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 define the direction of knowledge, and what it means to extend that direction. A special case is function inversion, and we give three algorithms for function inversion. Their performance on non-trivial problems and their shortcomings are demonstrated. All algorithms are implemented in Prolog

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...

We prove that product and sum types are de nable in polymorphic PCF with call-by-name evaluation strategy. This establishes polymorphic PCF and its extensions can be conservatively extended by product and sum types and recursive types.

We investigate a fast optimization method for determining the minimizer of the negative Poisson likelihood function for the global analysis of fluorescence lifetime microscopy. Using the alternating optimization strategy, we iteratively solve a non-convex optimization problem to estimate the lifetime parameters and a convex optimization problem to estimate the concentration parameters. We effec...

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

In this equation, n represents the number of Figures of Merit (FoMs) that are to be combined, xi represents the output of the scoring function for the ith FoM and wi represents the weight of importance assigned to the ith FoM. The Product Tradeoff Function is commonly used for example in computing cost to benefit ratios and in doing risk analyses. Unfortunately, the published equation on page 1...