Torsion coefficient

The torsion part can be represented as a direct product of cyclic groups of prime power order. The exponents of this groups are called the torsion coefficients. The nth Betti number and the torsion coefficients are uniquely determined by the group Hn A]Z). 

MM 1 -CARB is the MM 1 force field modified to include the carbohydrate parameters reported by Jeffrey and Taylor in 1980. MMl itself contained all the valence parameters necessary for carbohydrates, and the MMl-CARB version represents a modest tuning of the MMl parameters. As in the cases of MM2 and MM3, the values of the glycosidic torsion coefficients (V 3) in MMl were chosen so as to reproduce the experimentally determined AH for the axial to equatorial conversion of 2a to 2e. In contrast to MM2, MMl lacks a torsion-stretch cross term, and the MMl-GARB parameterization introduced different valence stretching and... 

So far we have introduced four Miesowicz viscosities. Two other viscosities can be proposed by considering the following. The director n in Fig. 4.1(a), if free to move, will rotate due to a viscous torque the viscosity coefficient 71 is introduced to describe this situation and characterises the torque associated with a rotation of n. For this reason 71 is often called the rotational viscosity or twist viscosity. The coefficient 71 generally determines the rate of relaxation of the director. Also, a rotation of n due to body forces will induce a flow. The viscosity coefficient 72 characterises the contribution to the torque due to a shear velocity gradient in the nematic and is sometimes referred to as the torsion coefficient in the velocity gradient it leads to a coupling between the orientation of the director and shear flow. The two viscosities 71 and 72 have no counterpart in isotropic fluids. We therefore have a total of six viscosities four Miesowicz viscosities plus 71 and 72. It turns out, as will be seen in the problems to be discussed in later Sections, that 7i and 72 are precisely the viscosities introduced in the constitutive theory at equations (4.78) and (4.79), namely. 

Inserting this value into equations (5.239) yields equation (5.250) again. If we suppose a 0, then it is evident that 0(r) must take a constant value determined by a relation which involves only the rotational viscosity 71 and the torsion coefficient 72. This value is given by the relation... 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

A torsional spring of stiffness K, a mass of moment of inertia / and a fluid damper with damping coefficient C are connected together as shown in Figure 3.25. The angular displacement of the free end of the spring is 0 ( ) and the angular displacement of the mass and damper is 6a t). 

The main features of this device are shown in Fig. 3.9a. The z-axis is directed along the field (the vertical), and the lever with a small mass m rotates around a horizontal axes, passing through the point 0. The length of the lever is a. The torsion spring is installed on the axis, and its torsion constant is equal fi. This is a coefficient of proportionality between the moment t and the rotation angle or. 

The transition between crystalline and amorphous polymers is characterized by the so-called glass transition temperature, Tg. This important quantity is defined as the temperature above which the polymer chains have acquired sufficient thermal energy for rotational or torsional oscillations to occur about the majority of bonds in the chain. Below 7"g, the polymer chain has a more or less fixed conformation. On heating through the temperature Tg, there is an abrupt change of the coefficient of thermal expansion (or), compressibility, specific heat, diffusion coefficient, solubility of gases, refractive index, and many other properties including the chemical reactivity. 

The stretch-bend, torsional, electrostatic and VdW terms in MM3 are identical in form to the corresponding ones in MM2 (although the electrostatic treatment in MM3 also includes charge-dipole interactions and the VdW terms have slightly different numerical coefficients) and will not be further discussed here. 

Pure crystalline silicon is a brittle material with a gray metallic appearance. Its mechanical properties, such as Knoop hardness (950-1150 kg mm-2), Young s modulus (190 GPa for (111), 170 GPa for (110), 130 GPa for (100)), torsion modulus (4050 kg mm-2) and compression breaking strength (5000 kg cm-2) vary slightly with crystal orientation. Silicon has a low thermal expansion coefficient (2.33x 1(T6 K-1) and a high thermal conductivity (148 W K-1m-1). Crystalline silicon melts at 1413 °C (1686 K). 

Top Dplu versus superhelix density. >plat is the apparent DLS diffusion coefficient at A2 = 20 x 10 ° cm 2. The lower values at a = —0.015, —0.020, and -0.025 suggest that, in this intermediate range, DNA exhibits a different secondary structure with decreased torsional and/or bending rigidity. 

Here cp represents the torsional coordinate, I denotes the moment of inertia, and Vf is the first coefficient of the Fourier series expansion of the isomerization potential of periodicity nn. 

A prerequisite to these calculations is the knowledge of bending and torsional rigidity coefficients (A and C, respectively) of the naked DNA. A was calculated from a persistence length a = A/kT = 50 nm [58], and C through the following equation, valid only for a naked DNA minicircle near relaxation [59]... 

Such a distribution has a plausible physical basis, since the driving force for phenyl rotation into the porphyrin plane provided by the electronic excitation (the eg orbital has particularly large coefficients at the meso carbon atoms ( )) encounters steric resistance from the non-bonded interactions between the protons at the ortho positions of the phenyl groups and those on the outer pyrrole carbon atoms (20). Consequently the phenyl torsion potential in the excited states may be relatively flat. Nevertheless, the vibrational frequencies are expected to be sensitive to the torsion angle for orientation close to co-planar because of the effect of conjugation. 

By considering the variation with torsion angle of both the mixing coefficient and the component of the charge flow magnetic moment perpendicular to the... 

Eq. 2 V/S is the volume to surface ratio Phi, the torsion angle between the C2 atom and the B ring o, the Hammett coefficient of the B ring dV, volume difference between substrate and flavonoid dL, length of the C3 side chain (p, dipole moment and C3, C5, and C4 the electron density occurring at these atoms. 

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