Distance Sets Relating to Orthogonal Exponentials

The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi＇s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.

Big Data Clustering Optimization Based on Intuitionistic Fuzzy Set Distance and Particle Swarm Optimization forWireless Sensor Networks

Big data clustering plays an important role in the field of data processing in wireless sensor networks.However,there are some problems such as poor clustering effect and low Jaccard coefficient.This paper proposes a novel big data clustering optimization method based on intuitionistic fuzzy set distance and particle swarm optimization for wireless sensor networks.This method combines principal component analysis method and information entropy dimensionality reduction to process big data and reduce the time required for data clustering.A new distance measurement method of intuitionistic fuzzy sets is defined,which not only considers membership and non-membership information,but also considers the allocation of hesitancy to membership and non-membership,thereby indirectly introducing hesitancy into intuitionistic fuzzy set distance.The intuitionistic fuzzy kernel clustering algorithm is used to cluster big data,and particle swarm optimization is introduced to optimize the intuitionistic fuzzy kernel clustering method.The optimized algorithm is used to obtain the optimization results of wireless sensor network big data clustering,and the big data clustering is realized.Simulation results show that the proposed method has good clustering effect by comparing with other state-of-the-art clustering methods.

Three-dimensional inversion of knot defects recognition in timber cutting

The comprehensive utilization of wood is the main goal of log cutting,but knot defects increase the diffi-culty of rationally optimizing cutting.Due to the lack of real shape data of knot defects in logs,it is diffi cult for detection methods to establish a correlation between signal and defect morphology.An image-processing method is proposed for knot inversion based on distance regularized level set segmentation(DRLSE)and spatial vertex clustering,and with the inversion of the defects existing relative board position in the log,an inversion model of the knot defect is established.First,the defect edges of the top and bottom images of the boards are extracted by DRLSE and ellipse fi tting,and the major axes of the ellipses made coplanar by angle correction;second,the coordinate points of the top and bottom ellipse edges are extracted to form a spatial straight line;third,to solve the intersection dispersion of spatial straight lines and the major axis plane,K-medoids clustering is used to locate the vertex.Finally,with the vertex and the large ellipse,a 3D cone model is constructed which can be used to invert the shape of knots in the board.The experiment was conducted on ten defective larch boards,and the experimental results showed that this method can accurately invert the shapes of defects in solid wood boards with the advantages of low cost and easy operation.

B-vex Fuzzy Mappings and Its Application to Fuzzy Optimization Problems

The concept of b-vex and logarithmic b-vex for fuzzy mappings are introduced by relaxing the definition of convexity of a fuzzy mapping. Most of the basic properties of b-vex fuzzy mapping are discussed and established for the nondifferentiable case. Necessary and sufficient conditions for b-vex fuzzy mapping are presented. Sevaral important results are given for nonlinear fuzzy optimization problems assuming that the objective and constraint functions are b-vex fuzzy mappings.

Liver Segmentation in CT Images Based on DRLSE Model

Liver segmentation in CT images is an important step for liver volumetry and vascular evaluation in liver pre-surgical planning. In this paper, a segmentation method based on distance regularized level set evolution(DRLSE) model was proposed, which incorporated a distance regularization term into the conventional Chan-Vese (C-V) model. In addition, the region growing method was utilized to generate the initial liver mask for each slice, which could decrease the computation time for level-set propagation. The experimental results show that the method can dramatically decrease the evolving time and keep the accuracy of segmentation. The new method is averagely 15 times faster than the method based on conventional C-V model in segmenting a slice.

Distance domination of generalized de Bruijn and Kautz digraphs

Let G= （V,A） be adigraph and k ≥ 1 an integer. For u,v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V ／ D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γk（G）, is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs GB（n, d） and generalized Kautz digraphs Gg（n, d） are good candidates for interconnection k networks. Denote △k ：=（∑j^k=0 d^j）^-1. F. Tian and J. Xu showed that [n△k] ≤ γk（GB（n,d）） ≤ [n/d^k] and [n△k] ≤ γk（GK（n,d）） ≤ [n/d^k]. In this paper, we prove that every generalized de Bruijn digraph GB（n, d） has the distance k- domination number [n△k] or [n△k] ＋ 1, and the distance k-domination number of every generalized Kautz digraph GK（n, d） bounded above by [n/ （d^k-1 ＋d^k）]. Additionally, we present various sufficient conditions for γk（GB（n, d）） = [n△k] and γk（GK（n, d）） = [n△k].

Growth in Infinite Groups of Infinite Subsets

Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.