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We address several problems of coordination and consensus on and that can be formulated as minimization problems on these Lie groups. Then, gradient descent methods for minimization of the corresponding functions provide distributed algorithms for coordination and consensus in a multi-agent system. We point out main differences in convergence of algorithms on the two groups. We discuss advantages and effects of representing 3D rotations by quaternions and applications to the coordinated motion in space. In some situations (and depending on the concrete problem and goals) it is advantageous to run algorithms on and map trajectories onto via the double cover map , instead of working directly on .

We address several problems of coordination and consensus on and that can be formulated as minimization problems on these Lie groups. Then, gradient descent methods for minimization of the corresponding functions provide distributed algorithms for coordination and consensus in a multi-agent system. We point out main differences in convergence of algorithms on the two groups. We discuss advantages and effects of representing 3D rotations by quaternions and applications to the coordinated motion in space. In some situations (and depending on the concrete problem and goals) it is advantageous to run algorithms on and map trajectories onto via the double cover map , instead of working directly on .

This paper examines consensus building in AHP-group decision making from a Bayesian perspective. In accordance with the multicriteria procedural rationality paradigm , the methodology employed in this study permits the automatic identification, in a local context, of \"agreement\" and \"disagreement\" zones among the actors involved. This approach is based on the analysis of the pairwise comparison matrices provided by the actors themselves. In addition, the study integrates the attitudes of the actors implicated in the decision-making process and puts forward a number of semiautomatic initiatives for establishing consensus. This information is given to the actors as the first step in the negotiation processes. The knowledge obtained will be incorporated into the system via the learning process developed during the resolution of the problem. The proposed methodology, valid for the analysis of incomplete or imprecise pairwise comparison matrices, is illustrated by an example.

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics of the network are organized by a transcritical singularity. As such, we detail a slow passage through a transcritical singularity for a simple network, and we observe that an exchange between consensus and clustering of the nodes is possible. In contrast to the classical planar fast–slow transcritical singularity, the network structure of the system under consideration induces the presence of a maximal canard. Our main tool of analysis is the blow-up method. Thus, we also focus on tracking the effects of the blow-up transformation on the network’s structure. We show that on each blow-up chart one recovers a particular dynamic network related to the original one. We further indicate a numerical issue produced by the slow passage through the transcritical singularity.

Recently, tampering in digital images considered the main challenge in image forensic analyses. Hence, copying a part and pasting it in the same image became the most crucial action in image forgery. It threatens the integrity and authenticity of image ownership. The intruder utilizes the development tools of image processing programs to make the forged image the same as the authentic one and strict for detection. This work as copy-move forgery detection (CMFD) manipulates the problem of a few key points in small-size and homogenous digital images by adopting a merging scheme to detects sufficient key points by using a linear scale-space detector based on Speedup Robust Feature(SURF) and curvature scale space detector based on Maximally Stable Extremal Region (MSER). Afterward, these key points are described distinctively by extract unique vectors, and matching these vectors to find duplicated regions. In the post-processing stage, we propose new filters, the first is called Parallel Filter and the other called Distance Ratio Filter. These Filters aim to remove false-positive results and boost true positive results thus improving the accuracy of the detection scheme. The experimental results on standard data sets (MICC 220, F8 Multi) show that CMFD is efficient and insensitive against simple and combination post-processing attacks like photometric and geometric transformations. Also, it is invariant against non-uniform transformation like (skew, wrap), and detects multi cloning efficiently with a high true-positive ratio (TPR=98.5) and low false-positive ratio (FPR=4).

The practical Byzantine fault tolerant (PBFT) consensus mechanism is one of the most basic consensus algorithms (or protocols) in blockchain technologies. Thus its performance evaluation is an interesting and challenging topic due to the higher complexity of its consensus work in a peer-to-peer network. This study describes a simple stochastic performance model of the PBFT consensus mechanism. This model is refined not only as a queuing system with complicated service times but also as a level-independent quasi-birth-and-death (QBD) process. With regard to the level-independent QBD process, we apply the matrix-geometric solution to obtain the necessary and sufficient condition under which the PBFT consensus system is stable and then numerically compute the stationary probability vector of the QBD process. Thus, we provide four useful performance measures for the PBFT consensus mechanism, and we can numerically calculate these performance measures. Finally, we use numerical examples to verify the validity of our theoretical results and demonstrate how the four performance measures are influenced by certain key parameters of the PBFT consensus. Considering theory of multi-dimensional Markov processes, we are optimistic that the methodology and results presented in this study are applicable to a wide range of PBFT consensus mechanism and even other types of consensus mechanisms.

Robust geometric fitting is one of the crucial and fundamental problems in computer vision and pattern recognition. While random sampling and consensus maximization have been popular strategies for robust fitting, finding a balance between optimization quality and computational efficiency remains a persistent obstacle. In this paper, we adopt an optimization perspective and introduce a novel maximum consensus robust fitting algorithm that incorporates the maximum entropy framework into the consensus maximization problem. Specifically, we incorporate the probability distribution of inliers calculated using maximum entropy with consensus constraints. Furthermore, we introduce an improved relaxed and accelerated alternating direction method of multipliers (R-A-ADMMs) strategy tailored to our framework, facilitating an efficient solution to the optimization problem. Our proposed algorithm demonstrates superior performance compared to state-of-the-art methods on both synthetic and contaminated real datasets, particularly when dealing with contaminated datasets containing a high proportion of outliers.

We consider a set N = { 1 , … , n } of interacting agents whose individual opinions are denoted by x i , i ∈ N in some domain D ⊆ R . The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1). In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity μ , where the capacity value μ ( S ) of a coalition of agents S ⊆ N is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated with S . The capacity μ is obtained in terms of its 2-additive Möbius transform m μ , and the corresponding Shapley power and interaction indices are identified. We then discuss two types of consensus dynamics, both of which refer significantly to the notion of context opinion. The second type converges simply the plain mean, whereas the first type produces the Shapley mean as the asymptotic consensual opinion. In this way, it provides a dynamical realization of Shapley aggregation.

Failure mode and effects analysis is an effective and powerful risk evaluation technique in the field of risk management, and it has been extensively used in various industries for identifying and decreasing known and potential failure modes in systems, processes, products, and services. Traditionally, a risk priority number is applied to capture the ranking order of failure modes in failure mode and effects analysis. However, this method has several drawbacks and deficiencies, which need to be improved for enhancing its application capability. For instance, this method ignores the consensus-reaching process and the correlations among the experts’ preferences. Therefore, the aim of this study was to present a new risk priority method to determine the risk priority of failure modes under an interval-valued Pythagorean fuzzy environment, which combines the extended Geometric Bonferroni mean operator, a consensus-reaching process, and an improved Multi-Attributive Border Approximation area Comparison approach. Finally, a case study concerning product development is described to demonstrate the feasibility and effectiveness of the proposed method. The results show that the risk priority of failure modes obtained by the proposed method is more reasonable in practical application compared with other failure mode and effects analysis methods.

The evaluation of different Internet marketing channels from the perspective of organizations is important to gain understanding of managerial preferences and assist in decision-making. Few studies have examined multi-channel customer management. However, the evaluation of different Internet marketing channels in the context of organizations remains a challenge. This study focuses on identifying organizational preferences in marketing products for different Internet marketing channels. The authors have used a hybrid MCDM approach in their study. The study involves four steps of computation: calculating the relative importance of factors that are critical for different organizations in marketing their products on the Internet using fuzzy extension of DEMATEL, calculating priorities of users for each channel using fuzzy extension of AHP, reaching consensus achievement using ordinal consensus improvement approach through geometric ordinal consensus index (GOCI), and ranking different Internet marketing channel using TOPSIS. The approach is illustrated through a case study.