Vol 39, No 3 (2018)

In this paper we study the general properties of non-commutative operator graphs. The problem of the existence of quantum anticliques is considered. The covariant property for the resolution of the identity which generates the graph is investigated.

A well known result in probability is the Poisson limit for rare independent Bernoulli trials. The asymptotic result also holds for a multinomial setting where events are dependent. An interesting data type is a survival process in which one observes many individuals over time periods. The risk set is all those available at the beginning of each time period, thus the same individual can appear in successive risk sets. Individuals exit rarely. In our motivating example these are corporations who exit a public trading system by default or merger, so there are several exit types, hence the multinomial setting. There are also covariates available at the beginning of each period. We study the numbers of exits over time. Under rare multinomial conditions we show that the exits types converge to independent Poisson laws with respect to the exit types and also with respect to time. An immediate application is the construction of one step ahead predictions which may then be tabulated or plotted, giving a convenient tool to study themodel behaviour with respect to time. Thus one can obtain one step ahead prediction intervals for the number of exits of each type, in our case bankruptcy or merger. This is a tool that is useful for large institutional investors such as pension plans.

The firefly algorithm is one of the best latest bio-inspired algorithms, which proved its performance in solving continuous and discrete optimization problems. This paper presents a more detailed comparison study using a set of test functions. The main goal is the application of Firefly algorithm (FA) to solve Lot size optimization in supply chain management which is the most complex part of stock management process. A complexity that comes from the conflict between the minimization of the costs and the maximization of the level of service. For these reason, the traditional methods of lot size control have to deal with the explosion of new needs related to supply chain evolution. The optimal solutions obtained by FA are far better than the best solutions obtained by deterministic methods analyzed in the literature.

We consider the problem of an estimation of the mean value of the normal distribution with a prior information that this parameter is positive and very small. The prior information is implemented in terms of the exponential prior distribution. The estimation procedures are constructed for two cases: fixed sample size and sequential estimation that guarantee the given constraints on the precision and the d-risk of the estimator. An analytical review of the comprehensive literature for the problems of guaranteed statistical inference (d-risk and pFDR) is provided. For the practical applications of the proposed estimators with the unknown value of the prior distribution parameter, we solve the problem of choosing this parameter in the framework of empirical (parametric) Bayesian approach or in the framework of existing State Standards on the precision and output quality of the estimated parameter. As an implementation of the proposed statistical procedures, the problem of estimation of the chemical element of arsenic (As) in a food product is considered. The model parameters are chosen according to the State Standards for carrying out a laboratory tests for As detection. For the chosen values of the parameters, the probability of stopping for the experiment is estimated for each step by the method of statistical simulations. The histogram of the Bayesian estimate for the As content is presented.

Compositional data are met in many different fields, such as economics, archaeometry, ecology, geology and political sciences. Regression where the dependent variable is a composition is usually carried out via a log-ratio transformation of the composition or via the Dirichlet distribution. However, when there are zero values in the data these two ways are not readily applicable. Suggestions for this problem exist, but most of them rely on substituting the zero values. In this paper we adjust the Dirichlet distribution when covariates are present, in order to allow for zero values to be present in the data, without modifying any values. To do so, we modify the log-likelihood of the Dirichlet distribution to account for zero values. Examples and simulation studies exhibit the performance of the zero adjusted Dirichlet regression.

In this paper we derive explicit formula for the average run length (ARL) of Cumulative Sum (CUSUM) control chart of autoregressive integrated moving average ARIMA(p,d,q) process observations with exponential white noise. The explicit formula are derived and the numerical integrations algorithm is developed for comparing the accuracy. We derived the explicit formula for ARL by using the Integral equations (IE) which is based on Fredholm integral equation. Then we proof the existence and uniqueness of the solution by using the Banach’s fixed point theorem. For comparing the accuracy of the explicit formulas, the numerical integration (NI) is given by using the Gauss-Legendre quadrature rule. Finally, we compare numerical results obtained from the explicit formula for the ARL of ARIMA(1,1,1) processes with results obtained from NI. The results show that the ARL from explicit formula is close to the numerical integration with an absolute percentage difference less than 0.3% with m = 800 nodes. In addition, the computational time of the explicit formula are efficiently smaller compared with NI.

An extension of the concept of the guillotine layout function has been proposed for solving the problem of rectangular orthogonal packing; this extension is a function that assigns a triple of values to the sheet width. In addition to the standard effect for the guillotine layout function (the sheet with a given width has a minimum length), which is sufficient to arrange a given set of rectangles in a guillotine manner, two additional values have been used. They describe the method of cutting this sheet to uniquely form a guillotine cutting card and a guillotine layout card of the set of rectangles. These data involve the characteristics of the first cut of the sheet as well as the partition of the set of rectangles corresponding to the cut into two subsets, which is uniquely determined by the number of one of these subsets. The description of the first cut is modeled by a single numerical value that reflects both the size of the offset from the lower-left corner of the sheet and the orientation of the cut: a cut is required along or transverse of the sheet. It has been shown that this information is sufficient for the recovery of the guillotine cutting card and the guillotine layout card for a set of rectangles. Modifications of the algorithms for calculating the sum of two right-semicontinuous monotonically nonincreasing step functions with a finite number of steps and the minimum of two functions of this type have been proposed to determine additional information about the first cut and calculate the extension of the guillotine layout function. Also, an algorithm for the formation of a guillotine cutting card and a guillotine layout card for rectangles has been proposed that uses the calculated extensions of guillotine layout functions for all subsets of the required set of rectangles.

J. Koolen posed the problem of studying distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs whose second eigenvalue is at most t for a given positive integer t. This problem is reduced to the description of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with a nonprincipal eigenvalue t for t = 1, 2,.... In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, Makhnev and Paduchikh found intersection arrays of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue t, where 2 < t ≤ 3. Graphs with intersection arrays {125, 96, 1; 1, 48, 125}, {176, 150, 1; 1, 25, 176}, and {256, 204, 1; 1, 51, 256} remained unexplored. In this paper, possible orders and fixed-point subgraphs of automorphisms are found for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125}. It is proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4, 6). Composition factors of the automorphism group for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125} are determined.