Thesis (Ph.D.)--McGill University. / Written for the Dept. of Mathematics. Typewritten MS. Bibliography: leaves [216]-217.

We consider the differential equation d²u/dz² - zⁿu = 0 (z, u complex variables; n a positive integer), which is the simplest second order ordinary differential equation with a turning point of order n. The solutions which we study, herein called Aռ functions, are generalizations of Airy functions. Most of their properties are then deduced from those of related Bessel functions of order [formula omitted], but in the discussion of the zeros in section 3, results are deduced directly from the differential equation. It is easy to see that the Aռ functions are special cases of functions studied by Turrittin [9]. The relation of the former to Bessel functions, however,, enables us to use methods not available in [9] to obtain uniform asymptotic representations for large z. We obtain new results on the distribution of the zeros which extend a property [6] of Airy functions, that is, of A₁functions,, to all positive integers n. A similar remark applies to bounds [8] for Airy functions and their reciprocals. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis deals with the calculation of Hartree-Fock wave functions satisfying an off-diagonal hypervirial relation as a constraint. The constraint in this case implies that the dipole length form and the dipole velocity form of the transition probability give identical values. Mathematically, this is equivalent to forcing the approximate eigenfunctions of the Hamiltonian of the system to satisfy a relation which is true for exact eigenfunctions. The method of constrained variation is used to solve this problem. The constrained Hartree-Fock system of equations is solved numerically. The Z-expansions of radial wave functions, the diagonal and the off-diagonal energy parameters and the parameter of constraint are carried out. The effect of the constraint on the total energy E of the system, defined as the change in the Hartree-Fock total energy due to the constraint, is estimated. The method of constrained variation is then applied to a few two, three and four electron systems to calculate the constrained total energy E of the system and also the oscillator strengths of a few of the transitions of the system. The results indicate that the oscillator strengths can be calculated more accurately, at practically no cost of the total energy E, with the aid of the constrained Hartree-Fock functions than with the standard Hartree-Fock functions in all those cases where the correlation effects are not too strong to invalidate the single configuration approximation. / Science, Faculty of / Mathematics, Department of / Graduate

Three different constraints are considered in this thesis, namely: the integral electron cusp condition as a constraint; the off-diagonal hyper-virial theorems as constraints; and perturbation-induced constraints. Eleven approximate configuration-interaction wavefunctions for the ground state of helium are used to test the application of the integral electron cusp condition as a constraint. The results indicate that, if the approximate wavefunction is flexible enough, the calculated electron density at the nucleus is improved when the cusp constraint is imposed. However, the expectation values of rˉ ¹and r ˉ² do not change significantly. Further investigation is made on the use of off-diagonal hypervirial theorems as constraints. The transition from the 1¹ S state to the 2¹ P state of helium is chosen as an example. Firstly, it is found that an energy-independent off-diagonal constraint is not useful in improving calculation of transition probabilities. Secondly, when the approximate wavefunction is flexible enough, iteration on the transition energy converges very rapidly. Finally, the study is extended to the isoelectronic species Li⁺ and Be⁺⁺. A five-term approximate configuration-interaction wavefunction for the ground state of helium is used to test the validity of the perturbation-induced constraints scheme. Different constraint operators are constructed for different properties. The properties studied are expectation values of rˉ² , rˉ¹ , r and r⁺² . Two methods of the perturbation-induced constraints are tested: in one, the first-order wavefunction is fixed and constant for all constrained properties; in the other, the first-order wavefunction varies with the constrained properties. It is disappointing to find that for this particular wavefunction chosen, both methods fail to improve the properties studied when one imposes the constraints. / Science, Faculty of / Chemistry, Department of / Graduate

The purpose of this thesis is to attempt to evaluate the permanent function of a n×n complex matrix with entries aij = θij being a primitive n root of unity. If this matrix is denoted by An then its permanent function is given by per An = [formula omitted] In this thesis the following results are proved. Per An is always an integer; with per An ≡ 0 mod n. If n is even per An = 0. For n odd however, the problem is in general not resolved. It is shown that if n=p² with p a prime, that per An = 0 mod p⁴ and that for any prime n, per An can be narrowed down to be one of a restricted class of numbers. / Science, Faculty of / Mathematics, Department of / Graduate

The notion of recursiveness is treated in a model-theoretical way by using a particular instance of Kreisel's definition of 'invariant definability'. Naming the chosen notion 'finite describability', a number of basic definitions and properties are defined and proved. As one would expect, these properties coincide with the ones for recursion theory. The equivalences of finite describability and recursiveness bring model theory and recursion theory slightly together. / Science, Faculty of / Mathematics, Department of / Graduate

The transient stability of power systems is investigated using Liapunov's direct method. Willems' method is applied to three-and four-machine power systems with the effect of damping included. The distribution of damping among the machines of a multi-machine system is studied, and optimum ratios are derived. An extension of Willems' method is used to include governor action in the system representation. Finally, the effect of flux decay on stability regions is studied using Chen's method. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

We are going to study the remote points in βR and the P-points in βR - R. A remote point in βR is a point which is not in the βR chosure of any discrete subset of R. A point p ε βR - R is a P-point of βR - R if every Gδ-set containing p is a neighbourhood of p. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis, we discuss the problem of estimating a characteristic function and its derivatives. We obtain estimates which are consistent and asympototically normal, and uniformly consistent with probability one. The methods employed here are similar to the methods used in estimating a probability density function and its derivatives (see [7], [9] for references). / Science, Faculty of / Mathematics, Department of / Graduate

In this paper, we study continuous functions with no finite or infinite one-sided derivative anywhere. In 1925, A. S. Beskovitch published an example of such a function. Since then we call them Beskovitch functions. This construction is presented in chapter 2, The example was simple enough to clear the doubts about the existence of Besicovitch functions. In 1932, S. Saks showed that the set of Besicovitch functions is only a meager set in C[0,1]. Thus the Baire category method for showing the existence of Besicovitch functions cannot be directly applied. A. P. Morse in 1938 constructed Besicovitch functions. In 1984, Maly revived the Baire category method by finding a non-empty compact subspace of (C[0,1], || • ||) with respect to which the set of Morse-Besicovitch functions is comeager.