First critical case

In view of this condition, the equilibrium states associated with the first critical case are also called degenerate. Since the implicit function theorem may no longer be applied here, the persistence of such an equilibrium state in a neighboring system is not necessarily guaranteed. Thus, a transition through the stability boundary in the first critical case may result in the disappearance of the equilibrium state. 

Chap. 10. The Behavior of Dynamical Systems on. .. 10.2. The first critical case... 

Expansion strains may be taken up in three ways by bending, by torsion, or by axial compression. In the first two cases maximum stress occurs at the extreme fibers of the cross section at the critical location. In the third case the entire cross-sectional area over the entire length is for practical purposes equally stressed. 

All turbines are variable-speed drivers and operate near or above one of the rotor s critical speeds. Narrowbands should be established that track each of the critical speeds defined for the turbine s rotor. In most applications, steam turbines operate above the first critical speed and in some cases above the second. A movable narrowband window should be established to track the fundamental (1 x), second (2x), and third (3x) harmonics of actual shaft speed. The best method is to use orders analysis and a tachometer to adjust the window location. 

In most cases, running speed is the forcing function that excites the natural frequency of the dynamic component. As a result, rotating equipment is designed to operate at primary rotor speeds that do not coincide with the rotor assembly s natural frequencies. Most low- to moderate-speed machines are designed to operate below the first critical speed of the rotor assembly. 

While many of the observed events of the MBH reaction could be included in this scheme, the mechanism failed in some critical cases [47]. First, the mechanism did not provide any clue as to why stereocontrol is so difficult in MBH reactions. Privileged nucleophilic chiral catalysts [48], which in the past have usually allowed good results in related asymmetric transformations, afforded only modest asymmetric induction. This fact was surprising, and pointed to lack of understanding of the basic factors governing the selectivity of the reaction. Other obser-... 

Consider first the case of clean steps. The curvature of the step is then entirely determined by its macroscopic radius r. Consequently, c(r) increases continuously with r reaching rather rapidly the saturation concentration ce. If it is now assumed that for a new step to be nucleated c(0)/ce has to be equal to the critical value c0/ce given by Eq. (2), it is clear that r/a has to become macroscopically large whatever the value given to c in other words, the first step has to make room for the following one to be nucleated. In particular, if c0/ce 11 0.20 as appears to be the case for LiF in water solutions considered above, one deduces from Eq. (4) that r/a -106 even for c 0. Needless to say, such a large value of r/a corresponds to an extremely flat crystal surface and no visible pit. 

It is essential to prepare single cell suspensions, which exist naturally in very few cases (culmred cells such as HeLa, Hel, and blood cells). In most critical cases (sohd rnmors, tissues, and neural cells), a preliminary step is necessary and classically requires transformation of the tissue in suspension. Two methods are used the first is mechanical, whereas the second requires enzymatic action. It is essential to obtain a monocell suspension, whose characteristics have to be controlled by FC. Once this suspension is obtained, another critical parameter is... 

Calculation of the ground state energy of the helium atom is a critical case as well because it is the first example of correlation energy, the difference between the Hartree-Fock energy and the exact value. The energy required to remove one electron from a neutral He atom is the first ionization potential... 

In the case of electrolytes that are capable of causing the reversal of surface charge of dispersed particles, coagulation occurs only at electrolyte concentrations corresponding to rather small values of the -potential. As electrolyte concentration becomes higher, the C, -potential decreases coagulation starts as soon as the C, -potential drops below the critical value. This corresponds to the first critical coagulation concentration, C.C.C. -1, (c, point in the curves shown in Fig. VIII-19). Further increase in electrolyte... 

The required sampling frequency dictates that the second dimension separation should be as fast as possible while providing adequate resolution and the first dimension separation should be slowed down to accommodate the sampling frequency and second dimension separation time. The total separation time is the product of the second dimension separation time and the total number of fractions injected into the second dimension. Thus, the separation time of the second dimension separation is a major factor in determining the total separation time of comprehensive two-dimensional separations. When more than one separation dimension is utilized in a sequential coupled column mode, a larger dilution of the original injection concentration occurs with loss of sample detectability [88]. The column dilution factors and split ratios used to compute limits of detection are multiplicative per dimension. In critical cases information may be lost when a fraction transferred from the first dimension falls below the detection threshold after separation in the second dimension. 

The first safety cases prepared for the process industries were those developed for North Sea offshore oil and gas operations following the Piper Alpha disaster that occurred in the year 1988. The Cullen report (Cullen, 1990) that was written following that accident was highly critical of offshore operating practices and recommended that a safety case approach be implemented. The Seveso incident that occurred in Italy further prompted the increased use of safety cases. Since that time the use of safety cases has spread to other industries (such as mining and railway operations) and to other nations, primarily in Europe and Australasia. (It is notable that the safety case regime approach has not been taken up for offshore oil and gas operations in the Gulf of Mexico—instead a more prescriptive approach based on industry consensus standards is used.)... 

In the case of the first critical point the amplitudes A and B are identically equal to zero, which corresponds to the uniformly propagating wave in the original problem. The second and third critical points correspond to waves traveling along the front, which are right- and left-traveling waves, respectively. The last critical point... 

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