1. ### Perfect Repetition Codes of Ideals of the Polynomial Ring F2n[x]mod(xn-1) for Error Control In Computer Applicatons

Olege Fanuel, Okaka Akinyi Colleta, Owino Maurice Oduor, Aywa Shem

Abstract: The study of perfect codes has attracted a lot of interest among researchers in coding theory in view of the fact that many authors have indicated that this type of codes is rare. These codes are considered the best for theoretical and practical reasons. In this paper we demonstrate the determination of perfect codes from ideals of polynomial rings and characterize them for error control in computer applications. GAP software has been used to generate these codes and to confirm that they are indeed perfect. The Mathematical Structure of the generating polynomial ring has been fully discussed and the corresponding perfect codes have been characterized.

Keywords: Polynomial Ring, Ideals, Perfect codes, Error Control.

2. ### On Statistical Analysis of Competing Risks with Application to the Time of First Goal

Petr Volf

Abstract: The contribution deals with the analysis and application of the competing risks model. First, the problem of identifiability of marginal and joint distributions of competing random variables is discussed. Then, the notion of copula is recalled and used to express the dependence in the competing risks scheme. Copula model is then utilized to the analysis of times of scoring the first goal in a football (soccer) match. It is assumed that competing latent times are exponentially distributed with parameters depending on attack and defence strengths of teams, while their mutual dependence is described with the aid of a conveniently chosen copula ensuring the model identifiability. As a real example the data from the season 2014-2015 of the Czech premier league (called Synot League) are analyzed. It is shown that the correlation in the framework of proposed model is, as a rule, negative, and its value is related to the first goal importance. The outcomes of analysis are discussed both from theoretical and practical point of view.

Keywords: survival analysis, competing risks, copula, sports statistics.

3. ### Multiple Imputation for a Continuous Variable

Seppo Laaksonen

Abstract: Multiple imputation (MI) is invented by Rubin in 1970’s. He recommends to create imputations through a Bayesian process. Most software’s are respectively following Bayesian principles. Since MI is not much used in official statistics, on the contrary to single imputation, any Bayesian framework is not there necessarily appropriate. Nevertheless, MI can be considered to be useful for several reasons. Björnstad suggests another approach, non-Bayesian imputation. It thus does not require any Bayesian rules for imputing the missing values several times. This paper first presents an approach to imputation that is not much applied. It strictly follows a two-stage strategy, so that the imputation model is first estimated and then completed by the imputation task. These tasks are called model-donor and real-donor that are clearer than those used traditionally. The approach is concretized with apattern of examples in which the variable being imputed is income. Bayesian and non-Bayesian methods are compared. The main result is that a Bayesian framework is not necessarily superior, at least if the applications are done by software’s such as SAS and SPSS. If a user himself can create imputations using an appropriate non-Bayesian framework, the performance is often better.

Keywords: Imputation model, Imputation variance, Model-donor imputation, Poor vs rich fit, Real-donor imputation, SAS, SPSS.

4. ### A System Reliability Model with Priority to Repair over Preventive Maintenance Under Different Weather Conditions

Neeraj and M.S. Barak

Abstract: The main emphasis of this paper is to obtained profit analysis of a two unit cold standby system with priority to repair activity over the preventive maintenance of the units working under different weather conditions. The both units are identical in nature which may fail directly from normal mode. There is a single server who visits the system immediately whenever required and works only in normal weather conditions. The operative unit under goes for preventive maintenance after a maximum operation time. Repair of the unit is done by the server at its complete failure. The unit works as new after maintenance and repair. The time to failure of the unit follows negative exponential distribution while the distributions of preventive maintenance and repair times are taken as arbitrary with different probability density functions. All random variables are statistically independent. The expressions for several reliability measures are derived in steady state using regenerative point technique and semi-Markov process. The graphical behavior of MTSF, availability and profit function have been observed with respect to preventive maintenance rate for arbitrary values of other parameters and costs.

Keywords: Cold Standby system, Reliability Model, Preventive Maintenance, Repair, Priority, Weather Conditions and Profit Analysis.