This work presents the nonlinear properties of six Sn(IV) Phthalocyanines. Three of the phthalocyanines are linked by an alkylthiol substituent and the rest are linked with a phenoxy substituent. For all six compounds non-linear optic analysis was carried out in four solvents: chloroform, toluene, dichloromethane, and tetrahydrofuran, and their differences were recorded. Calculation of the linear, singlet excited, triplet excited and two photon absorption cross-sections were also carried out and the results compared. To form a comparison the first order hyperpolarizabilities, DFT calculations were also performed and the results compared to see if the behaviour between the two properties can be predicted using DFT.

The present study is concerned with the stability of a flow of viscous conducting liquid driven by a pressure gradient between two parallel walls in the presence of a transverse magnetic field, which is investigated using a Chebyshev collocation method. This magnetohydrodynamic counterpart of the classic plane Poiseuille flow is generally known as Hartmann flow. Although the magnetic field has a strong stabilizing effect, the turbulence is known to set in this flow similarly to its hydrodynamic counterpart well below the threshold predicted by the linear stability theory. Such a nonlinear transition to turbulence is thought to be mediated by unstable equilibrium flow states which may exist in addition to the base flow. Firstly, the weakly nonlinear stability analysis carried out in this study shows that Hartmann flow is subcritically unstable to small finite-amplitude disturbances regardless of the magnetic field strength. Secondly, two-dimensional nonlinear travelling wave states are found to exist in Hartmann flow at substantially subcritical Reynolds numbers starting from Ren = 2939 without the magnetic field and from Ren ∼ 6.50 × 103Ha in a sufficiently strong magnetic field defined by the Hartmann number Ha. Although the latter value is by a factor of seven lower than the linear stability threshold Rel ∼ 4.83×104Ha and by almost a factor of two lower than the value predicted by the mean-field (monoharmonic) approximation, it is still more than an order of magnitude higher than the experimentally observed value for the onset of turbulence in this flow. Three-dimensional disturbances are expected to bifurcate from these two-dimensional travelling waves or infinity and to extend to significantly lower Reynolds numbers. The by-product of this study are two developments of numerical techniques for linear and weakly nonlinear stability analysis. Firstly, a simple technique for avoiding spurious eigenvalues is developed for the solution of the Orr-Sommerfeld equation. Secondly, an efficient numerical method for evaluating Landau coefficients which describe small amplitude states in the vicinity of the linear stability threshold is introduced. The method differs from the standard approach by applying the solvability condition to the discretised rather than the continuous problem.

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The bright squeezed vacuum state of light is a macroscopic nonclassical state found at the output of a strongly pumped unseeded travelling-wave optical parametric amplifier. It has been applied to quantum imaging, quantum communication, and phase supersensitivity, to name a few. Bright squeezed states are in general highly multimode, while most applications require a single mode. We separated two nonlinear crystals in the direction of propagation of the pump in order to narrow the angular spectrum down to a nearly-single angular mode. We observed noise reduction in the photon number difference between the two down-converted channels, which constitutes of proof of nonclassicality. By introducing a dispersive medium between the two nonlinear crystals, we were able to tailor the frequency spectrum of bright squeezed vacuum and to dramatically reduce the number of frequency modes down to 1.82 ± 0.02, bringing us closer to truly single-mode bright squeezed vacuum.

A method is presented for determining approximate solutions to a class of grossly nonlinear, non-autonomous second order differential equations characterized by [formula omitted] with the restriction that resonance effects be negligible. Solutions are developed in terms of the Jacobian elliptic functions, and may be related directly to the degree, of non-linearity in the differential equation. An integral error definition, which can be applied to any particular differential equation, is used to portray regions of validity of the approximate solution in terms of equation parameters. In practice the approximate solution is shown to be of greater accuracy than would be expected from the error analysis, and use of the error diagram leads to a pessimistic estimate of solution accuracy. Two autonomous equations are considered to facilitate comparison between the elliptic function approximation and that obtained from the method of Kryloff and Bogoliuboff. The elliptic function solution is shown to be accurate even for heavily damped nonlinear autonomous equations, when the quasi-linear approximation of Kyrloff-Bogoliuboff cannot with validity be applied. Four examples are chosen, from the fields of astrophysics, mechanics, circuit theory and control systems to illustrate, some areas to which the general approximation method relates. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

The subject matter of this thesis consists of a qualitative study of the stability and asymptotic stability of the zero solution of certain types of nonlinear differential equations, for arbitrary initial perturbations, and the construction of a periodic solution for a Hamiltonian system with n( ≥ 2) degrees of freedom. The material is divided into three chapters. The stability of the system (1) ẋ = xh₁(y) + ay, ẏ = f(x) + yh₂(x) with some restrictions on the functions h₁ (y), h₂(x) and f(x), is discussed in the first chapter. It turns out that some of the results proved by I.H. MUFTI ([l], [2], [3]), for the systems (2) ẋ = xh₁(y) + ay, ẏ = xh₂(x) + by and (3) ẋ = xh₁(y) + ay, ẏ = bx + yh₂(x) become particular cases of our results for system (1). Consequently an answer in the affirmative has been given to a problem proposed by I.H. MUFTI [1]. In the same chapter a generalization to the problem of M. A. AIZERMAN [l] for the case n = 2 is given in the form (4) ẋ = f₁(x) + f₂(y), ẏ = ax + f₃(y). This system has been discussed first by a qualitative method and second by constructing a LYAPUNOV function. In chapter II, stability of a quasilinear equation (5) [formula omitted] is discussed, by using LYAPUNOV's second method. It has been proved that if (i) [formula omitted] (ii) [formula omitted] for all values of x and y = ẋ (iii) [formula omitted] for all x,y (iv) [formula omitted] (where G,g and w are defined in Theorem 2.1) (v) [formula omitted] then the zero solution of (5) is asymptotically stable for arbitrary initial perturbations. In the same chapter certain equations of third order have also been discussed for "complete stability". These equations are special cases of (5) and are more general than those considered by SHIMANOV [l] and BARBASHIN [l]. AIZERMAN's [l] problem for the case n = 3 is generalized to two different forms, one of which is (6) [formula omitted] which is more general than the forms considered by V.A. PLISS [4] and N.N. KRASOVSKII [l]. Under a non-singular linear transformation equations(6) assume the form (7) [formula omitted] It has been proved that if in addition to the usual existence and uniqueness requirements, the conditions (i) [formula omitted] (ii) [formula omitted] (iii) [formula omitted] are fulfilled, then the zero solution of (7) is asymptotically stable in the large. In the third chapter a Hamiltonian system with n (≥ 2) degrees of freedom is considered in the normalized form (8)[formula omitted] where fĸ are power series in zk beginning with quadratic terms. A periodic solution for system (8) is constructed in the form (9) [formula omitted] where [formula omitted] is a homogeneous polynomial of degree [formula omitted] in terms of four time dependent variables a, B, y, õ. C. L. SIEGEL [l] constructs a periodic solution in terms of two variables [formula omitted] under the assumption that the corresponding linear system has a pair of purely imaginary eigenvalues. Here it is assumed that the linear system possesses two distinct pairs of purely imaginary eigenvalues and this necessitates the consideration of four time dependent variables in the construction of the periodic solution. / Science, Faculty of / Mathematics, Department of / Graduate

In the study of many engineering systems involving nonlinear elements such as a saturating inductor in an electrical circuit or a hard spring in a mechanical system, we face the problem of solving the equation ẍ + 2εẋ + x + μx³ = 0 which does not have an exact analytical solution,. Because a consistent framework is desirable in the course of the study, we can assume that the initial conditions are x(0) = 1 and ẋ(0) = 0 without loss of generality. This equation is studied in detail by using numerical solutions obtained from a digital computer. When ε and μ are small, classical methods such as the method of variation of parameters and averaging methods based on residuals provide analytical approximations to the equation and enable the engineer to gain useful insight into the system. However, when ε and μ are not small, these classical methods fail to yield acceptable results because they are all based on the assumption that the equation is quasi-linear. Therefore, two new analytical methods, namely: the parabolic phase approximation and the correction term approximation, are developed according to whether ε < 1 or ε ≥1, and are proven to be applicable for values of ε and μ far beyond the limit of classical methods. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

New mono-chelate complexes with ferrocenyl (Fc), octamethylferrocenyl (Me8Fc) and diaminophenyl (Dap) donor groups, connected through a conjugated bridge, to ZnCl2, Zn(OAc)2 or ReCl(CO)3 acceptor groups, are described. A thorough characterisation of the complexes is provided, including single crystal structures for one pro-ligand and three complexes. Visible d(FeII)→π* metal-to-ligand charge-transfer (MLCT) bands accompany π→π* intraligand charge-transfer absorptions in the UV region. TD-DFT calculations confirm the nature of these absorptions and indicate transitions at higher energies also contain some d(FeII)→π* character. Fc and Me8Fc containing chromophores display a fully reversible oxidation process when probed electrochemically. Molecular quadratic nonlinear optical (NLO) responses are determined using hyper-Rayleigh scattering (HRS) and Stark spectroscopy. Larger β values are determined for complexes with Dap donors and ReICl(CO)3 acceptors. A family of novel fac-ReI(Lax)(CO)3(N-N) complexes, where N-N is 4,4′-dicyano-2,2′-bipyridyl (dcnbpy) or 4,4′-bis-(p-cyanophenyl)-2,2′-bipyridine (bbnbpy), with various axial ligands (Lax) are reported. The ReI complexes are useful precursors for metalation with electron-rich RuII ammine donor groups, to prepare novel tri-metallic V-shaped molecules. Single crystal X-ray structures are reported for five ReI complexes, confirming the fac geometry of the carbonyl ligands; the ReI complexes also display stretching frequencies typical of fac-ReI(Lax)(CO)3(N-N) complexes. The UV-visible spectra contain a low intensity band due to a d(ReI)→π*(bpy) transition, along with a more intense π→π* band. 1H NMR studies reveal the formation of trimetallic complexes, upon treatment of ReI complexes with a molar excess of [RuII(NH3)5(H2O)][PF6]2.The synthesis of octupolar heptametallic complexes containing RuII ammine donor groups has been investigated. The ligand 4,4′-bis-[(E)-2-(4-cyanophenyl)ethenyl]-2,2′-bipyridine (bbnpe) was used to prepare tris-chelate complexes of various transition metals, in order to understand its complexation behaviour. The ZnII tris-chelate BPh4– salt was treated with a 10 molar excess of a RuII ammine aquo complex, to produce the heptametallic complex as a mixed anion salt. HRS and Stark spectroscopy have been used to determine the quadratic NLO response for the heptametallic mixed anion complex salts, the latter gives large β0 values, approximately 10–27 esu. Density functional theory (DFT) calculations have been carried out on twelve cationic, 2D NLO chromophores with pyrazinylbis(pyridinium) electron acceptors with either 4-(methoxy/dimethylamino)-phenyl or pyridyl coordinated {RuII(NH3)5}2+/trans-{RuII(NH3)4(py)}2+ electron donor groups and the results compared with data previously obtained experimentally. The B3LYP/6-311G(d) level of theory was used to model the absorption spectra and to calculate static hyperpolarisability (β0) values, whilst the B3LYP/LANL2DZ/6-311G(d) level was used for the complexes. The extent of prediction of trends in ICT bands and β0 is partial, with the main discrepancies relating to the progression from one to two electron donor groups. The quantitative accuracy of predictions diminishes as the systems depart from a relatively simple one-dimensional (1D) dipolar motif.

Many problems in economics, statistics and numerical analysis can be formulated as the optimization of a convex quadratic function over a polyhedral set. A polynomial algorithm for solving convex quadratic programming problems was first developed by Kozlov at al. (1979). Tardos (1986) was the first to present a polynomial algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients. In the first part of the thesis we extended Tardos' results to strictly convex quadratic programming of the form max {cTx-½xTDx : Ax ≤ b, x ≥0} with D being symmetric positive definite matrix. In our algorithm the number of arithmetic steps is independent of c and b but depends on the size of the entries of the matrices A and D. Another part of the thesis is concerned with proximity and sensitivity of integer and mixed-integer quadratic programs. We have shown that for any optimal solution z̅ for a given separable quadratic integer programming problem there exist an optimal solution x̅ for its continuous relaxation such that / z̅ - x̅ / ∞≤n∆(A) where n is the number of variables and ∆(A) is the largest absolute sub-determinant of the integer constraint matrix A . We have further shown that for any feasible solution z, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution z̅ having greater objective function value and with / z - z̅ / ∞≤n∆(A). Under some additional assumptions the distance between a pair of optimal solutions to the integer quadratic programming problem with right hand side vectors b and b', respectively, depends linearly on / b — b' / ₁. The extension to the mixed-integer nonseparable quadratic case is also given. Some sensitivity analysis results for nonlinear integer programming problems are given. We assume that the nonlinear 0 — 1 problem was solved by implicit enumeration and that some small changes have been made in the right hand side or objective function coefficients. We then established what additional information to keep in the implicit enumeration tree, when solving the original problem, in order to provide us with bounds on the optimal value of a perturbed problem. Also, suppose that after solving the original problem to optimality the problem was enlarged by introducing a new 0 — 1 variable, say xn+1. We determined a lower bound on the added objective function coefficients for which the new integer variable xn+1 remains at zero level in the optimal solution for the modified integer nonlinear program. We discuss the extensions to the mixed-integer case as well as to the case when integer variables are not restricted to be 0 or 1. The computational results for an example with quadratic objective function, linear constraints and 0—1 variables are provided. Finally, we have shown how to replace the objective function of a quadratic program with 0—1 variables ( by an integer objective function whose size is polynomially bounded by the number of variables) without changing the set of optimal solutions. This was done by making use of the algorithm given by Frank and Tardos (1985) which in turn uses the simultaneous approximation algorithm of Lenstra, Lenstra and Lovász (1982). / Business, Sauder School of / Graduate

A necessary and sufficient condition is established for the existence of an invertible mapping of a system of nonlinear differential equations to a system of linear differential equations based on a group analysis of differential equations. It is shown how to construct the mapping, when it exists, from the invariance group of the nonlinear system. It is demonstrated that the hodograph transformation, the Legendre transformation and Lie's transformation of the Monge-Ampere equation are obtained from this theorem. The equation (ux)Puxx-uyy=0 is studied and it is determined for what values of p this equation is transformable to a linear equation by an invertible mapping. Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance groups of equations associated with the given nonlinear equations. A number of such; examples are given, including Burgers' equation uxx +uuz-ut=0 a nonlinear diffusion equation (u⁻²ux ) x -ut =0, equations of wave propagation {Vy-wx=0, Vy-avw-bv-cw=0}, equations of a fluid flow {wy+vx=0, wx -v⁻¹wP=0} and the Liouville equation uxy=eu. As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the invariance groups of the respective equations. All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate