Heuristics, definition

The preceding four sections summarize only part of the content of the following four chapters several interesting results have not been mentioned. To keep the paper within bounds, I had to forego detailed discussion of aspects I deemed less important. For this subject matter, definitions and even formal calculations and heuristic deductions seem to me often more important than complete proofs. Thus, proofs were eliminated first in particular, in the case of several analogous propositions the proof of only one theorem is... 

Also, we consider the total approximation method as a constructive method for creating economical difference schemes for the multidimensional equations of mathematical physics. The notion of additive scheme is introduced as a system of operator difference equations that approximates the original differential equation in the total sense. Two quite general heuristic methods (proposed earlier by the author) for obtaining additive economical schemes are discussed in full details. The additive schemes require a new technique for investigating convergence and a new type of a priori estimates that take into account the definition of the property of approximation. 

Intermolecular recognition and self-assembly processes both in the solid, liquid, and gas phases are the result of the balanced action of steric and electronic factors related to shape complementarity, size compatibility, and specific anisotropic interactions. Rather than pursuing specific and definitive rules for recognition and self-assembly processes, we will afford some heuristic principles that can be used as guidelines in XB-based supramolecular chemistry. 

We briefly present a heuristic argument how quark matter becomes ferromagnetic by the use of above definition [8], The Fock exchange interaction, /kc.qC between two quarks is defined by... 

The term ionization may refer to different processes depending on the context. For radiation effects in the gas phase, it usually implies the removal of the least bound electron to infinity. Such a theoretical definition is not feasible in the condensed phase and it is necessary use a heuristic or operational procedure. Thus, in liquid hydrocarbons, one may use the electron scavenging reaction or a conductivity current to quantify the electrons liberated from molecules. It has only been possible to extrapolate the conductivity current at a low irradiation dose and at a relatively low external field to saturation in the cases of liquefied... 

Remark 4 The termination criterion in step 6 is based on obtaining a larger value in the primal problems for consecutive iterations. However, there is no theoretical reason why this should be the criterion for termination since the primal problems by their definition do not need to satisfy any monotonicity property. (i.e they do not need to be nonincreasing). Based on the above, this termination criterion can be viewed only as a heuristic and may result in premature termination of the OA/ER/AP. Therefore, the OA/ER/AP can fail to identify the global solution of the MINLP problem (6.33). 

Constitutional formulae were designed "on paper", primarily to be "in harmony" with known chemical properties and without pretension to "represent the symmetrical or spatial arrangement of the atom in a compound" [22], Not only was this stipulation gradually relaxed to represent three-dimensional structures, but the connecting lines were also soon after assumed to represent definite electronic links between atoms. This assumption opened the door for the introduction of semi-empirical quantum-mechanical characterization of chemical bonds. It is important to realize that chemical bonds have never been observed in any experiment and that they only exist as conjectures to interpret primitive molecular graphs. Their value as heuristic aids in the study of chemical change and composition is beyond dispute, but as a basis for the theoretical understanding of chemical cohesion they are of little value. 

By comparing the heats of melting and sublimation one finds that only 13% of H-bonds in ice are broken upon melting A similar result (19%) was recently suggested on the basis of a heuristic density-functional method. Many other estimates of the percentage of broken H-bonds are available in the literature. " These estimates, based both on experimental results obtained by various techniques and on theoretical models, provided values ranging from 2 to 72%, " which are dependent on the definition used for an H-bond in liquid water. 

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