Model reduction for discrete-time switched linear time-delayed systems with finite-frequency specifications

The paper focusses on the model reduction of discrete-time switched linear systems with finite-frequency specifications. First, we propose a performance index that can estimate systematic performance over finite frequency. And then, we use GKYP (generalized Kalman–Yakubovich–Popov lemma) and introduce some relaxation matrices with variable parameters to analyse the problem in different situations. Based on the above conditions, sufficient conditions can be obtained for meeting the index of finite frequency. Finally, we illustrate two numerical examples. The results show that the proposed method can improve approximation performance.

Weighted amplifiers and inapproximability results for Travelling Salesman problem

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from $$\frac{123}{122}$$ 123 122 (Karpinski et al. in J Comput Syst Sci 81(8):1665–1677, 2015) to $$\frac{117}{116}$$ 117 116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.

APPROXIMATION OF DISCONTINUOUS FUNCTIONS OF TWO VARIABLES BY DISCONTINUOUS INTERLINATION SPLINES USING TRIANGULAR ELEMENTS

The paper develops a method for approximation of the discontinuous functions of two variables by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function given along a system of lines (such data are commonly used in remote methods, in particular in tomography). The paper also proposes a method for approximating the discontinuous functions of two variables taking into account triangular elements having one curved side. The proposed methods improve approximation of the discontinuous functions, allowing an application to complex domains of definition and avoiding the Gibbs phenomenon.

A Generalized Approach to Improve Approximation of Inverse Langevin Function

The inverse Langevin function has a crucial role in different research fields, such as polymer physics, para- or superpara-magnetism materials, molecular dynamics simulations, turbulence modeling, and solar energy conversion. The inverse Langevin function cannot be explicitly derived and thus, its inverse function is usually approximated using rational functions. Here, a generalized approach is proposed that can provide multiple approximation functions with a different degree of complexity/accuracy for the inverse Langevin function. While some special cases of our approach have already been proposed as approximation function, a generic approach to provide a family of solutions to a wide range of accuracy/complexity trade-off problems has not been available so far. By coupling a recurrent procedure with current estimation functions, a hybrid function with adjustable accuracy and complexity is developed. Four different estimation families based four estimation functions are presented here and their relative error is calculated with respect to the exact inverse Langevin function. The level of error for these simple and easy-to-use formulas can be reduced as low as 0.1%.

Approximating natural landscape pattern using aggregated harvest

Successful implementation of the natural disturbance model for timber harvest is hindered by the lack of strategies to approximate landscape fire pattern. In the forests of Alberta, Canada, the fire regime is dominated by large fires that create large regions of same-aged forest. Current forestry practices disperse harvest blocks across the landscape, causing increased fragmentation as compared with fire. Aggregating harvest blocks is one potential strategy to improve approximation of natural landscape pattern. We used a simulation approach to compare landscape pattern created by aggregated harvest strategies, the current dispersed harvest approach, and the natural disturbance regime for a 270 000 ha forest landscape in northeastern Alberta. Compared with dispersed harvest, aggregated strategies increased compatibility with natural landscape pattern by reducing fragmentation. Capacity to aggregate harvest declined when the constraint of maintaining a constant proportion of deciduous to coniferous harvest was included. We conclude that aggregated harvest can improve implementation of the natural disturbance model by bringing several landscape metrics closer to the conditions that fall within the natural range of variability. Aggregated harvest alone, however, performed poorly at maintaining interior old forest, emphasizing that an explicit old-forest strategy is also required.