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

The Ubiquity of Sidon Sets That Are Not I0I_0

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$

Misappropriation Theory: How the World’s Two Largest Economies Regulate Insider Trading

Prior to the government adopting policies of economic reform in the late 1970s, the People’s Republic of China (“the PRC” or “China”) did not have a formal securities market or an accompanying regulatory scheme. For the most part, it was not operationally feasible for a market to develop and flourish in China because the PRC had a centrally planned economy with state-owned enterprises as the primary form of business ownership. However, economic reform brokered conditions where stock trades casually began in markets located in Shanghai, Shenzhen, Chengdu and several other cities in the early 1980s. This informal trading persisted until the formal establishment of modern stock exchanges in Shanghai and Shenzhen in December 1990. Historically, the securities market in China has been prone to fraud and corruption. In fact, there have been occasions where Chinese officials have been openly involved in the fraud and corruption that has taken place. To address these issues in its securities market, Chinese regulators sought assistance from abroad. Specifically, in April of 1994, China’s securities regulatory authority, the China Securities Regulatory Commission (“CSRC”), signed a Memorandum of Understanding with the United States Securities and Exchange Commission (“SEC”) in a move designed to obtain access to technical and enforcement assistance from its American counterpart. Pursuant to the guidance it received under the Memorandum of Understanding (“1994 MOU”), China implemented many of the policies used by the SEC in its securities regulation efforts. One example of such implementation was China’s revamping of its regulatory configuration from two independent levels (the CSRC and the Securities Commission of the State Council or “SCSC”) to a single-level structure in which the CSRC is subordinate to the SCSC. Another example is the Securities Law of the People’s Republic of China (“1998 Securities Law”), which implements many securities laws that are also used in America. China is the world’s second-largest economy in terms of nominal gross domestic product (GDP) and has the fourth-largest stock exchange in the world in terms of market capitalization. The United States is the largest economy in the world in terms of nominal GDP with the two largest stock exchanges in the world in terms of market capitalization. Because of these facts and China’s obtainment of significant technical and enforcement assistance from the United States in the area of securities law, this paper will address the issue of misappropriation of inside information from a comparative perspective by discussing the issue under both Chinese and American securities laws. After a general introduction to insider trading theories in Part I of this paper, Part II will discuss the American approach to regulating misappropriation and Part III will discuss China’s basic approach to the issue. Part IV will consider and answer the question of whether a breach of fiduciary duty requirement is the proper approach for regulating misappropriation or whether applying misappropriation liability without a fiduciary duty requirement is the better approach. Part V will provide suggested solutions for addressing the problems with both the American and Chinese approaches to regulating misappropriation

The identification of the domestic waste collection system associated with the least operative musculoskeletal disorders using human resource absence data

With increasing pressures around public sector costs, UK Local Authorities (LAs) and waste collection companies, are under pressure to reduce absence rates due to ill health. The identification of the ‘safest’ method of waste collection in the UK has been largely unresolved with many different types of waste and recycling receptacles used and deemed acceptable. The purpose of the study was to investigate the relationships between domestic waste collection methods and absence due to Musculoskeletal Disorders (MSDs) through the comparison of absence rates for different activity. Studies based upon ergonomic theory had suggested the use of wheeled bins is better than the use of boxes, but this has not been tested empirically. Absence data was obtained from 15 LAs who allocated a more detailed activity role to their records, allowing for activity absence rates to be calculated. The outputs were collated and analysed using SPSS to identify statistically significant relationships between types of waste collection services. The results confirm that wheeled bins are associated with less proxy measures of MSD than boxes, baskets and sacks with even lower absence rates associated with 1100 litre capacity bins, when handled by two workers. Findings also indicates that there is a level where MSD absence interventions are unlikely to be sustainable. In conclusion these findings should help LAs better understand some critical factors regarding waste collection strategies and MSD absence and inform HSE enforcement strategies. Employers should interrogate their own ill health data and seek to move to systems that create less MSDs

A comparison of manual handling risks in different domestic waste collection systems using three separate evaluation methods

One of the greatest challenges for Local Authorities (LAs) is to operate sustainable systems of work. The most significant causes of physical absence is attributed to that of musculoskeletal disorders (MSDs). The study’s aim was to compare the reliability of the manual handling element of the Health and Safety Executives (HSEs) risk comparator tool for different domestic waste collection methods with self-reported pain via body-mapping and MSD ill health absence rates. Participatory body-mapping exercises were carried out in five authorities with one authority resurveyed, six months after the move from 35-50 litre containers to a wheeled bin recycling service. The lowest levels of self-reported pain were for services designed with 240l wheeled bins excluding glass; the highest levels were for services that included 100l of garden waste sacks and recycling boxes. Industry data supports previous laboratory studies showing wheeled bins to be associated with less MSD outcomes than boxes, baskets and sacks. Triangulation of data established a statistically significant correlation of 0.85 (Pearson) between average pain count (APC) and the mean MSD absence rates, with a strong correlation of 0.77 (Spearman) between APC and risk rating. The correlation is moderate, 0.49, (Spearman) between MSD absence and risk rating, reflecting possible intervening variables and a low participation rate by LAs. The contribution of this study is to improve the design of sustainable waste collection strategies to minimise MSD associated absence. In the absence of reliable absence data, body mapping should be used as a proxy method of assessing MSD risk

Using body mapping as part of the risk assessment process - a case study

This paper reports on a study undertaken to identify levels of MSD in relation to methods of waste collection. The need to quantify and eliminate ill health arising out of work is vital to reduce workplace absence leading to debate on associated relationships between the methods of waste collection and musculoskeletal disorders (MSDs). Body mapping is a participatory research tool that has been successfully used to investigate workplace ill health problems. Participatory body mapping exercises were carried out using staff at a UK District Council 2 years before and after the move from boxes and baskets to a wheeled bin recycling service. The study introduces the concept of Average Pain Count (APC). The data, supports previous studies showing wheeled bin based services (APC 2.07 & 2.80) are associated with less MSD outcomes than services including boxes, baskets and sacks (APC 4.02).The surveys provided compelling evidence to suggest that there are associations between age and self-reported pain although there appeared to be no patterns with regards length of service. These findings should help Local Authorities better understand critical factors regarding waste collection strategies and self-reported pain. There are recommendations regarding the use of body mapping and for industry practice