The main purpose of this paper is using the elementary method and analytic method to study the asymptotic properties of the integer part of the k-th root positive integer, and give two interesting asymptotic formulae.

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 ...

cranial injury, as it is known today, is not a new concern of modern medicine. on stepping on the earth, the man was in reality encountered with various types of injuries, particularly those of a cranial nature. leading a life, whether wild or civilized, has always been associated with injuries for human race from the very beginning of birth. therefore, managing cases of this type has gradually...