First invariant

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

A group of transformations can be characterized by its invariants, if any. There are two invariants for one parameter group with three variables. The first invariant is found by adding Eqs. (72) and (73) to eliminate the... 

Its first invariant A] is equal to zero by definition. The second and third invariants of this tensor arc A — atJaJt and A3 — atjajkakt, respectively. The range of physically allowed values of A2 and A3 is bounded and represented by the so-called Lumley triangle in the (A3, A2) plane (Lumley, 1978). The distanced = (Ay + Af) from the isotropic state, i.e., from the origin (A2 — 0, A3 — 0), is a measure of the degree of anisotropy. See also Escudie and Line (2006) for a more extensive discussion as to how to quantify and visualize how different from isotropic turbulence a stirred vessel is. 

A further result of general interest is obtained when the trace (or the first invariant) of the stress contribution tensor, p k is calculated. According to eq. (2.27) one obtains ... 

A close inspection of the right-hand side of eq. (2.34) reveals that it is simply equal to twice the excess free energy of the considered chain in a flowing system compared with that in a system at rest. By summing up the contributions of all chains per unit of volume, one obtains for the first invariant of the macroscopic stress tensor ... 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

One approach to the problem of establishing a knot s chirality or achirality is through the use of knot invariants. The first invariant capable of distinguishing between enantiomorphs, a one-variable Laurent polynomial (a polynomial that has both positive and negative powers), was discovered only as recently as 1985, by Vaughan Jones. 00 More powerful two-variable polynomials have subsequently been developed by others.101 102... 

The relaxation time r can be considered to be a function of the first invariant of the tensor of additional stresses... 

Equations (9.63)-(9.65) were used, in fact, to evaluate the shear viscosity coefficient and the relaxation times which reveal the nature of dependence on the velocity gradient v 12 or shear stress oyi (Isayev 1973). It is convenient to consider the shear viscosity coefficient and the relaxation time as a generalised function of the first invariant of the tensor of additional stresses... 

First invariant leading eigenvalue of the adjacency matrix A Second invariant leading eigenvalue ofGG matrix... 

Finally, a second graph invariant, which can be different from the first invariant, is applied to the graphical matrix in the numerical form to obtain the double invariant. 

The shear rate is often calculated as the second invariant (the first invariant is the trace) of the rate-of-strain tensor ... 

In the case of an incompressible fluid, the first invariant is zero, I - div v = 0. For simple one- and two-dimensional flows (such as flows in thin films, longitudinal flow in a tube, and tangential flow between concentric cylinders), the third invariant h is identically zero. 

KR domains are found in most modules. KR domains contain a potential motif for NADP(H)-binding (GxGxxAxxxA) [33]. The first invariant Ala in this motif had been found to be substituted by Gly in many polyketide synthases [21, 23,... 

Terms on the main diagonal are strain velocities, side elements represent angle rates of initially cuboid elements. The first invariant of D, sp(D), is equal to div v, the volumetric strain rate, which must be zero for incompressible flow. 

In formula (1), /, respectively represent the first invariant of the stress tensor partial second invariant in the stress tensor. Different a, k are constants related to c and tp, different a, k represents different circles on plane n, Zhao Shangyi, Zheng Yingren, Deng Weidong gave elaboration and derivation in the relevant articles. 

Definition (7) has been invoked within the parentheses that extend the right hand side of equation (6). Divergence represents the trace (first invariant) of the transient total strain tensor of the porous granular structure. It signifies a change in volume of this saturated porous structure. This contrasts to the deviatoric component of the total structural strain tensor, which signifies a change in shape. 

The first invariant represents rate of change of volume, which is zero for incompressible fluids. The third invariant IIIo is zero for plane flows. The second invariant IIo represents a mean rate of deformation including all shearing and extensional components. It is convenient to define, for all flows, a generalized strain rate as... 

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