Mathematical methods, discrete

The model developed was based on the framework and model developed by Majozi and Zhu (2001) for the following reasons. Firstly, the models that exploit the structure of the SSN result in fewer binary variables than those derived from other mathematical methods, because the SSN only takes states into account while tasks are implicitly incorporated. Secondly, this model is based on the non-uniform discretization of the time horizon, thus resulting in fewer binary variables. Thirdly, the model is a MILP, thus solutions are globally optimal. 

Many industrial users of batch distillation (Chen, 1998 Greaves, 2003) find it difficult to implement the optimum reflux ratio profiles, obtained using rigorous mathematical methods, in their pilot plants. This is due to the fact that most models for batch distillation available in the literature treat the reflux ratio as a continuous variable (either constant or variable) while most pilot plants use an on-off type (switch between total reflux and total distillate operation) reflux ratio controller. In Greaves et al. 2001) a relationship between the continuous reflux ratio used in a model and the discrete reflux ratio used in the pilot plant is developed. This allows easy comparison between the model and the plant on a common basis. 

The various mathematical methods for determining optimum conditions, as presented in this chapter, represent on a theoretical basis the conditions that best meet the requirements. However, factors that cannot easily be quantitized or practical considerations may change the final recommendation to other than the theoretically correct optimum condition. Thus, a determination of an optimum condition, as described in this chapter, serves as a base point for a cost or design analysis, and it can often be quantitized in specific mathematical form. From this point, the engineer must apply judgment to take into account other important practical factors, such as return on investment or the fact that commercial equipment is often available in discrete intervals of size. 

Mathematical chemistry, the new challenging discipline of chemistry has established itself in recent years. Its main goal is to develop formal (mathematical) methods for chemical theory and (to some extent) for data analysis. Its history may be traced back to Caley s attempt, more than 100 years ago, to use the graph theoretical representation and interpretation of the chemical constitution of molecules for the enumeration of acyclic chemical structures. Graph theory and related areas of discrete mathematics are the main tools of qualitative theoretical treatment of chemistry [1,2]. However, attempts to contemplate connections between mathematics and chemistry and to predict new chemical facts with the help of formal mathematics have been scarce throughout the entire history of chemistry. 

However, before mathematical methods could be employed in chemistry, the latter had to reach sufficient theoretical maturity. Thus, the graph theory and some other divisions of discrete mathematics could be applied to the solution of chemical problems only after the chemical structure theory had emerged. In the next section we shall briefly discuss some important landmarks in the development of ideas concerning the structure of chemical compounds. 

P. A. Jensen and J. F. Bard, Mathematics of discrete-time Markov chains, in Operations Research Models and Methods. Wiley, Hoboken, NJ, 2002, pp. 466-492. 

Fonnal methods involve the use of formal mathematical logic, discrete mathematics and computer languages (including a formalised grammar and vocabulary) to provide evidence that the system is complete and correct with respect to its requirements, and a determination of which code, S/W requirements or S/W architecture satisfy the next higher level of S/W requirements DO-178C. 

Quantification of the related phenomena proceeds by the definition of a model that describes the physical processes (e.g. mass transport, chemical reactivity, etc.) in terms of a set of mathematical expressions. Traditionally, workers have attempted to solve these relationships directly via standard mathematical methods however, in many cases the complexity of the problem does not permit this analytical-solution approach. Digital simulation breaks down (discretizes) the problem into a series of steps that can be solved sequentially by the composition of a suitable computer program. This discretization process gives rise to a variety of different digital strategies with which electrochemical or related problems can be solved. 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

Such Bayesian models could be couched in terms of parametric distributions, but the mathematics for real problems becomes intractable, so discrete distributions, estimated with the aid of computers, are used instead. The calculation of probability of outcomes from assumptions (inference) can be performed through exhaustive multiplication of conditional probabilities, or with large problems estimates can be obtained through stochastic methods (Monte Carlo techniques) that sample over possible futures. 

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

The discretization studied here is related to that of the Euler-McLaurin method well-known in nnmerical mathematics (see e.g. [23]). The difference is that in this method one approximates the mean value of/(x) in the interval by the average of the values at the boundaries of the interval, while we approximate it by its value at the center of the interval. This choice is more closely related to the expansion of a fnnction in a basis. 

The various contributions can also be classified in accordance with the optimization techniques used. However, this method of organization gives rise to an even more diverse classification, since the techniques used range all the way from rules of thumb (A3-A5, M6-M8, Ol, T2) and analytical solution (S8) to the more recent developments in mathematical programming. Most of the techniques reported are continuous, but some are discrete (C8, R5) and still others are of mixed integer types (G3). Table VI shows such a classification for the papers reviewed. It is clearly beyond the scope of this review to delve into the mathematical bases of these methods. We shall... 

Any analysis of risk should recognize these distinctions in all of their essential features. A typical approach to acute risk separates the stochastic nature of discrete causal events from the deterministic consequences which are treated using engineering methods such as mathematical models. Another tool if risk analysis is a risk profile that graphs the probability of occurrence versus the severity of the consequences (e.g., probability, of a fish dying or probability of a person contracting liver cancer either as a result of exposure to a specified environmental contaminant). In a way, this profile shows the functional relationship between the probabilistic and the deterministic parts of the problem by showing probability versus consequences. 

A mathematical formulation based on uneven discretization of the time horizon for the reduction of freshwater utilization and wastewater production in batch processes has been developed. The formulation, which is founded on the exploitation of water reuse and recycle opportunities within one or more processes with a common single contaminant, is applicable to both multipurpose and multiproduct batch facilities. The main advantages of the formulation are its ability to capture the essence of time with relative exactness, adaptability to various performance indices (objective functions) and its structure that renders it solvable within a reasonable CPU time. Capturing the essence of time sets this formulation apart from most published methods in the field of batch process integration. The latter are based on the assumption that scheduling of the entire process is known a priori, thereby specifying the start and/or end times for the operations of interest. This assumption is not necessary in the model presented in this chapter, since water reuse/recycle opportunities can be explored within a broader scheduling framework. In this instance, only duration rather start/end time is necessary. Moreover, the removal of this assumption allows problem analysis to be performed over an unlimited time horizon. The specification of start and end times invariably sets limitations on the time horizon over which water reuse/recycle opportunities can be explored. In the four scenarios explored in... 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

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