Adjacent Subsets Research Articles

On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples

In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p≥2 of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed.

Turing Pattern Formation Under Heterogeneous Distributions of Parameters for an Activator-Depleted Reaction Model

The purpose of this article is to study Turing pattern formation in one- and two-dimensional domains under heterogeneous distributions of the parameters for an activator-depleted model. Unlike previous studies of this nature, the choice of the heterogeneous distributions of the parameters is closely linked and estimated by use of rigorous wave mode selection in order to excite different modes in different subsets of the domains. This allows us to relate the numerical solutions with theoretical linear stability analytical results. Our most revealing results show that the wave modes of adjacent subsets evolve locally and yet possess continuity across the interface. These local patches of the solutions result in a globally heterogeneous solution stable only in the presence of heterogeneous distributions of the parameters. Furthermore, our results show that initial conditions continue to play a crucial role in the selection of excitable wave modes and consequently the formation of the inhomogeneous patterns formed. In particular, initial conditions influence pattern orientation and polarity, and yet with a prepattern, the patterns conserved orientation and polarity. Numerical solutions are obtained by the use of the finite element method and the backward Euler scheme to deal with the spatial and the time discretisations, respectively.

Quality Assessment Method of Speckle Patterns Based on Correlation Coefficient of Adjacent Subsets

Quality Assessment Method of Speckle Patterns Based on Correlation Coefficient of Adjacent Subsets

Deployment optimization of multi-hop wireless networks based on substitution graph

In multi-hop wireless networks, the maximum number of hops between clients and the service gateway (GW) significantly impacts the quality of network service, and thus GW deployment optimization plays a critical role in network design and planning. It is well know that the optimal GW deployment can be formulated as a vertex k-centre problem. However, achieving the optimal solution of a k-centre problem is highly complex due to its large solution space. To address this issue we propose a new algorithm based on the substitution principle of network by exploring the inclusion relationship of adjacent node subsets, to reduce the original network to a smaller scale substitution graph. The proposed algorithm can eliminate a large number of redundant nodes, thus reducing the solution space of the optimization problem, and improving the probability of achieving a globally optimal solution. The performance of the proposed algorithm is also analysed. Simulation results show that the proposed t-step substitution algorithm can significantly reduce the solution space of the k-centre problem by up to 80%. We then apply the proposed algorithms to traditional optimization methods, such as genetic (GA), artificial immune (AIA), and K-means for solving discrete space problems, and it is shown that the substitution based algorithm can significantly improve the performance of respective traditional GA, AIA and K-means methods,yielding a better GW deployment scheme with a smaller covering radius.

On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-calledr-weaker Meir-Keeler orr,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.

A systematic method of smooth switching LPV controllers design for a morphing aircraft

This paper is concerned with a systematic method of smooth switching linear parameter-varying (LPV) controllers design for a morphing aircraft with a variable wing sweep angle. The morphing aircraft is modeled as an LPV system, whose scheduling parameter is the variation rate of the wing sweep angle. By dividing the scheduling parameter set into subsets with overlaps, output feedback controllers which consider smooth switching are designed and the controllers in overlapped subsets are interpolated from two adjacent subsets. A switching law without constraint on the average dwell time is obtained which makes the conclusion less conservative. Furthermore, a systematic algorithm is developed to improve the efficiency of the controllers design process. The parameter set is divided into the fewest subsets on the premise that the closed-loop system has a desired performance. Simulation results demonstrate the effectiveness of this approach.

Smooth switching linear parameter‐varying control for hypersonic vehicles via a parameter set automatic partition method

This study presents a systematic approach to design smooth switching linear parameter-varying (LPV) controllers for hypersonic vehicles (HVs). Considering the switched LPV model of HVs, the smooth switching controllers are designed by means of dividing the scheduling parameter set into subsets with overlaps and the controllers in the overlapped subsets are smoothly scheduled between controllers in adjacent subsets. A switching law without constraint on average dwell time is obtained which makes the conclusion less conservative. To divide the parameter set automatically, a novel set partition method, in which the selecting of datum vertex and the handling of unpartitioned subset are innovative, is proposed. The switching LPV controllers that achieve a desired performance are constructed efficiently by solving a convex optimisation problem. Simulation results for HVs demonstrate the effectiveness of the proposed smooth switching LPV controllers and parameter set division method.

On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

AbstractThis paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.

On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.

Fixed and Best Proximity Points of Cyclic Jointly Accretive and Contractive Self‐Mappings

p(≥2)‐cyclic and contractive self‐mappings on a set of subsets of a metric space which are simultaneously accretive on the whole metric space are investigated. The joint fulfilment of the p‐cyclic contractiveness and accretive properties is formulated as well as potential relationships with cyclic self‐mappings in order to be Kannan self‐mappings. The existence and uniqueness of best proximity points and fixed points is also investigated as well as some related properties of composed self‐mappings from the union of any two adjacent subsets, belonging to the initial set of subsets, to themselves.

A parameter set division and switching gain-scheduling controllers design method for time-varying plants

This paper presents a new technique to design switching gain-scheduling controllers for plants with measurable time-varying parameters. By dividing the parameter set into a sufficient number of subsets, and by designing a robust controller to each subset, the designed switching gain-scheduling controllers achieve a desired L 2 -gain performance for each subset, while ensuring stability whenever a controller switching occurs due to the crossing of the time-varying parameters between any two adjacent subsets. Based on integral quadratic constraints theory and Lyapunov stability theory, a switching gain-scheduling controllers design problem amounts to solving optimization problems. Each optimization problem is to be solved by a combination of the bisection search and the numerical nonsmooth optimization method. The main advantage of the proposed technique is that the division of the parameter region is determined automatically, without any prespecified parameter set division which is required in most of previously developed switching gain-scheduling controllers design methods. A numerical example illustrates the validity of the proposed technique.

Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition

AbstractThis article discusses a more general contractive condition for a class of extended (p ≥ 2) -cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are non-empty, closed and convex, then all the iterates converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduces to a unique fixed point if all such subsets intersect.

Genotype by environment interaction for milk yield in Sarda dairy sheep

Genotype by environment interaction (GEI) for milk yield (ME) wasinvestigated by analyzing 40 140 first lactation records performed in 538 flocks. Flock–year effects from national genetic evaluation were used to define low (L), medium (M) and high (H) yielding environments. Differences observed between adjacent subsets were approximately 20% of the trait overall mean. Sire variance components estimated using multitrait analysis were 58, 82 and 149% of the estimate from the complete dataset (L, M and H, respectively) and genetic correlation coefficient between extreme subsets was 0.66, lower than genetic correlations obtained from randomly sampled subsets. Rank correlation coefficients between estimated transmitting abilities (ETA), obtained independently within L, M and H subsets for 71 AI rams having at least 10 daughters per subset, were lower than one (ranging from 0.42 to 0.56). These results strongly supported the existence of GEI for milk yield in Sarda sheep flocks. The influence of GEI on phenotypic responses in flocks differing in yield level could be due to a reduced expression of genes under conditions of restricted environmental opportunity. Splitting the selection scheme according to farming conditions does not seem a suitable solution for the Sarda breeding program. Thus, to provide farmers with genotypes suitable to exhibit a good phenotypic expression in different conditions, the selection across all environments, in which the descendants are expected to perform, still seems the best option.

An Extremal Problem in the Hypercube and Optimization of Asynchronous Circuits

We prove that if m≥ 2, then the minimumk∈N such that the k -cube {0, 1}kcan be decomposed as the disjoint union of m connected adjacent subsets satisfies 2log2m−log2log2m− 1 ≤k≤ 2⌈log2m⌉ − ⌊log2log2m⌋ + 5.

Diets for fat sets

In this paper we describe the discrete interval encoding tree for storing subsets of types having a total order and a predecessor and a successor function. In the following, we consider for simplicity only the case for integer sets; the generalization is not difficult.The discrete interval encoding tree is based on the observation that the set of integers {i[mid ]a[les ]i[les ]b} can be perfectly represented by the closed interval [a, b]. The general idea is to represent a set by a binary search tree of integers in which maximal adjacent subsets are each represented by an interval. For example, inserting the sequence of numbers 6, 9, 2, 13, 8, 14, 10, 7, 5 into a binary search tree, respectively, into a discrete interval encoding tree results in the tree structures shown in figure 1.The efficiency of the interval representation, both in terms of space and time, improves with the density of the set, i.e. with the number of adjacencies between set elements. So what we propose is a ‘diet’ (discrete interval encoding tree) for ‘fat’ sets in the sense of ‘the same amount of information with less nodes’.

Energy-based smoothing of data

Given a set of large noisy data, a technique which generates line segments and provides an optimal smooth connection of the line segments to form a smoothing function is proposed. In this technique data are recursively subdivided into two subsets, means of the data in each subset are computed and a line segment is defined between the means of two adjacent subsets. An energy function is defined as the sum of variances of all the subsets added with a weighted (by a weighting parameter) sum of squaring the difference of means between two adjacent subsets. Optimal connections of line segments are obtained by the simulated annealing technique. The weighting parameter is introduced in the energy function to express the relative importance of minimizing local variances of data versus the smoothness of connections of line segments. By adjusting the weighting parameter, a second optimization is obtained in the sense that the smoothing function formed by the connected line segments provides the best compromise between the closeness of data to the smoothing function and the optimal smoothness of the function.

A boundary finding algorithm and its applications

An algorithm for locating gray level and/or texture edges in digitized pictures is presented. The algorithm is based on the concept of hypothesis testing. The digitized picture is first subdivided into subsets of picture elements, e.g., 2 \times 2 arrays. The algorithm then compares the first- and second-order statistics of adjacent subsets; adjacent subsets having similar first- and/or second-order statistics are merged into blobs. By continuing this process, the entire picture is segmented into blobs such that the picture elements within each blob have similar characteristics. The boundaries between the blobs comprise the boundaries. The algorithm always generates closed boundaries. The algorithm was developed for multispectral imagery of the earth's surface. Application of this algorithm to various image processing techniques such as efficient coding, information extraction (terrain classification), and pattern recognition (feature selection) are included.