Open Access Short Research Article

R. Frühwirth, R. Malina, W. Mitaroff

The rules of a game of dice are extended to a ``hyper-die'' with \(n\in\mathbb{N}\) equally probable faces, numbered from 1 to \(n\). We derive recursive and explicit expressions for the probability mass function and the cumulative distribution function of the gain \(G_n\) for arbitrary values of \(n\). A numerical study suggests the conjecture that for \(n \to \infty\) the expectation of the scaled gain \(\mathbb{E}[{H_n}]=\mathbb{E} [{G_n/\sqrt{n}\,}]\) converges to \(\sqrt{\pi/\,2}\).

The conjecture is proved by deriving an analytic expression of the expected gain \(\mathbb{E} [{G_n}]\).

An analytic expression of the variance of the gain \(G_n\) is derived by a similar technique. Finally, it is proved that \(H_n\) converges weakly to the Rayleigh distribution with scale parameter~1.

Open Access Original Research Article

Chamilanka Wanigasekara, Lakshika S. Nawarathna, V. S. N. Vithanaarachchi

The main goal of this study is to create a suitable model to predict the tooth eruption pattern of Sri Lankan children. Also, we identify the relationship between variables associated with eruption sequence, compare the eruption sequence between sexes, compare the eruption sequences between upper and lower jaws, identifying common polymorphisms of tooth eruption sequences of children and determine the frequencies of occurrence of emergence polymorphisms for different tooth pairs. This analysis was performed using the data of the extent of tooth eruption of all 28 teeth at 10 different time points in each year. Welch two-sample t-test was used to identify the relationship between variables associated with eruption sequence. Frailty models and Cox-Proportional Hazard models developed for each tooth type separately and the model selection procedures Akaike information criterion (AIC), Bayesian information criterion (BIC) and Root Mean Square Error (RMSE) values are measured for each model. Since Gamma Frailty models have the smallest AIC and BIC for seven types of tooth which divide according to the eruption stage of the each tooth, we choose Gamma Frailty models as the best predictor for the tooth eruption. There is a significant difference between the eruption pattern of gender and jaw associated with time. However, no significant difference between sides associated with the eruption sequence was observed.

Open Access Original Research Article

Abdul Basit, Zafar Iqbal, En-Bing Lin

In this paper, two new generalized entropies have been introduced with their respective properties. The results of these entropies have been verified for the exponential and weighted exponential distributions. These two entropies produce the results in the form of simple entropy, generalized entropy, residual entropy, cumulative entropy and mixtures of all these entropies. Some characteristics of residual & past entropy have been derived and special cases have also been obtained. These cases indicate that new generalized entropies are more comprehensive and useful. The main advantage of this study is to derive different types of generalization of entropies using the different parameter values of α and β.

Open Access Original Research Article

Sylva Ligeiaziba, Maxwell A. Ijoma, Emmanuel I. Biu

Prostate cancer is the second most common cause of cancer related deaths in men. It is detected using many screening methods. Like every other cancer, there are risk factors associated with prostate cancer. This include but not limited to, Family History (FH) of the disease, smoking habit, alcohol intake, age and Body Mass Index (BMI). The survival of prostate cancer patients is dependent on many factors such as, early detection of the disease, age of patient and the aggressiveness of the cancer. Gleason score is used to measure the level of aggressiveness of a prostate cancer in a patient. the score ranges from 6 to 10. It is made up of two Gleason grades that ranges from 3 to 5. This study was carried out to determine whether there are significant differences in the mean of Gleason score by the various categories of BMI and FH of patients while controlling for the number of hospital visits. Gleason score was used as the dependent variable while FH and BMI and Number of hospital visits were used as the independent variables. Descriptive statistical measures were used to summarize the basic features of the data. Spearman correlation coefficient was used to measure if there is a significant statistical relationship between the Gleason score, age and BMI, while Analysis of Covariance (ANCOVA) was used to measure the differences in the mean of Gleason score by the categories of FH and BMI while controlling for number of hospital visits. The analysis was done using Statistical Programme for Social Science (SPSS 25.0) and Intellectus Statistics software. Results from the analyses were presented in tabular form. The results showed a significant effect of Body Mass Index (BMI) on Gleason score and that Gleason score increases, as age tends to increase.

Open Access Original Research Article

Ahmed N. Shain, Mahiuob M. Q. Shubatah

Inverse domination is very much useful in network theory, Radio Stations, Electrical stations and several fields of mathematics. In This article, inverse domination in an interval-valued fuzzy graphs is defined and studied. Some bounds on inverse domination number *γ*8(*G*)*. *are provided for several interval-valued fuzzy graphs, such as complete, complete bipartite, star,.. etc. Furthermore, the relationship of *γ*8(*G*)*. *with some others known parameters in interval-valued fuzzy graphs investigated with some suitable examples.