Minimum condition

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

Equation B3.5.5 is, in turn, equivalent to the minimum condition on the rational fiinction... 

As discussed in 2-6.3 and 6-4.2, powder ignition via bulking brush discharge requires the coincidence of several factors, each of which has an associated probability. The minimum conditions are... 

A number of gases may decompose (self-react) and propagate flames in the absence of any oxidant provided that they are above minimum conditions of pressure, temperature, and pipe diameter. Common examples are acetylene, ethylene oxide, and ethylene. Some, like acetylene, can decompose in a detonative manner, while ethylene cannot detonate in the absence of an oxidant, whatever the run-up length (CCPS 1993). Thus, detonation arresters must be used for acetylene, but deflagration arresters may be used for ethylene, even for in-line applications. 

Decomposition Flames Flames that are produced hy exothermic decomposition of certain gases in the absence of any oxidant, provided that they are above minimum conditions of pressure, temperature, and pipe diameter. Common examples include acetylene, ethylene oxide, and ethylene. 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

Note than when (L/D) min is used as the starting basis, the Lr, Lg, Vr, Vg and their ratios will be for the minimum condition, and correspondingly so when the operating reflux is used. 

To this end, the numbers r and 9 are so chosen as to satisfy the minimum condition that we have mentioned above. As far as j E < Aq < j E for a self-adjoint operator Aq, we might have... 

It was concluded that, as a minimum condition, the receptor has sites that respond to hydroxyl, alkoxyl, and the glycosyl (or adjacent) parts of the molecule (R) (see Fig. 23). The orientation of the B-ring is critical. It must be in quasiaxial orientation for active interaction with the receptor. ... 

According to Shepherd,2 The minimum conditions for effective competition are reasonably clear, both from theory and business experience . Shepherd2 states that any of the following conditions is sufficient to ensure that competition is effective first, at least five comparable competitors second, no single company with a dominant position (the share of the market leader should be no greater than 40 per cent, and there should be no substitute product in the close vicinity) and third, freedom of entry into the market and the various market segments. 

The minimum condition for an ignition is 0.10 mJ therefore the operating conditions for this vessel are extremely hazardous. 

These recommendations are produced to formulate minimum conditions for research workers,organisations and industries, who wish to have consistent results and measurements which can be compared with odour measurements in other laboratories and other countries. Also laboratories beginning with odour measurements are advised to start at least at this level. 

The designer must specify the maximum flow rate that is required under the worst conditions and the minimum flow rate that is required. Then the valve flow equations for the maximum and minimum conditions give two equations and two unknowns the pressure head of the centrifugal pump APj> and the control valve size C . 

The control valve/pump sizing procedure proposed above is not without its limitations. The two design equations for the maximum and minimum conditions in general terms are ... 

An equivalent method was used by Kitaigorodski and Ahmed (11) to predict the equilibrium habit of anthracene. Starting from a nucleus of just one molecule, the growth was simulated by adding one molecule at a time at positions such that the free energy minimum condition was always satisfied. A simple 6-exp two-parameter potential function was used to estimate... 

Riba, J.P., Routie, R. and Couderc, J.P., Conditions minimales de mise en fluidisation per une liquide [Minimum conditions for initiation of fluidisation with a liquid]. Can. ]. Chem. Eng., 56 (1978) 26-34. 

From (8) the estimate zn < Tn z0 immediately follows, in which the quantity Tn depends on the parameters and rj2 Roughly speaking, a proper choice of such parameters is stipulated by the minimum condition for the norm Tn in connection with a minimal number of the necessary iterations. To be more specific, when making a substantiated choice, we have at our disposal two collections of parameters Ti t21 ... 

It is generally assumed that humidity is not important in most rubber tests and, hence, conditioning in an atmosphere with control of temperature only is usually specified. However, control of humidity is considered necessary in certain cases, for example testing latex rubber and electrical tests. In many instances the 16h minimum conditioning period will not be sufficient for equilibrium to have been reached, especially with relatively thick test pieces. Hence, all that this conditioning can hope to achieve is to bring test pieces having similar dimensions into more nearly comparable conditions than they would otherwise be. To reach complete moisture equilibrium would in many cases take several days and for thicker test pieces probably weeks. 

After the cell and test piece have been assembled and the high pressure side filled with gas at the test pressure, the increase in pressure on the low pressure side is measured as a function of time. The standard suggests a conditioning period of at least 16 h to reach steady state conditions unless an approximate value of the diffusion coefficient is known, when the minimum conditioning time can be estimated from ... 

The roots of eq. (39) with v and y considered as parameters determine branches of candidates for butterfly singularity points, which advance to that status if certain conditions are not violated. We have already seen how the conditions A 0, e 0, w —y arise naturally in the derivation of eq. (39) as guarantees that solutions of the latter equation are valid solutions of (28a)-(28e). A separate class of conditions arise from the theory of Golubitsky and Schaeffer, allowing one to relate the quasi-global behaviour of the system of interest to that of simple polynomial functions. For example, the minimum conditions from the theory for a butterfly point are that eqs (28a)-(28e) be satisfied and also... 

Spitael and Kinget [5] investigated the influence of ionic states on the dissolution of different enteric film coatings. They pointed out that buffers have to be carefully selected for dissolution studies and in vitro-in vivo correlation analysis. Ideally, their ionic state should be as close as possible to the in vivo conditions. As a minimum condition, the nature and concentration of cations should be matched as closely as possible. 

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