Gap Problems for Integer Part and Fractional Part Sequences

Part VI Example Problems

An object is cloaked if its presence cannot be detected by an observer using electromagnetic or other forms of imaging. In fact, the observer should not notice that cloaking is even occurring. Cloaking has a long history in science fiction, but recent developments have put the idea on a firmer mathematical and physical basis. Suppose that an observer seeks to image some bounded region Ω in spac...

Part II: Problems

Part-Products of 1-Free Integer Compositions

If ~λ is a composition of the positive integer n, define B(~λ) to be the product of the parts of ~λ. We present a modified version of Hitczenko’s stopped sequence construction that leads to a proof of the asymptotic lognormality of B for random 1-free compositions (compositions containing no parts of size 1).

Part-products of S-restricted Integer Compositions

If S is a cofinite set of positive integers, an “S-restricted composition of n” is a sequence of elements of S, denoted ~λ = (λ1, λ2, . . . ), whose sum is n. For uniform random S-restricted compositions, the random variable B(~λ) = ∏ i λi is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinat...

Fractal Sequences , Part 1 : Overview

Infinity is where things happen that don't. — S. Knight Ever since Benoit Mandelbrot published his book on fractals [1], we've become accustomed to seeing fantastic and beautiful fractal images such as the ones at the bottom of the pages of this article. A fractal is a mathematical object that exhibits self similarity — it looks the same at any scale. If you zoom in on an image of a fractal, yo...