The perimeter of uniform and geometric words: a probabilistic analysis

Let a word be a sequence of $n$ i.i.d. integer random variables. The perimeter $P$ of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of $P$. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of $P$ is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motionComment: 13 pages, 7 figure

The perimeter of uniform and geometric words: a probabilistic analysis

Let a word be a sequence of $n$ i.i.d. integer random variables. The perimeter $P$ of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of $P$. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of $P$ is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motionComment: 13 pages, 7 figure

Horizontal and Vertical Mathematization Processes of Junior High School Students in Solving Open-Ended Problems

Mathematization is converting information from problems into mathematical models. The mathematization process is divided into horizontal and vertical mathematization. This descriptive qualitative research aimed to describe junior high school students' horizontal and vertical mathematization process in solving open-ended problems. The subjects are three students with good, medium, and poor mathematical problem-solving abilities. The instruments used were interview guidelines, mathematical problem-solving ability tests, and open-ended problem tests with topics area and perimeter of rectangles and circles. This research shows the horizontal and vertical mathematization process in solving open-ended problems. The horizontal mathematization process was; identifying the information and topics area and perimeter from the problem; representing the problem into some rectangle and circle figures and expressing the problem in the subject’s own words; writing the mathematics language; finding the regularity of the relations to find the possible solutions; and making mathematical models. The vertical mathematization process was; using mathematical representations with symbols and formulas related to the area and perimeter of rectangles and circles; using formal algorithms; customizing and combining some models to get the correct answers; making logical arguments to support the solution and other possible solutions that suit the problem; and generalizing the solution using the concepts of area and perimeter of rectangles and circles to solve similar problems. Every student may have different strategies and solutions when solving open-ended problems

Developmental morphology of the small intestine in Yangzhou goslings

The objective of this study was to investigate the development of the weight and the morphological development of the small intestine in Yangzhou geese. The weight, length and perimeter of the small intestine, height and width of the villi, depth of the crypts were measured when geese were 1, 14, 28, 42, 56 and 70 days of age. The results revealed that the weight of the duodenum, jejunum, and ileum (relative to the BW) increased, peaked on day 14 and tended to decline thereafter with age. The weight of the duodenum, jejunum, and ileum (relative to the BW) kept steady-going on 56, 42, 42 days, respectively. A 3-fold increase in length and 2-fold increase in perimeter for the three segments during the period were obtained. The duodenum increased little in length, whereas both jejunum and ileum increased 2 and 3-folds from 1 to 70 days examined. The increase in perimeter of the duodenum, jejunum and ileum was greater from day 1 to 14 than from day 14 to 28. The length and perimeter of the intestinal segments of the gastro-intestinal of Yangzhou geese (relative to the BW) peaked on day 1, then decreased with age and kept steady-going on 42 days, except the relative perimeter of the duodenum. The villus height, surface area and crypt depth in the small intestine were positively correlated with the age of the geese. The ratio of the villus height to the crypt depth (V/C) differed among the segments of the small intestine and at the different time points. The V/C in the ileum increased from day 1 to 70, whereas in the duodenum the ratio first decreased, then rose and descended. However, in the jejunum the ratio increased first, then dropped and rose again, and descended eventually.Key words: Geese, small intestine, morphology

Yew (Taxus baccata L.) population dynamics in the Iberian Mediterranean Mountains: natural regeneration and expansion in East Central System (Spain)

Aim of study: To evaluate demographical dynamics of yew populations in the Iberian Mediterranean mountains in order to assess population trends and a hypothetical natural regeneration of the species.Area of study: Ayllón Mountain Range, Eastern end of Central Mountain Range, Spain.Material and methods: Afield survey was conducted on three yew groves, taking data of altitude, orientation, companion woody species, UTM coordinates, height, perimeter (at breast height, 120cm) and sex. Individuals were classified in four age groups, considering height and trunk perimeter: AgeGroup1, Seedling; AgeGroup2, Juvenile; AgeGroup3, Adult and AgeGroup4, Old. Finally Recruitment rate was calculated for each population.Main results: Surveyed yew populations include more than 1600 trees, showing elevated Recruitment rates, which indicates that there are more seedling and juvenile individuals (Group1 and Group2) than adult and old ones (Group3 and Group4).Research highlights: Surveyed yew populations show vigorous regeneration. These results lead us to reconsider the relictic character assigned to the species in the Iberian Mediterranean mountains.Key words: Ayllón; microclimatic; regeneration rate; relictic; survey