Lee Larson

ℐ-density continuous functions

The I-density topology is a generalization of the ordinary density topology to the setting of category instead of measure. This work involves functions which are continuous when combinations of the I-density, deep-I-density, density and ordinary topology are used on the domain and range. In the process of examining these functions, the I-density and deep-I-density topologies are deeply explored and the properties of these function classes as semigroups are considered.

The space of density continuous functions

We denote by Rd the set of real numbers, R, endowed with the density topology. A function f : Rd ~ Rd is said to be density continuous, if it is continuous with respect to the topology on Rd in both the domain and range. The set of density continuous functions has been studied in several limited ways. Bruckner [1] and Niewiarowski [3] have studied density continuous functions which are homeomorphisms under the standard topology on R. Ostaszewski has investigated the local behavior of density continuous functions [4] and has investigated their behavior as a semigroup [5]. In this paper, we consider the composition of the set of density continuous functions. The structure of this set seems to be quite complicated. Ostaszewski [5] has noted that it is not dosed under uniform convergence. In Example 2 we show that it is not a vector space. Corollary 3 shows that each real-analytic function is density continuous, but Example 1 is a C ~ function which is not density continuous. It is not difficult to construct a density continuous function which is not continuous. On the other hand, every density continuous function must be approximately continuous. In what follows, the right (left) unilateral derivatives of a function f are represented as f+ ( f ) . The Lebesgue measure of a set A is denoted by IAI and the Lebesuge density (right, left Lebesgue density) of A at a point x is written as d(A ,x ) (d+(A ,x ) ,d (A ,x ) ) . The set of functions which are infinitely differentiable on R is written as C ~176 Finally, if A and B are two sets such that sup A __< inf B, then we write A << B. Before stating the main result, we first present the following lemma.

Texture classification through level lines

The paper presents a method for texture classification through level lines. Level lines provide a non-redundant and full representation of an image, and they can be regarded as a signature for the image. By using earth movers distance (EMD), we can classify texture images based on EMD between their level lines. Our method is a geometric approach for texture classification, which has the advantage of being contrast-, rotation- and scale-invariant.

Numerical solutions of the orbital equations for diatomic molecules

A basis of Hermite splines is used in conjunction with the collocation method to solve the orbital equations for diatomic molecules. Accurate solutions of the Hartree-Fock equations are obtained using iterative methods over most regions of space, while solving the equations by Gaussian elimination near the nuclear centres. In order to improve the speed and accuracy of our iterative scheme, a new self-adjoint form of the Hartree-Fock equation is derived. Using this new equation, our iterative subroutines solve the Hartree-Fock equations to one part in 106. The Gaussian elimination routines are accurate to better than one part in 108.

Spline collocation calculation for

A basis of Hermite splines is used in conjunction with the collocation method to obtain accurate solutions of the Schrodinger equation for . The spectral transform variation of the Lanczos method is found to be the most suitable method for solving the matrix eigenvalue problem, while the preconditioned conjugate gradient method is very effective for solving the systems of linear equations necessary to evaluate the matrix - vector products that occur for each Lanczos iteration. The lowest eigenvalue is obtained to more than six significant figures without taking advantage of the separability of the orbital equations.

Density Continuity versus Continuity.

Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if it is a nowhere dense Fσ set. The corresponding characterization for the approximately continuous functions is a AErst category Fσ set. An alternative proof of that result is given. Density continuous functions belong to the class Baire*1, unlike the approximately continuous functions.

Baire classification of -density and -approximately continuous functions

Let 9~e be the ordinary topology, ̂ be the ./-density topology and «^ be the deep «/-density topology on the real numbers, (R. Any continuous function/: (IR, ̂ ) -+ (IR, «^) is a Darboux function of the first Baire class. Any unilaterally continuous function/: (IR, «^) -> (IR, #i) is a Darboux function of the Baire*! class. 1991 Mathematics Subject Classification: 26A03, 26A21

Refinements of the density and ℐ-density topologies

Given an arbitrary ideal f on the real numbers, two topologies are defined that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of f . This shows why the restriction to Baire sets is necessary in the usual definition of the J>-density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal. In studying real functions it is often helpful to endow the real numbers 11R with a topology finer than the natural one, denoted by T&, arising from its order. The most common examples of such topologies are the density topology Tr and its category analog, the >J-density topology . To recall the definitions of these topologies we need the following notions. A point x E JR is said to be a density point of A c JR if (1) lim m(A n (x l/n, x + 1/n))1 n--+oo 2/n

Level sets of density continuous functions

A function f: R--R is density continuous if it is continuous when both its range and domain are endowed with the density topology. The level sets of density continuous functions are characterised as those sets which are density closed and ambiguous.