On the distribution of sums of residues

We generalize and solve the \roman{mod}\,q analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the $2^n$ sums of the form $\sum_{i=1}^n\varepsilon_ia_i$, where each $\varepsilon_i$ is $0$ or $1$. For all $q$, $n$, $k$ we determine the maximum, over all reduced residues $a_i$ and all sets $P$ consisting of $k$ arbitrary residues, of the number of these sums that belong to $P$.Comment: 5 page

Power and influence in Human Resource Development: teaching the politics of HRD on a professional programme

Addressing organisation politics is problematic in all parts of the management curriculum. It alludes to the dark side of organisational life and requires engagement with contentious issues of power and interest. Yet, at the same time, it potentially provides a critical perspective or window through which a richer understanding of management can be achieved. Set in this context it provides a challenge for curriculum and associated professional development. This paper reports upon a research based teaching and learning initiative undertaken in the context of one branch of management, human resource development, and considers its application for other professions. The initiative sought to raise the profile of the politics of HRD within the curriculum. The paper discusses the research undertaken to generate teaching material and how we have subsequently deployed this research within a process of curriculum development. Outcomes are discussed at two levels. First, conventionally, in terms of our use, for example, of a number of depth case studies within the curriculum. Secondly, in terms of the impact of the initiative upon our own self development and professional practice. It is in respect of the latter where we lay claim to more 'benefits'; though questions are raised about the ease with which such benefits may be transferred into curriculum and professional development in higher education management teaching more generally

The practice of HRD in the voluntary sector: towards an understanding of impact

This paper addresses the role of learning in assisting voluntary sector organisations achieve their organisational objectives. Specifically it seeks to develop a platform from which to position necessary research in order to understand the link between how learning is organised, managed and delivered within voluntary sector organisations and its impact upon performance. The paper is thus a step towards a more robust theoretical and evidential understanding of a relatively under-researched domain of HRD practice

Poset-free Families and Lubell-boundedness

Given a finite poset $P$, we consider the largest size \lanp of a family \F of subsets of $[n]:=\{1,...,n\}$ that contains no subposet $P$. This continues the study of the asymptotic growth of \lanp; it has been conjectured that for all $P$, \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn exists and equals a certain integer, $e(P)$. While this is known to be true for paths, and several more general families of posets, for the simple diamond poset \D_2, the existence of $\pi$ frustratingly remains open. Here we develop theory to show that $\pi(P)$ exists and equals the conjectured value $e(P)$ for many new posets $P$. We introduce a hierarchy of properties for posets, each of which implies $\pi=e$, and some implying more precise information about \lanp. The properties relate to the Lubell function of a family \F of subsets, which is the average number of times a random full chain meets \F. We present an array of examples and constructions that possess the properties

Extremal Values of the Interval Number of a Graph

The interval number $i( G )$ of a simple graph $G$ is the smallest number $t$ such that to each vertex in $G$ there can be assigned a collection of at most $t$ finite closed intervals on the real line so that there is an edge between vertices $v$ and $w$ in $G$ if and only if some interval for $v$ intersects some interval for $w$. The well known interval graphs are precisely those graphs $G$ with $i ( G )\leqq 1$. We prove here that for any graph $G$ with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil$. This bound is attained by every regular graph of degree $d$ with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil$ for graphs on $n$ vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor$ for graphs with $e$ edges

A design handbook for phase change thermal control and energy storage devices

Comprehensive survey is given of the thermal aspects of phase change material devices. Fundamental mechanisms of heat transfer within the phase change device are discussed. Performance in zero-g and one-g fields are examined as it relates to such a device. Computer models for phase change materials, with metal fillers, undergoing conductive and convective processes are detailed. Using these models, extensive parametric data are presented for a hypothetical configuration with a rectangular phase change housing, using straight fins as the filler, and paraffin as the phase change material. These data are generated over a range of realistic sizes, material properties, and thermal boundary conditions. A number of illustrative examples are given to demonstrate use of the parametric data. Also, a complete listing of phase change material property data are reproduced herein as an aid to the reader

Study of air pollutant detection by remote sensors

Air pollution detection using satellite observatio

Diamond-free Families

Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of $[n]:=\{1,...,n\}$ that contains no subposet P. This problem has been studied intensively in recent years, and it is conjectured that $\pi(P):= \lim_{n\rightarrow\infty} La(n,P)/{n choose n/2}$ exists for general posets P, and, moreover, it is an integer. For $k\ge2$ let \D_k denote the $k$-diamond poset $\{A< B_1,...,B_k < C\}$. We study the average number of times a random full chain meets a $P$-free family, called the Lubell function, and use it for P=\D_k to determine \pi(\D_k) for infinitely many values $k$. A stubborn open problem is to show that \pi(\D_2)=2; here we make progress by proving \pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 page