Closure principle

Integration of the ODEs results in the concentration profiles for all reacting species as a function of the reaction time and the initial concentrations. The explicit solutions for the ODEs above (3.75) are given below (3.76). We list the equations for one concentration only. The remaining concentrations can be calculated from the closure principle, which is nothing else but the law of conservation of mass (e.g. in the first example [B]=[A]o-[A], where [A]o is the concentration of A at time zero). Only in example d) two concentrations need... 

Meyers, T.P., and Paw, U.K.T. (1987) Modelling the plant canopy micrometeorology with higher-order closure principles, J. Agric. Forest Meteorol. 41, 143-163. 

This closure principle and its explanatory analogue are quite weak. All they state is that any physical event that has a cause at a given time will have a physical cause at that time, and that any physical event that has a causal explanation at a time will have a physical causal explanation at that time. As stated, they do not require that this physical cause is a sufficient cause or the exclusive cause, nor do they require that the causal explanation will be a sufficient causal explanation or that it will exclude other causal explanations. 

Similarly, Kims point that existential quantification over a set of first-order properties doesn t guarantee the specification of a new property is an argument against a closure principle, and not a general argument that higher-order predicates cannot express genuine properties just because they are definable in terms of quantification over lower-order properties. 

Gibb S (2010) Closure principles and the laws of conservation of energy and momentum. 

This deterministic reasoning is only possible in a closed world. The closure principle permeates every thought concerning industrial safety. The safety of an automated system is possible only in the case of a closed system whenever there is interaction, there are unpredictable risks. 

V (the potential) is identified with the enthalpy, i.e. the number n of base pairings (contacts), and T corresponds to the entropy. At each stage in the folding process, as many as possible new favourable intramolecular interactions are fonned, while minimizing the loss of confonnational freedom (the principle of sequential minimization of entropy loss, SMEL). The entropy loss associated with loop closure is (and the rate of loop closure exp... 

CannedFoods, Principles of Thermal Process Control, Acidification, and Container Closure Evaluation, 5th ed.. The Food Processors Institute, Washington, D.C., 1988. D. M. Considine and G. D. Considine, eds.. Foods and Food Production Fnyclopedia, Van Nostrand Reinhold Co., New York, 1982. 

Equipment. A typical commercial quartz-growing autoclave is Hlustrated in Figure 1. The material of constmction for use at 17 MPa (25,000 psi) and 400°C can be a low carbon steel, such as 4140, or various types of low aHoy steel. The closure, a modified Bridgeman closure, is based on the unsupported area principle (12). That is, the pressure in the vessel is transmitted through the plunger to the steel surfaces which initially are nearly line contacts. Thus, the pressure in the seal surface gready exceeds the pressure in the vessel because most of the area of the plunger is unsupported. Hydrothermal equipment has been further discussed (10). 

Measurements and Audits. The enabling element of continuous improvement is measurement. An old rule of thumb says that increased accuracy in measuring an energy use ultimately yields a reduction in use equal to 10% of the increased closure of the balance. A basic principle of economics is that any thing that is free is used in excess, ie, an unmetered electrical use is bigger than expected by at least 10%. Metering of the cost elements at each unit in a chemical plant provides effective accountabhity. Measurements should be linked via computer software to production as weh as to weather to result in maximum feedback. 

The principle of microscopic reversibility requires that the reverse process, ring closure of a butadiene to a cyclobutene, must also be a coiuotatory process. Usually, this is thermodynamically unfavorable, but a case in which the ring closure is energetically favorable is conversion of tra s,cis-2,4-cyclooctadiene (1) to bicyclo[4.2.0]oct-7-ene (2). The ring closure is favorable in this case because of the strain associated with the trans double bond. The ring closure occurs by a coiuotatory process. 

The ring opening reactions of unsaturated azlactones and other lactones - by methanol in the presence of diazomethane are analogous in principle. (The ring closure of pseudouric acid which occurs under the influence of diazomethane can also be understood as an example of base catalysis.)... 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

A classical topic for illustration of the above principles is the analysis of the behaviour of the system trimethylene cyclopropane methylene + ethylene (11). Ring closure of trimethylene is a simple example of an electrocyclic reaction (8a) ... 

On orbital symmetry grounds, the addition of ethylene to ethylene with ring closure (cycloaddition) should be thermally forbidden. If one compares this reaction with the reaction of trimethylene with approaching ethylene and butadiene (Fig.4), it is readily seen that, the A level being below the S level in trimethylene, the behaviour with respect to cycloaddition to olefins is reversed, that is, trimethylene is essentially an anti-ethylene structure. This principle can be generalized for instance (16) ... 

The theoretically interesting phenyl hydroxy cyclopropenone (57) was prepared by Famums1, s2 according to the general principle of cyclopropene ring closure developed by Closss3) from 53 via the vinyl carbene 54 and phenyl trichloro cyclopropene (55). 

The photocyclization of enamides to quinolines or isoquinolines has become an important reaction in the synthesis of alkaloids 219,358). It has recently been applied in the preparation of the isoquinoline alkaloid polycarpine 359). The principle of the reaction is demonstrated in the preparation of dihydroquinolines 360> (3.39) and of spirocyclohexaneisoquinoline derivatives (3.40) 361K In each case the electrocyclic ring closure product undergoes a subsequent 1,5-hydrogen shift. 

Anderson and Jackson (1967, 1968, 1969) and Ishii (1975) have separately derived the governing equations for TFMs from first principles. Although the details of constructing the averaged equations are different, the final equations are essentially the same. The TFMs differ significantly from each other as different closures for the solid stress tensor are used. 

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