On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

Multilabel Video Classification Model of Navigation Mark’s Lights Based on Deep Learning

At night, buoys and other navigation marks disappear to be replaced by fixed or flashing lights. Navigation marks are seen as a set of lights in various colors rather than their familiar outline. Deciphering that the meaning of the lights is a burden to navigators, it is also a new challenging research direction of intelligent sensing of navigation environment. The study studied initiatively the intelligent recognition of lights on navigation marks at night based on multilabel video classification methods. To capture effectively the characteristics of navigation mark’s lights, including both color and flashing phase, three different multilabel classification models based on binary relevance, label power set, and adapted algorithm were investigated and compared. According to the experiment’s results performed on a data set with 8000 minutes video, the model based on binary relevance, named NMLNet, has highest accuracy about 99.23% to classify 9 types of navigation mark’s lights. It also has the fastest computation speed with least network parameters. In the NMLNet, there are two branches for the classifications of color and flashing, respectively, and for the flashing classification, an improved MobileNet-v2 was used to capture the brightness characteristic of lights in each video frame, and an LSTM is used to capture the temporal dynamics of lights. Aiming to run on mobile devices on vessel, the MobileNet-v2 was used as backbone, and with the improvement of spatial attention mechanism, it achieved the accuracy near Resnet-50 while keeping its high speed.

The number of set-orbits of a solvable permutation group

A permutation group 𝐺 acting on a set Ω induces a permutation action on the power set P ⁢ ( Ω ) \mathscr{P}(\Omega) (the set of all subsets of Ω). Let 𝐺 be a finite permutation group of degree 𝑛, and let s ⁢ ( G ) s(G) denote the number of orbits of 𝐺 on P ⁢ ( Ω ) \mathscr{P}(\Omega) . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | \log_{2}s(G)/{\log_{2}\lvert G\rvert} over all solvable groups 𝐺. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ p^{\prime} -degree irreducible representations of a solvable group.

Harnessing Multi-label Classification Approaches for Economic Phenomena Categorization

One fashion to report a country’s economic state is by compiling economic phenomena from several sources. The collected data may be explored based on their sentiments and economic categories. This research attempted to perform and analyze multiple approaches to multi-label text classification in addition to providing sentiment analysis on the economic phenomena. The sentiment and single-label category classification was performed utilizing the logistic regression model. Meanwhile, the multi-label category classification was fulfilled using a combination of logistic regression, support vector machines, k-nearest neighbor, naïve Bayes, and decision trees as base classifiers, with binary relevance, classifier chain, and label power set as the implementation approaches. The results showed that logistic regression works well in sentiment and single-label classification, with a classification accuracy of 80.08% and 92.71%, respectively. However, it was also discovered that it works poorly as a base classifier in multi-label classification, indicated by the classification accuracy dropping to 13.35%, 15.40%, and 30.65% for binary relevance, classifier chain, and label power set, respectively. Alternatively, naïve Bayes works best as a base classifier in the label power set approach for multi-label classification, with a classification accuracy of 63.22%, followed by decision trees and support vector machines.

Compliance of Distribution System Reactive Flows with Transmission System Requirements

Transmission system operators (TSOs) often set requirements to distribution system operators (DSOs) regarding the exchange of reactive power on the interface between the two parts of the system they operate, typically High Voltage and Medium Voltage. The presence of increasing amounts of Distributed Energy Resources (DERs) at the distribution networks complicates the problem, but provides control opportunities in order to keep the exchange within the prescribed limits. Typical DER control methods, such as constant cosϕ or Q/V functions, cannot adequately address these limits, while power electronics interfaced DERs provide to DSOs reactive power control capabilities for complying more effectively with TSO requirements. This paper proposes an optimisation method to provide power set-points to DERs in order to control the hourly reactive power exchanges with the transmission network. The method is tested via simulations using real data from the distribution substation at the Sundom Smart Grid, in Finland, using the operating guidelines imposed by the Finnish TSO. Results show the advantages of the proposed method compared to traditional methods for reactive power compensation from DERs. The application of more advanced Model Predictive Control techniques is further explored.

Entropic Explanation of Power Set

A power set of a set S is defined as the set of all subsets of S, including set S itself and empty set, denoted as P(S) or 2S. Given a finite set S with |S|=n hypothesis, one property of power set is that the amount of subsets of S is |P(S)| = 2n. However, the physica meaning of power set needs exploration. To address this issue, a possible explanation of power set is proposed in this paper. A power set of n events can be seen as all possible k-combination, where k ranges from 0 to n. It means the power set extends the event space in probability theory into all possible combination of the single basic event. From the view of power set, all subsets or all combination of basic events, are created equal. These subsets are assigned with the mass function, whose uncertainty can be measured by Deng entropy. The relationship between combinatorial number, Pascal's triangle and power set is revealed by Deng entropy quantitively from the view of information measure.

Online Learning Algorithms for the Real-Time Set-Point Tracking Problem

With the recent advent of technology within the smart grid, many conventional concepts of power systems have undergone drastic changes. Owing to technological developments, even small customers can monitor their energy consumption and schedule household applications with the utilization of smart meters and mobile devices. In this paper, we address the power set-point tracking problem for an aggregator that participates in a real-time ancillary program. Fast communication of data and control signal is possible, and the end-user side can exploit the provided signals through demand response programs benefiting both customers and the power grid. However, the existing optimization approaches rely on heavy computation and future parameter predictions, making them ineffective regarding real-time decision-making. As an alternative to the fixed control rules and offline optimization models, we propose the use of an online optimization decision-making framework for the power set-point tracking problem. For the introduced decision-making framework, two types of online algorithms are investigated with and without projections. The former is based on the standard online gradient descent (OGD) algorithm, while the latter is based on the Online Frank–Wolfe (OFW) algorithm. The results demonstrated that both algorithms could achieve sub-linear regret where the OGD approach reached approximately 2.4-times lower average losses. However, the OFW-based demand response algorithm performed up to twenty-nine percent faster when the number of loads increased for each round of optimization.