a motion will be allowed for a relatively thick water/oil interface (i.e., three-dimensional) [11,12]. Thus, by observing rotational freedom of a dye molecule (i.e., excited dipole), one can discuss the thickness of a water/oil interface the correlation time(s) provides information about the chemi-cal/physical characteristics of the interface, including the dynamical behavioiu of the interfacial structure. Dynamic fluorescence anisotropy measurements are thus expected... 

Here the coefficients fl) (/) are the elements of a real, three-dimensional rotation matrix. Because the rows and the columns of a rotation matrix form a set of three orthonormal vectors, respectively, the following relations are fulfilled by the coefficients a (0 ... 

Prior to discussing the properties of octopolar molecules, it is instructive to consider first some of the basic properties of tensors. In general, any tensor of rank n can be decomposed in a sum of so-called irreducible tensors that are invariant under three-dimensional rotation [99] ... 

Equation (30j is called the reduction spectrum of the tensor T n). Each irreducible tensor of rank n and weight J has 3" components labeled by n indices, and its 3" components form the basis of an irreducible representation of the three-dimensional rotation group of degree 27 -h 1. Elence, each irreducible tensor with weight J has U 4- 1 independent components. The superscript y distinguishes between different components of identical weight. For example, the reduction spectrum of the hyperpolarizability tensor f3 (rank three) is ... 

Reconstruction of images from tomographic methods are performed using the reverse Radon transform (Herman, 1980) which uses the series of angular projections to reconstruct images. The resulting data set can be displayed as a rotating three-dimensional movie or resliced in any direction to display a series of tw o-dimensional slices. 

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