There will be a strong increase in the number of people affected by most movement disorders between 2010 and 2050. This increase will mostly depend on the future aging of populations in terms of their age structure and future life expectancy.

One challenging problem in biology is to understand the mechanism of DNA packing in a confined volume such as a cell. It is known that confined circular DNA is often knotted and hence the topology of the extracted (and relaxed) circular DNA can be used as a probe of the DNA packing mechanism. However, in order to properly estimate the topological properties of the confined circular DNA structures using mathematical models, it is necessary to generate large ensembles of simulated closed chains (i.e., polygons) of equal edge lengths that are confined in a volume such as a sphere of certain fixed radius. Finding efficient algorithms that properly sample the space of such confined equilateral random polygons is a difficult problem. In this paper we propose a method that generates confined equilateral random polygons based on their probability distribution. This method requires the creation of a large database initially. However, once the database has been created, a confined equilateral random polygon of length n can be generated in linear time in terms of n. The errors introduced by the method can be controlled and reduced by the refinement of the database. Furthermore, our numerical simulations indicate that these errors are unbiased and tend to cancel each other in a long polygon.

So far all calculations of the number of demented people are based on rates from meta-analyses, mean rates of meta-analyses or spatial analyses. This article presents age- and gender-specific prevalence and incidence rates of dementia that are based on a large sample of the German Sick Funds (Stichprobendaten von Versicherten der gesetzlichen Krankenversicherung (GKV)) with 2.3 million people from the year 2002. Prevalence rates increase from 0.8% and 0.6% for 60-64 year old men and women to 30% and 43% for men and women aged 100 or older, respectively. Incidence rates increase from 0.18 and 0.14 cases per 100 person-years for 60-64-year old men and women to 9.9 and 10.9 for over 95 year old men and women, respectively. Our results confirm rates from earlier studies on the basis of meta-analyses. Regional differences show for the first time that higher prevalence rates exist for East German women and men above age 85. In 2007 about 1.07 million moderately or severely demented people live in Germany of which about 244 000 are incident cases when we extrapolate our rates to the population of this year.

In this paper we continue our earlier studies [5, 6] on the generation methods of random equilateral polygons confined in a sphere. The first half of the paper is concerned with the generation of confined equilateral random walks. We show that if the selection of a vertex is uniform subject to the position of its previous vertex and the confining condition, then the distributions of the vertices are not uniform, although there exists a distribution such that if the initial vertex is selected following this distribution, then all vertices of the random walk follow this same distribution. Thus in order to generate a confined equilateral random walk, the selection of a vertex cannot be uniform subject to the position of its previous vertex and the confining condition. We provide a simple algorithm capable of generating confined equilateral random walks whose vertex distribution is almost uniform in the confinement sphere. In the second half of the paper we show that any process generating confined equilateral random walks can be turned into a process generating confined equilateral random polygons with the property that the vertex distribution of the polygons approaches the vertex distribution of the walks as the polygons get longer and longer. In our earlier studies, the starting point of the confined polygon is fixed at the center of the sphere. The new approach here allows us to move the the starting point of the confined polygon off the center of the sphere.

In this paper we continue our earlier studies (Diao et al 2011