Nested recursions with ceiling function solutions

Consider a nested, non-homogeneous recursion R(n) defined by R(n) = \sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the parameters are integers satisfying k > 0, p_i > 0 and a_ij > 0. We develop an algorithm to answer the following question: for an arbitrary rational number r/q, is there any set of values for k, p_i, s_i, a_ij and nu such that the ceiling function ceiling{rn/q} is the unique solution generated by R(n) with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to R(n). The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form ceiling{n/q}$ is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if ceiling{rn/q} is the solution generated by any recursion R(n) of the form above, then r=1. We also use our ceiling function methodology to derive the first known connection between the recursion R(n) and a natural generalization of Conway's recursion.Comment: Published in Journal of Difference Equations and Applications, 2010. 11 pages, 1 tabl

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

How hand movements and speech tip the balance in cognitive development:A story about children, complexity, coordination, and affordances

When someone asks us to explain something, such as how a lever or balance scale works, we spontaneously move our hands and gesture. This is also true for children. Furthermore, children use their hands to discover things and to find out how something works. Previous research has shown that children’s hand movements hereby are ahead of speech, and play a leading role in cognitive development. Explanations for this assumed that cognitive understanding takes place in one’s head, and that hand movements and speech (only) reflect this. However, cognitive understanding arises and consists of the constant interplay between (hand) movements and speech, and someone’s physical and social environment. The physical environment includes task properties, for example, and the social environment includes other people. Therefore, I focused on this constant interplay between hand movements, speech, and the environment, to better understand hand movements’ role in cognitive development. Using science and technology tasks, we found that children’s speech affects hand movements more than the other way around. During difficult tasks the coupling between hand movements and speech becomes even stronger than in easy tasks. Interim changes in task properties differently affect hand movements and speech. Collaborating children coordinate their hand movements and speech, and even their head movements together. The coupling between hand movements and speech is related to age and (school) performance. It is important that teachers attend to children’s hand movements and speech, and arrange their lessons and classrooms such that there is room for both