Root, defined

In our approach, the concept of ensemble of equilibrium subsystems introduced in our earlier chapters (see in detail Lazman and Yablonskii, 1991) was used as a very efficient tool of mathematical analysis and physico-chemical understanding. The equilibrium subsystem is such a system that corresponds to the following assumption (n—1) steps are considered to be under equilibrium conditions, one step is limiting, where n is a number of steps. In fact, the concept of "equilibrium subsystems" is a generalization of the concept of "equilibrium step", which is well known in chemical kinetics. Then, we take n of these equilibrium subsystems an ensemble of equilibrium subsystems). It was shown that solutions of these subsystems ("roots", "all roots", not just one "root") define coefficients of the kinetic polynomial. 

These roots define a regular figure in 3D space that is a tetrahedron. The vertices of this tetrahedron correspond to... 

The moment of inertia of a ring of particles, mrki was used as the criterion for stability to define a closed orbit that combines circular motion with simple harmonic displacements. A more general discussion that substantiates the derivation is given by Goldstein [11]. The frequency of revolution is obtained in the form of a square root, defined by a set of integers,... 

Chao (2002) developed a sophisticated tabu search method consisting of an initial constmction heuristic and a tabu search improvement heuristic to search the neighborhood of an initial solution and hence refine them. The method contains standard VRP moves as well as a new TTRP-specific root-defining move where the roots of sub-tours are changed to new customers. Four years later, Scheuerer (2006) extended Chao s work to develop a better tabu search heuristic that outperforms the Chao s method. Scheurer made use of two new construction heuristics called the T-Cluster and T-Sweep. The new construction heuristics proved to successfiilly find good initial solutions for TTRP. He was able to obtain better results in each of the 21 instances Chao (2002) adapted. 

The geometric mean of a set of N numbers is the A th root of the product of the numbers The root mean square (RMS) or quadratic mean of a set of numbers is defined by ... 

The hazard posed can be limited by maintaining a zone free of people and property around a storage area of explosive material. The minimum radius of the zone depends on the type and quantity of explosive, the extent and type of barrica ding, and the magnitude of loss that would be encountered if an explosive incident occurred. The maximum distance to which hazardous explosive effects propagate depends on the blast overpressure created, which as a first approximation is a function of the cube root of the explosive weight, W. This is termed the quantity distance and is defined as... 

In a free jet the absence of a pressure gradient makes the momentum flux at any cross section equal to the momentum flux at the inlet, ie, equations 16 and 17 define jet velocity at all points. For a cylindrical jet this leads to a center-line velocity that varies inversely with (x — aig), whereas for slot jets it varies inversely with the square root of (x — Xq As the jet proceeds still further downstream the turbulent entrainment initiated by the jet is gradually subordinated to the turbulence level in the surrounding stream and the jet, as such, disappears. 

The clinical performance of a hemodialy2er is usually described in terms of clearance, a term having its roots in renal physiology, which is defined as the rate of solute removal divided by the inlet flow concentration as shown in equation 7, where Cl is clearance in ml,/min and all other terms are as defined previously except that, in deference to convention, flow rates are now expressed in minutes rather than seconds and feed side (/) is now synonymous with blood flow on the luminal side. 

The fixed capital estimate depends on the definition of the plant. A grass-roots plant is a complete faciUty at a new location, including all utihties, services, storage faciUties, land, and improvements. If a process plant is located at an existing processing complex, it can usually share some of these auxihary faciUties. A battery-limits plant is defined as the process faciUty itself, so that the auxiUaries, off-site, and land-related items are excluded from the fixed capital estimation. However, a battery-limits plant maybe assigned allocated capital charges for the share of common utihty and service faciUties used by the plant. 

In these equations, is the molar volume and 5 is the solubiUty parameter of the pure Hquid k. The solubiUty parameter is the square root of an energy density, defined as... 

Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Tmn.s. Am. Soc. Mech. Eng., 67,621 (1945)]. 

The relative volatiHties Ot) are defined by Eq. (13-33), is the mini-mum-reflux ratio (L v + i/D)min,. nd q describes the thermal condition of the feed (e.g., 1.0 for a bubble-point feed and 0.0 for a saturated-vapor feed). The Xi p values are available from the given feed composition. The 0 is the common root for the top-section equations and the bottom-section equations developed by Underwood for a column at minimum reflux with separate zones of constant composition in each section. The common root value must fall between 06/, and Ot/, where hk and Ik stand for heavy key and light key respectively. The key components are the ones that the designer wants to separate. In the butane-pentane splitter problem used in Example 1, the light key is /1-C4 and the heavy key is i-C. ... 

The parameter used to design rapid mix and flocculation systems is the root mean square velocity gradient G, which is defined by equation... 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

To make contact with the diffusion-in-a-sphere model, we have defined the spherical radius as the root-mean-square fluctuation of the protons averaged over 100 ps. The varia-... 

The standard deviation of the sample is defined as the square root of the variance. For the example,... 

The eoneept of an Amplifieation Faetor (AF) is introdueed in the new API 616 standard. Amplifieation faetor is defined as the ratio of the eritieal speed to the speed ehange at the root mean square of the eritieal amplitudes. 

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