In this paper we prove that, if a collection of sets [Ascr] is union-closed, then the average set size of $\cal A is at least $\frac{1}{2}\log_2 (\vert{\cal A}\vert)$.

10.1 Derived Instances of Eq and Ord  140

10.2 Derived Instances of Enum  140

10.3 Derived Instances of Bounded  141

10.4 Derived Instances of Read and Show  142

10.5 An Example  143

24.1 Library Time  227

The four papers encompassing this special section on Robot Mediated Neuro-motor Therapy introduce a new area for robotics in the healthcare industry, that of assisting in the delivery of neuro-motor therapies, for patients recovering from a stroke. As life expectancy increases, so do problems associated with old age. In the paper by Krebs et al. the authors not only summarise the need for action in stroke rehabilitation but also makes a useful distinction between assistive technologies that provide enhanced abilities for the patient, and therapy tools for the clinician, that seek to enhance the patients recovery.

14.1 Showing Functions  158

14.2 Reading Functions  159

14.3 Miscellaneous  159

14.4 Library Numeric  160

16.1 Array Construction  174

16.2 Incremental Array Updates  175

16.3 Derived Arrays  176

16.4 Library Array  176

A general spectral bound for the sizes of some vertex subsets, which are mutually at a given minimum distance in a graph, is derived. This unifies and improves some previous results. Some applications to the study of certain metric parameters of the graph are then discussed.

5.1 Module Structure  68

5.2 Export Lists  69

5.3 Import Declarations  71

5.4 Importing and Exporting Instance Declarations  74

5.5 Name Clashes and Closure  74

5.6 Standard Prelude  77

5.7 Separate Compilation  78

5.8 Abstract Datatypes  78

It has been shown [2] that if n is odd and m1,…,mt are integers with mi[ges ]3 and [sum ]i=1tmi=|E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph <?TeX \displaystyle{\mathop{K}^{\raise-2pt\hbox{$\scriptstyle\leftrightarrow$}}}_{n}?> can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mi[ges ]2 and <?TeX \sum_{i=1}^t m_{i}=\vert E({\leftrightarrow}_{n})\vert ?>, except for the single case when n=6 and all mi=3. We also show that sufficiently dense Eulerian digraphs can be decomposed in a similar manner, and we prove corresponding results for (undirected) complete multigraphs.

8.1 Module Prelude  104

8.2 Module PreludeList  114

8.3 Module PreludeText  119

8.4 Module PreludeIO  123

2.1 Notational Conventions  7

2.2 Lexical Program Structure  8

2.3 Comments  9

2.4 Identifiers and Operators  10

2.5 Numerical Literals  11

2.6 Character and String Literals  12

2.7 Layout  13

15.1 Deriving Instances of Ix  170

15.2 Library Ix  171

 CPU Time  233

7.1 Standard I/O Functions  97

7.2 Sequencing I/O Operations  99

7.3 Exception Handling in the I/O Monad  100

25.1 Library Locale  231

3.1 Errors  19

3.2 Variables, Constructors, Operators, and Literals  20

3.3 Curried Applications and Lambda Abstractions  21

3.4 Operator Applications  21

3.5 Sections  22

3.6 Conditionals  23

3.7 Lists  23

3.8 Tuples  24

3.9 Unit Expressions and Parenthesized Expessions  25

3.10 Arithmetic Sequences  25

3.11 List Comprehensions  25

3.12 Let Expressions  27

3.13 Case Expressions  27

3.14 Do Expressions  29

3.15 Datatypes with Field Labels  29

3.16 Expression Type Signatures  32

3.17 Pattern Matching  32

This paper presents the implementation of an explicit model reference adaptive control (MRAC) for position tracking of a dynamically unknown robot. An auto regressive exogenous (ARX) model is chosen to define the plant model and the control input is optimised in a H2 norm to reduce computational time and to simplify the algorithm. The theory of MRAC falls into a description of the various forms of controllers and parameter estimation techniques, therefore, applications may require very complicated solution methods depending on the selected laws. However, in this study, the proposed MRAC shows that applications may be as easy as classical control methods, such as PID, by guaranteeing the stability and achieving the convergency of the plant parameters. Despite the selected simple control model, simple optimisation method and drawbacks of the robot the experimental results show that MRAC provides an excellent position tracking compared with conventional control (PID). Many experimental implementations have been done on the robot and one of them is included in the paper.

19.1 Library Char  195

The Rehabilitation Technologies Division of Applied Resources Corp. (RTD-ARC) has engaged in a Phase I effort to commercialize a robotic bi-manual therapy machine for use in stroke rehabilitation, in cooperation with the VA Rehabilitation R&D Center in Palo Alto. The robotic therapy device, called ARCMIME here in order to differentiate it from its clinical predecessor, has the potential to improve rehabilitation outcomes significantly for individuals who have upper limb impairments due to stroke and other brain injuries.

This paper describes design considerations and clinical outcomes with regards to the Phase I system. It was found that the kinematically simpler system adequately replicated the data outcomes of the more sophisticated PUMA-based experimental test rig.