Let K(α), 0≤α< 1, denote the class of functions g(z)= z+∑∞n=2anzn which are regular and univalently convex of order α in the unit disc U . Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U , f(0) = 0, and f(z)+zf ′(z) < g(z)+zg′(z) in U , then (i) f(z) < g(z) at least in |z| < r0, r0 = √ 5/3 = 0.745 . . . if f ∈ K; and (ii) f...

Schwarz and Weck-Schwarz have shown that a T2-ordered space (X, τ, ) whose underlying topological space (X, τ) is completely regular need not be a completely regularly ordered space (that is, T3.5 + T2-ordered T3.5-ordered). We show that a completely regular T2-ordered space need not be completely regularly ordered even under more stringent assumptions such as convexity of the topology. One exa...

A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: • Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. • Every (not necessarily unital) countable von Neumann regular ring R has a map ε ...

An n-dimensional cross comprises 2n + 1 unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of R n by crosses for all n. AlBdaiwi and the first author proved that if 2n + 1 is not a prime then there are 2 ℵ 0 non-congruent regular (= face-to-face) tilings of R n by crosses, while there is a unique tiling of R n by crosses for n = 2, 3. They conj...

From the regular hexagon of unit size remove circular holes of small radius ε > 0 centered at the vertices, obtaining a region Hε of area 3 √ 3 2 − πε2. For each pair (x, ω) ∈ Hε × [0, 2π] consider a point particle moving at unit speed on a linear trajectory with specular reflections when meeting the boundary. Denote by τhex ε (x, ω) the time it takes the particle to reach one of the holes. Thi...

State-of-the art techniques for automated test generation focus on generating executions that cover program behavior. As they do not generate oracles, it is up to the developer to figure out what a test does and how to check the correctness of the observed behavior. In this paper, we present an approach to generate parameterized unit tests—unit tests containing symbolic preand postconditions ch...