Thesis--Göttingen. / Vita.

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1990.

The Lielmezs-Merriman equation of state has been modified in such a way that it can be applied over the entire PVT surface except along the critical isotherm. The dimensionless T* coordinate has been defined according to the two regions on the PVT surface as: [Formula Omitted] The two substance-dependent constants p and q are generated from the vapor pressure data. The applicability of the proposed modification has been tested by comparing its predictions of various pure compound physical and thermodynamic properties with known experimental data and with predictions from the Soave-Redlich-Kwong and Peng-Robinson equations of state. The proposed equation is the most accurate equation of state for calculating vapor pressure, and saturated vapor and liquid volumes. The Peng-Robinson equation is the best for enthalpy and entropy of vaporization estimations. The Soave-Redlich-Kwong equation is the least accurate equation for pressure and volume predictions in the single phase regions. For temperature prediction, all three equations of state give similar results in the subcritical and supercritical regions. None of the three equations is capable of representing all departure functions accurately. The Peng-Robinson equation and the proposed equation are very similar in accuracy except in the region where the temperature is near the critical. That is, between 0.95 ≤ Tr ≤ 1.05, the proposed equation gives rather poor results. For isobaric heat capacity calculation, both Soave-Redlich-Kwong and Peng-Robinson equations are adequate. The Soave-Redlich-Kwong equation gives the lowest overall average RMS % error for Joule-Thomson coefficient estimation. The Soave-Redlich-Kwong equation also provides the most reliable prediction for the Joule-Thomson inversion curve right up to the maximum inversion pressure. None of the cubic equations of state studied in this work is recommended for second virial coefficient calculation below Tr = 0.8. An α-function specifically designed for the calculation of second virial coefficient has been included in this work. The estimation from the proposed function gives equal, if not better, accuracy than the Tsonopoulos correlation. / Applied Science, Faculty of / Chemical and Biological Engineering, Department of / Graduate

This dissertation addresses optimization of cubic polynomial problems. Heuristics for finding good quality feasible solutions and for improving on existing feasible solutions for a complex industrial problem, involving cubic and pooling constraints among other complicating constraints, have been developed. The heuristics for finding feasible solutions are developed based on linear approximations to the original problem that enforce a subset of the original problem constraints while it tries to provide good approximations for the remaining constraints, obtaining in this way nearly feasible solutions. The performance of these heuristics has been tested by using industrial case studies that are of appropriate size, scale and structure. Furthermore, the quality of the solutions can be quantified by comparing the obtained feasible solutions against upper bounds on the value of the problem. In order to obtain these upper bounds we have extended efficient existing techniques for bilinear problems for this class of cubic polynomial problems. Despite the efficiency of the upper bound techniques good upper bounds for the industrial case problem could not be computed efficiently within a reasonable time limit (one hour). We have applied the same techniques to subproblems with the same structure but about one fifth of the size and in this case, on average, the gap between the obtained solutions and the computed upper bounds is about 3%. In the remaining part of the thesis we look at global optimization of cubic polynomial problems with non-negative bounded variables via branch and bound. A theoretical study on the properties of convex underestimators for non-linear terms which are quadratic in one of the variables and linear on the other variable is presented. A new underestimator is introduced for this class of terms. The final part of the thesis describes the numerical testing of the previously mentioned underestimators together with approximations obtained by considering lifted approximations of the convex hull of the (x x y) terms. Two sets of instances are generated for this test and the descriptions of the procedures to generate the instances are detailed here. By analyzing the numerical results we can conclude that our proposed underestimator has the best behavior in the family of instances where the only non-linear terms present are of the form (x x y). Problems originating from least squares are much harder to solve than the other class of problems. In this class of problems the efficiency of linear programming solvers plays a big role and on average the methods that use these solvers perform better than the others.

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’s analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.