Optimization of the task of forming a management system of hierarchical multilevel complex organizational systems

A hierarchical configuration procedure in the process of analyzing the functioning of organizational structures using algorithms for solving problems using discrete programming methods is considered in detail in this paper. On the basis of the research, the authors propose a methodology for the combined application of an initial approximation algorithm with certain properties and an algorithm for a recursive procedure for a directed search for the optimal hierarchy, which, in the future, solving the traveling salesman problem in a minimum time much more efficient than using the methods of dynamic programming of branches and boundaries.

A novel framework to finite-time tracking of nonlinear MIMO systems based on fast terminal sliding-mode method

In this paper, a novel approach is proposed to design a robust finite-time tracking controller for uncertain affine multi-input multi-output (MIMO) systems. In the proposed approach, a fast terminal sliding mode (FTSM) controller is designed, based on the input–output relation of the MIMO nonlinear systems. The main advantage of the proposed method is its capability to handle the internal dynamics of the system. Moreover, in the proposed controller, the state variables of the internal dynamics of the system are not necessary to be measured. To realize the finite-time convergence of the output variables, a set of switching manifolds with a recursive procedure is utilized. Finally, the robust stability and tracking efficiency of the proposed control law are shown through computer simulations.

How to use Fibonacci numbers to teach recursive programming

There are too many examples and programming guides (which, e.g., an internet search for "recursive procedure Fibonacci" will turn up) to count that use Fibonacci numbers as an example to illustrate recursive programming. The motivation for this article is to show why the naive way of doing this is a bad idea, as it is horrendously inefficient. We will exhibit much more efficient ways of computing Fibonacci numbers, both iterative and recursive, and analyze and compare worst case running times and memory usages. Using some mathematical properties of Fibonacci numbers leads to the most efficient method for their computation. For illustration and benchmarking, we will use Maple and its programming language, however, similar behaviour can be demonstrated in almost any other programming language. This exposition combines and explores the mathematical properties of Fibonacci numbers, notions of algorithmic complexity, and efficient Maple programming and profiling techniques, and may be used as an introduction to any of these three subjects. The techniques described can be readily generalized to more general types of linear recurrences with constant coefficients.

Rational Interpolation: Jacobi’s Approach Reminiscence

We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.

Eigenvalue Estimates via Pseudospectra

In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence μkk∈N which converges to some element of its spectrum. The scalar λ0 serves as initial term (μ0=λ0), while additional terms are constructed through a recursive procedure, exploiting the fact that each term μk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point μk. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices.

Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Adaptive covariate acquisition for minimizing total cost of classification

In some applications, acquiring covariates comes at a cost which is not negligible. For example in the medical domain, in order to classify whether a patient has diabetes or not, measuring glucose tolerance can be expensive. Assuming that the cost of each covariate, and the cost of misclassification can be specified by the user, our goal is to minimize the (expected) total cost of classification, i.e. the cost of misclassification plus the cost of the acquired covariates. We formalize this optimization goal using the (conditional) Bayes risk and describe the optimal solution using a recursive procedure. Since the procedure is computationally infeasible, we consequently introduce two assumptions: (1) the optimal classifier can be represented by a generalized additive model, (2) the optimal sets of covariates are limited to a sequence of sets of increasing size. We show that under these two assumptions, a computationally efficient solution exists. Furthermore, on several medical datasets, we show that the proposed method achieves in most situations the lowest total costs when compared to various previous methods. Finally, we weaken the requirement on the user to specify all misclassification costs by allowing the user to specify the minimally acceptable recall (target recall). Our experiments confirm that the proposed method achieves the target recall while minimizing the false discovery rate and the covariate acquisition costs better than previous methods.

Optimal Surplus-Dependent Reinsurance Under Regime-Switching in a Brownian Risk Model

In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation.

Contrast in syntax and contrast in phonology

This chapter investigates the parallels and asymmetries between contrastive features in phonology and syntax, with particular reference to the notion of a contrastive hierarchy. In phonology, contrastive features can be assigned to segments by a recursive procedure that applies to the underlying phonemic inventory of a language. To apply such a procedure to morphosyntactic features, it is necessary to identify the inventory of items that the features serve to distinguish. This chapter argues that the relevant morphosyntactic inventory is the inventory of functional lexical items (in the sense used in Distributed Morphology), and not the inventory of vocabulary items. It further proposes that contrastive specification of functional lexical items is done separately on different dimensions of contrast—for example, that person features are specified separately from number features. This approach is illustrated by application to phi-features in Mi’gmaq, where it is shown to be consistent with patterns of agreement.

European Countries Ranking and Clustering Solution by Children’s Physical Activity and Human Development Index Using Entropy-Based Methods

The aim of the present study is to propose a new approach for evaluating and comparing European countries using indicators of the children physical activity and the human development index. The Global Matrix 3.0 on physical activity for children and youth and human development index data on the 18 European countries were used. MADM (multi-attribute decision making) approach was applied for this task. The criteria weights calculated by applying the weight balancing method—weight balancing indicator ranks accordance (WEBIRA). New methodology of interval entropy is proposed for determining the priority of criteria separately in each group. The novel approach of α-cuts for recursive procedure of ranking the alternatives was used. For comparison, three alternative entropy-based methods—entropy method for determining the criterion weight (EMDCW), method of criteria impact LOSs and determination of objective weights (CILOS) and integrated determination of objective criteria weights (IDOCRIW) were applied to address this MADM problem. Cluster analysis of European countries carried out using results obtained by all above methods. Comparison of the MADM methods revealed that three alternative methods assigned negligible values to whole group of criteria. Meanwhile, WEBIRA family methods performed the ranking of European countries according to the interrelation of the two groups of criteria in a balanced way. Thus, when addressing MADM tasks with two or more naturally related sets of criteria, it is appropriate to apply criteria adapted for that purpose, such as WEBIRA.