On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

THE METHOD OF EVALUATING THE OPERATION OF RADIO TECHNICIANS OF SPECIAL PURPOSE IN EXTREME (CRISIS) SITUATIONS

In extreme (crisis) situations, tasks arise, requiring an answer to the question of the effectiveness of decisions made. When using radio technicians of special purpose, the results depend on both the ability of the equipment and its condition and from the service personnel. At the same time, it is necessary to take into account the presence of targeted interfering factors. The paper proposed a method for evaluating the operation of radio equipment for special purpose in extreme (crisis) situations, a distinguishing feature of which is to account for the mutual influence of the components of the system. The evolution of the basic component of operation is proposed to approximate polynomial dependence using a fuzzy logic apparatus when making resulting solutions.

Nonlinear Dynamic Study of Oscillation System Featuring Quasi-Zero and Asymmetric Stiffness

This paper will broaden our previous works about the asymmetric and quasi-zero stiffness oscillator named AQZSO. In this paper, the dynamic stiffness of the AQZSO will be investigated. Then, the condition for which the minimum dynamic stiffness is quasi-zero around the equilibrium position is also determined. By using Multi-Scale method, the fundamental resonance response of the AQZSO subjected to the vibrating base is analyzed, in which the dynamic stiffness is expressed as a fifth-order approximate polynomial through expanding Taylor series. The stability of the response is then found out via nonlinear Routh-Herwitz criterion. Moreover, because of existing the sliding friction between the cylinder and piston, the nonlinear and varying-time dynamic characteristics, the complex dynamic response of the AQZSO is the need for discovery by performing direct integration of the original dynamic equation through using 5th-order Runge-Kutta algorithm. In this work, the friction force model of cylinder will be identified through virtual prototyping technique and genetic algorithm. Additionally, the Poincáre map is also employed to analyze the bifurcation phenomenon, coexistence of multiple solutions. The traction basin of the period-1, period-2 and period-3 solution is determined, indicating that the attractor basin is influenced by the asymmetric of the stiffness curve. This research will offer a useful insight to design low frequency vibration isolation systems.

Robust stabilization and tracking control schemes for disturbed multi-input multi-output Hammerstein model in presence of approximate polynomial nonlinearities

This article presents robust stabilization and tracking control problems for multi-input multi-output Hammerstein model with external disturbances. This model is characterized by static nonlinear elements followed by a linear dynamic block. Moreover, the unknown parameters of the identified mathematical model are estimated using the multivariable output error state space subspace algorithm. Unlike the general control strategy that used the nonlinearity inversion method, the nonlinearities are supposed not bijective. In this context, inverse nonlinear functions of polynomial structure are suggested in this article. Furthermore, the composition of the static nonlinear elements and their approximate inverses in series with the linear dynamic block are then decomposed into a set of linear parts using the Takagi–Sugeno fuzzy representation. Consequently, new sufficient stability conditions with decay rate and disturbance attenuation using the [Formula: see text] criterion and linear matrix inequality tools are discussed. Finally, simulation studies are provided to illustrate the merit of our purpose.

Experimentalanalytical identification of a mathematical model of a dc motor using the least squares method

The urgency of the problem lies in the formation of mathematical models of electromechanical converters corresponding to the objects of study with high accuracy. An experimental-analytical assessment of transient modes of a DC motor based on an installation for the study of electrical machines has been carried out. Based on the results obtained, an approximation of transient process graphs was carried out using the least squares method and an approximate polynomial of the corresponding order was selected with the closest imminence to the dynamic properties of the object under study.

Mathematical Modeling of Suspended Microbial Fuel Cells and Electron Transfer Mediator using Homotopy Perturbation Method

The theoretical model of microbial fuel cells with suspended cells and an additional electron transfer mediator is analyzed. Many biological, chemical and electrochemical reactions occur in the bulk liquid and on the surface of the electrode with the substrate (glucose), oxidized mediators and reduced mediators. The homotopy perturbation method (HPM) is used to solve the nonlinear diffusion equations in microbial fuel cells. Direct and approximate polynomial expression of a substrate (glucose), oxidized mediator and reduced mediator concentration are obtained at the mass transfer balance layer. The results of the experiment are compared with the results of the analytical and simulation and satisfactory agreement is noted.

Hydraulic Conductivity of Compacted Lime-Softening Sludge Used as Landfill Liners

The research goal was to investigate the hydraulic conductivity of compacted lime-softening sludge as a material to be applied to landfill liners. In doing so, the effect of compaction and moulding moisture content on the sludge hydraulic conductivity was assessed. An approximate polynomial k10mean at hydraulic gradients ≥30 for degree of compaction (0.95–1.05) and moulding moisture content (28%–36%) was determined. The results of short-term tap water permeation tests revealed that all hydraulic conductivity values were less than 2.5•10–8 m/s. A lowest hydraulic conductivity of 6.5•10–9 m/s, as well as a corresponding moisture content of 31% were then established. The long-term hydraulic conductivity was measured with tap water, distilled water, NaOH and HCl solutions and municipal waste leachate. The factors of permeating liquids and permeation time significantly affected the initial hydraulic conductivity. The long-term hydraulic conductivity increased for NaOH and HCl solutions and decreased for tap and distilled water. A significant reduction of hydraulic conductivity was observed for leachate permeation. The investigated material met the requirements for the liner systems of inert landfill sites regardless of pH and the limit value for hazardous and non-hazardous waste landfills.

Sublinear Time Numerical Linear Algebra for Structured Matrices

We show how to solve a number of problems in numerical linear algebra, such as least squares regression, lp-regression for any p ≥ 1, low rank approximation, and kernel regression, in time T(A)poly(log(nd)), where for a given input matrix A ∈ Rn×d, T(A) is the time needed to compute A · y for an arbitrary vector y ∈ Rd. Since T(A) ≤ O(nnz(A)), where nnz(A) denotes the number of non-zero entries of A, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, T(A) can be much smaller than nnz(A), yielding significantly sublinear time algorithms. For example, in the overconstrained (1+ε)-approximate polynomial interpolation problem, A is a Vandermonde matrix and T(A) = O(n log n); in this case our running time is n · poly (log n) + poly (d/ε) and we recover the results of Avron, Sindhwani, and Woodruff (2013) as a special case. For overconstrained autoregression, which is a common problem arising in dynamical systems, T(A) = O(n log n), and we immediately obtain n· poly (log n) + poly(d/ε) time. For kernel autoregression, we significantly improve the running time of prior algorithms for general kernels. For the important case of autoregression with the polynomial kernel and arbitrary target vector b ∈ Rn, we obtain even faster algorithms. Our algorithms show that, perhaps surprisingly, most of these optimization problems do not require much more time than that of a polylogarithmic number of matrix-vector multiplications.