We determine all pairs of real numbers (α, β) such that the dilated floor functions bαxc and bβxc commute under composition, i.e., such that bαbβxcc = bβbαxcc holds for all real x.

and Applied Analysis 3 Definition 2.2. The Riemann-Liouville fractional derivative of order α > 0 of a function f : 0,∞ → R is given by D 0 f t 1 Γ n − α ( d dt )n ∫ t 0 f s t − s α−n 1 ds, 2.2 where n α 1 and α denotes the integer part of α. The following two lemmas can be found in 17, 22 . Lemma 2.3. Let α > 0 and u ∈ C 0, 1 ∩ L1 0, 1 . Then fractional differential equation D 0 u t 0 2.3

For positive integers m and r, one can easily show there exist integers N such that for every map ∆ : {1, 2, · · · , N} → {1, 2, · · · , r} there exist 2m integers x1 < · · · < xm < y1 < · · · < ym which satisfy: 1. ∆(x1) = · · · = ∆(xm), 2. ∆(y1) = · · · = ∆(ym), 3. 2(xm − x1) ≤ ym − x1. In this paper we investigate the minimal such integer, which we call g(m,r). We prove that g(m, 2) = 5(m − ...

The purpose of this report is to provide a perspective on future accelerator projects, and to identify the R&D activities necessary to prepare for these projects. The report summarizes the conclusions of accelerator studies made during the 2001 Snowmass Summer Study on the Future of Particle Physics. In doing so, it serves as a summary of the opinions on accelerator R&D expressed by the scienti...

Earlier this year, after three years of planning and negotiations, a collection of approximately 100,000 Inuit drawings and prints was transferred on loan from the West Baffin Eskimo Co-operative Ud. in the eastern Canadian Arctic to the McMichael Canadian Art Collection in southern Canada. The arrival of this collection marked the beginning of a multi-year co-operative project between the two ...

Growing evidence demonstrates that metabolism and chromatin dynamics are not separate processes but that they functionally intersect in many ways. For example, the lysine biosynthetic enzyme homocitrate synthase was recently shown to have unexpected functions in DNA damage repair, raising the question of whether other amino acid metabolic enzymes participate in chromatin regulation. Using an in...

many findings point to the presence of a close relationship between the immune, endocrine and neurologic systems. for example the suppressive effects of β-human chorionic gonadotropin (β-hcg) on il-l production, the potentiative effect of enkephalins on lymphocyte functions and the suppressive effect of leuteinizing hormone (lh) on natural killer (nk) cell activities have been clearly establish...

conclusions a review of the literature on the use of hypnotic drugs in patients with liver dysfunction showed that some hypnotic drugs used during anesthesia could be safely used in patients with impaired liver function. in these patients, certain drugs should be used with caution. results anesthesia should be performed with caution in patients with hepatic dysfunction and drugs with long half-...