Stolarsky's conjecture and the sum of digits of polynomial values

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$\liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0.$$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + ... + a_0 \in \Z[x]$ with $h\geq 2$ and $a_h>0$ and any base $q$, $$\liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.$$ For any $\epsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \epsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \epsilon$.Comment: 13 page

The sum of digits of nn and n2n^2

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 2005, Melfi examined the structure of $n$ such that $s_2(n) = s_2(n^2)$. We extend this study to the more general case of generic $q$ and polynomials $p(n)$, and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case $p(n) = n^2$, looking at the subsets of $n$ where $s_q(n) = s_q(n^2) = k$ for fixed $k$.Comment: 16 page

Optical waveguiding in proton-implanted GaAs

We have produced optical waveguides in n-type GaAs by implantation with 300-keV protons. The guiding is shown to be due to the elimination of charge carriers from the implanted region. Annealing of the waveguide leads to very large reductions in the 1.15-µ guided-wave absorption

RP Process Selection for Rapid Tooling in Sand Casting

The significant cycle-time improvements and geometrical capabilities of solid freeform fabrication systems have led to applications in sand casting industry for design verification and tooling. The time and cost effective deployment of rapid tooling processes using rapid prototyping technology has thus becoming an emerging area to be studied. To make full use of the advantages of rapid prototyping processes, the factors influencing the tooling approach must be identified and understood. This understanding is then used to develop a decision-making structure for RP process selection for rapid tooling in sand casting. In this manuscript we review our work in evaluating and building a framework for tooling process selection for sand castingMechanical Engineerin

A study of family attitudes: Their relation to the adjustment of twenty schizophrenic veterans on trial visit from Veterans Administration Hospitals.

Thesis (M.S.)--Boston Universit

The Properties of the Heterogeneous Shakhbazyan Groups of Galaxies in the SDSS

We present a systematic study of the sub-sample of Shakhbazyan groups (SHKs) covered by the Sloan Digital Sky Survey Data Release--5 (SDSS-5). SHKs probe an environment with characteristics which are intermediate between those of loose and very compact groups. Surprisingly, we found that several groups identifying algorithms (e.g. Berlind et al. 2006, Tago et al. 2008) miss this type of structures. Using the SDSS-5 spectroscopic data and the photometric redshifts derived in D'Abrusco et al. 2007, we identified possible group members in photometric redshift space and derived, for each group, several individual properties. We also combined pointed and stacked Rosat All Sky Survey data to investigate the X-ray luminosities of these systems. Our study confirms that the majority of groups are physical entities with richness in the range 3--13 galaxies, and properties ranging between those of loose and compact groups. We confirm that SHK groups are richer in early-type galaxies than the surrounding environment and the field, as expected from the morphology-density relation and from the selection of groups of red galaxies. Furthermore, our work supports the existence of two sub-classes of structures, the first one being formed by compact and isolated groups and the second formed by extended structures. We suggest that while the first class of objects dwells in less dense regions like the outer parts of clusters or the field, possibly sharing the properties of Hickson Compact Groups, the more extended structures represent a mixture of [core+halo] configurations and cores of rich clusters. X-ray luminosities for SHKs are generally consistent with these results and with the expectations for the L_X-sigma_v relation, but also suggest the velocity dispersions reported in literature are underestimated for some of the richest systems.Comment: 20 pages, 14 figures, 4 tables. Accepted for publication by MNRA