Approximations

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.

Many information measures are suggested in the literature. Among these measures are the Shannon H(n, b) and the Awad A(n, b) entropy measures. In this work we suggest a new entropy measure, B(n, b, estimated b) , which is based on the maximum likelihood function. These three entropy measures were calculated from the gamma distribution and its normal approximation, the binomial and its Poisson approximation, and the Poisson and its normal approximation. The relative losses in these three entropy measures are used as a criterion for the appropriateness of the approximation.

The most common explicit correlations for estimation of the friction factor in rough and smooth pipes are reviewed in this paper. Comparison of any friction factor equation with the Colebrook's equation was expressed trough the mean relative error, the maximal positive error, the ...

The most common explicit correlations for estimation of the friction factor in rough and smooth pipes are reviewed in this paper. Comparison of any friction factor equation with the Colebrook's equation was expressed trough the mean relative error, the maximal positive error, the maximal negative error, correlation ratio and standard deviation. The statistical comparison of different equations was also carried out using the "Model selection criterion" and "Akaike Information Criterion". It was found that the equation of Zigrang and Sylvester provides the most accurate value of friction factor, and that Haaland's equation is most suitable for hand calculations.

Many information measures are suggested in the literature. Among these measures are the
Shannon H,(O) and the Awad A,(O) entropies. In this work we suggest a new entropy measure, B~(0),
which is based on the maximum likelihood function.
These three entropies were calculated from the gamma distribution and its normal approximation, the
binomial and its Poisson approximation, and the Poisson and its normal approximation. The relative
losses in these three entropies are used as a criterion for the appropriateness of the approximation.
Copyright © 1996 Elsevier Science Ltd.

It is generally claimed that infinite idealizations are required for explaining phase transitions within statistical mechanics (e.g., Batterman 2011). Nevertheless, Menon and Callender (2013) have outlined theoretical approaches that describe phase transitions without using the infinite limit. This paper closely investigates one of these approaches, which consists of studying the complex zeros of the partition function (Borrmann et al. 2000). Based on this theory, we argue for the plausibility for eliminating the infinite limit for studying phase transitions. We offer a new account for phase transitions in finite systems, and we argue for the use of the infinite limit as an approximation for studying phase transitions in large systems.

Keywords infinite idealizations · phase transitions · finite systems · ineliminability · approximation · zeros of partition function

Tolerance analysis usually complements tolerance synthesis; however, analysis algorithms, especially the Monte Carlo method proves highly expensive. This study offers an alternative to the Monte Carlo method. The huge sample sizes are replaced by small orthogonal experiments. The mean and variance of critical design requirements are calculated via the two methods. Preliminary results indicate the need for larger size orthogonal arrays to conform to results by Monte Carlo simulation (MCS). Two new modifications are introduced and their effects on the optimum tolerances and process combinations are studied. Results indicate the potential of the algorithm for further future developments.

Surrogate modelling is becoming a necessary modelling tool in engineering analysis and synthesis. This is, especially, true when mathematical models are not available. The capability of response surface methodology is explored with single unconstrained/constrained linear and non-linear problems. Problems with special features or with highly non-linear nature are dealt with. A further development with multi-objective problems starts with bi-objective functions and continues to deal with higher number of objective functions. A simple multi-objective problem is taken to initiate the development. Orthogonal arrays and response surface models are combined and used to explore the sets of Pareto Frontiers. One important result is that various array sizes are able to explore the Pareto set at no or little cost change with comparable accuracy. Except for one array, the optimum solution is typical for all arrays employed. One development is using the first derivative of the two objective functions to obtain the starting point for model construction. Design of experiments and analysis of variance (ANOVA) are employed to verify the relative significance of variables. Results of ANOVA guide the modeller in the model building process. We attempt to achieve a 'local accurate model' rather than a 'global accurate model'. Computational experience with different classes of problems is given.

Surrogate modelling is becoming a necessary modelling tool in engineering analysis and synthesis. This is, especially, true when mathematical models are not available. The capability of response surface methodology is explored with single unconstrained/constrained linear and non-linear problems. Problems with special features or with highly non-linear nature are dealt with. A further development with multi-objective problems starts with bi-objective functions and continues to deal with higher number of objective functions. A simple multi-objective problem is taken to initiate the development. Orthogonal arrays and response surface models are combined and used to explore the sets of Pareto Frontiers. One important result is that various array sizes are able to explore the Pareto set at no or little cost change with comparable accuracy. Except for one array, the optimum solution is typical for all arrays employed. One development is using the first derivative of the two objective functions to obtain the starting point for model construction. Design of experiments and analysis of variance (ANOVA) are employed to verify the relative significance of variables. Results of ANOVA guide the modeller in the model building process. We attempt to achieve a 'local accurate model' rather than a 'global accurate model'. Computational experience with different classes of problems is given.

We introduce and investigate a new sort of stochastic equations and inclusions given in terms of mean derivatives defined with respect to conditional expecta-tion relative to the “past ” sigma-algebra of a process. Some existence of solution results are proved. A new type of approximations to an upper semi-continuous set-valued mapping with convex compact values, point-wise converging to a mea-surable selector, is constructed and applied to investigation of inclusions with the above-mentioned derivatives.

(Title in English: USING IDEALIZATIONS AND APPROXIMATIONS IN HYPOTHESIS TESTING IN SCIENCE)

According to the oldest account of empirical-hypothesis testing known as the hypothetico-deductive method, we deduce, from the hypothesis we want to test, a proposition whose truth or falsehood can be found out through observation.  If observation reveals that the proposition we deduced is true, we adopt the hypothesis (at least for the time being).  If observation shows the proposition to be false, then we reject the hypothesis.

As Duhem and Quine have pointed out, hypothesis testing is not all that simple.  In order to be able to validly derive an observational proposition from our hypothesis, we almost always have to employ in the derivation a number of auxiliary hypotheses in conjunction with our hypothesis.  A serious problem for the hypothetico-deductive method arises at this point, for very often there are idealizations and approximations among the auxiliary hypotheses used.  We know that idealizing and approximating assumptions are actually false, but without them it is impossible to mathematically analyze the phenomenon in question.

The fact that we must use idealizations and approximations to expedite computation of predictions from our hypotheses paralyzes the hypothetico-deductive method.  In this paper, I argue that the best account of the role of idealizations and approximations in hypothesis testing can be given by the Bayesian model.  The Bayesian model, which rests on the concept of probability, is the most popular alternative to the hypothetico-deductivist account of hypothesis testing among philosophers of science today.  Yet, Bayesians, like hypothetico-deductivists earlier, have typically been oblivious of the importance of idealizations and approximations in science.  The confirmation and disconfirmation criteria that I propose here are a version of the Bayesian criteria expanded in a way to accommodate the impact of idealizations and approximations in testing of hypotheses.

Digital speech signal is one of the most important types of data used due to the large number of computerized applications that are needed for mankind. Some computerized applications that use digital signals need a high level of security to protect the signal, and to turn it into a vague and incomprehensible signal to any third party, and that is why we have to seek a secure method of cryptography to protect the speech signal. In this research paper we will introduce a method based wavelet packet tree decomposition and reconstruction to generate an encrypted speech, the method will reorder the original speech using a secret number of decomposition levels, the obtained wavelet decomposition sizes will be used to form a speech segments, which must be rearranged in a secret order to get the encrypted speech, the introduced method will be implemented and evaluated to prove the security issues.

Exact values of the best one-sided approximation by Haar polynomials have been obtained in the integral and uniform metrics for some classes of the functions of several variables defined using modulus of continuity ω(f ; t) and ω_ρi (f ; δ).

Robust optimization (RO) has been successfully applied in a number of industrial engineering applications to address the presence of uncertainty about parts of the problem data. Unlike the stochastic programming approach, the RO framework does not require the characterization of a probability distribution and usually reduces to problems that are easier to solve, especially in the presence of integer decision variables. In this extended abstract, we study the application of adjustable robust optimization, with the so-called affine decision rules, to a robust location transportation problem with uncertain demand. Unlike previously proposed approximation formulations for this problem, the decision rules we employ allow us to exploit the fact that while strategic decisions such as the location and capacity of the facilities need to be immediately implemented, operational decisions, such as final production and distribution of goods, can be delayed until the actual demand is observed. Our proposed methods are evaluated numerically using a budgeted uncertainty set for the demand. The results leads us to conclude that accounting for operational flexibility is critical with respect to making truly robust strategic decisions.