For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln‖(z−T)−1x‖ln‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved ...that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓp(N), then Λ(T)={1} when ke0=1, where {en}n=0∞ be the canonical basis for ℓp(N); (2) There is a class of backward unilateral weighted shifts on ℓp(N) whose power set is 0,1; (3) There exists a bilateral weighted shift on ℓp(Z) with power set 12,1.
For a quasinilpotent operator T on a separable Hilbert space H, Douglas and Yang defined kx=limsup|z|→0log‖(z−T)−1x‖log‖(z−T)−1‖ for x∈H, x≠0, and called Λ(T)={kx:x∈H,x≠0} the power set of T. This ...paper computes the power set of cyclic quasinilpotent operators and studies the connection with their invariant subspaces. We first prove that the power set of the strictly cyclic quasinilpotent unilateral weighted shift only contains 1. Then we define the strongly strictly cyclic quasinilpotent operator, characterize its invariant subspaces, and show that its power set also only contains 1. Finally, we show that the Volterra integral operator V on L20,1, whose power set contains more than one point, is cyclic but not strictly cyclic.
Entropic Explanation of Power Set Song, Yutong; Deng, Yong
International journal of computers, communications & control,
08/2021, Volume:
16, Issue:
4
Journal Article
Peer reviewed
Open access
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.
In this paper we provide arguments supporting the idea that module theory in the symmetric, monoidal closed category of complete lattices and join-preserving maps is a mathematical framework for ...fuzzy set theory as developed by Zadeh. In this context, quantale-enriched category theory is a well established mathematical technique allowing an internal development of various concepts from fuzzy set theory – e.g. the fuzzy power set, Zadeh's forward operator or the fuzzy inclusion order.
For a GL-monoid L with a complete lattice L underneath, this paper considers the L-unbalanced power object monad on the category Set(L) of L-sets, and introduces a new monad on Set(Lop) called the ...L-power L-set monad. The Kleisli and Eilenberg-Moore categories of these monads are studied. Both monads are enhanced to partially ordered monads that allow us to apply the theory of topological space objects developed by Ulrich Höhle. We give the explicit descriptions of the category of topological space objects in Set(L) with respect to the partially ordered L-unbalanced power object monad and the category of topological space objects in Set(Lop) with respect to the partially ordered L-power L-set monad. Furthermore, it is shown that the L-power L-set monad on Set(Lop) determines a monad on Set(L) in a one-to-one manner, and all existing results can be directly converted into this monad.
In this study, a general framework for implementing a retail energy market based on the Nikaido–Isoda/relaxation algorithm is proposed as an electricity market structure with large distributed energy ...resources (DERs) penetration and demand side management of consumers. Moreover, the consumers are able to participate in the market as prosumers (i.e. producer and consumer at the same time). By considering the related uncertainties, the DERs can maximise their expected payoff or profit by undertaking strategies through the price bidding strategy, based on the proposed structure, considering Nash equilibrium. The results show the effectiveness and accuracy of the proposed framework in determining the optimal power set-points of players participating in the market to achieve the objectives.
•We propose a new DEA model for the semi-additive production technology.•Proposed model decreases the computational complexity of using the semi-additive production technology.•Proposed model is ...linear programming based.•Number of variables shrinks from 2n to n.
The semi-additive production technology allows for the aggregation of production units in the process of efficiency analysis using data envelopment analysis (DEA). This means that one must consider not only the full set of decision-making units (DMUs), but as well the entire power set, that is the set of all aggregations of the set of DMUs. Specifically, for n DMUs the number of variables involved is 2n, hence requiring one to solve a set of linear programming problems, each of size O(2n). In a big data sense, such a problem becomes prohibitive as n increases. The current paper proposes a new model that significantly decreases the computational complexity of models based on the semi-additive production technology. We show that the proposed semi-additive methodology allows the number of variables to be decreases from2nto n, and the complexity of the algorithm to be reduced from O(2n)to O(n). Three numerical and graphical examples are presented to illustrate the methodology.
The power setΛ(V) of quasinilpotent operator V on a Hilbert space H is defined in 4 to study the singularity of the non-Euclidean metrics ‖(V−z)−1x‖2dz⊗dz¯ at σ(V)={0}, and it is shown that if Λ(V) ...contains more than one point then V has a nontrivial hyperinvariant subspace. This paper first proves that the Volterra integral operator on the classical Hardy–Hilbert space has singleton power set, thus answering a question raised in 4. Then, it studies the length of circles under the metrics and its connection with power set. In particular, it determines the maximal length of the unit circle with respect to the change of x in the metrics. Moreover, it shows that an extremal value integral equation is able to detect representing functions for all the invariant subspaces of the Volterra operator.
This paper analyses the influence of the variation of meteorological and operational parameters on estimation of the power output of a dispatchable wind farm (WF). The active power set-points ...(APSPs), established to regulate the wind farm power output (WFPO), are one of the analysed parameters. The WF considered as case study is integrated in the Gorona del Viento wind-hydro power plant (El Hierro-Canary Islands-Spain), which supplies the primary energy demand of the island.
Statistical inference between the errors generated by different WFPO estimation models, each fed with different input features, is performed. These models are based on supervised machine learning (ML) regression algorithms, namely support vector regression, random forest, and a combination of the strengths of these two base learning algorithms constructed using stacked regression ensemble techniques. From the results obtained, the following conclusions are drawn: a) There is a marked difference between the errors obtained with the model that only considers wind speed and direction and that which additionally incorporates the APSP parameter, showing the importance of considering this particular parameter; b) the model which incorporates air density and turbulence intensity in addition to the APSP improves the values of all the metrics, independently of the ML technique employed.
•The active power set-point (APSP) parameter is very important in WFPO estimation.•Air density, turbulence intensity and APSP parameters improve WFPO estimation.•Eight models are compared for wind farm power output (WFPO) estimation.•Three machine learning techniques implemented in the models are analysed.•Non-parametric statistical inference is used to compare the models.
Background
The complexity of laparoscopy requires special training and assessment. Analyzing the streaming videos during the surgery can potentially improve surgical education. The tedium and cost of ...such an analysis can be dramatically reduced using an automated tool detection system, among other things. We propose a new multilabel classifier, called LapTool-Net to detect the presence of surgical tools in each frame of a laparoscopic video.
Methods
The novelty of LapTool-Net is the exploitation of the correlations among the usage of different tools and, the tools and tasks—i.e., the context of the tools' usage. Towards this goal, the pattern in the co-occurrence of the tools is utilized for designing a decision policy for the multilabel classifier based on a Recurrent Convolutional Neural Network (RCNN), which is trained in an end-to-end manner. In the post-processing step, the predictions are corrected by modeling the long-term tasks’ order with an RNN.
Results
LapTool-Net was trained using publicly available datasets of laparoscopic cholecystectomy, viz., M2CAI16 and Cholec80. For M2CAI16, our exact match accuracies (when all the tools in one frame are predicted correctly) in online and offline modes were 80.95% and 81.84% with per-class F1-score of 88.29% and 90.53%. For Cholec80, the accuracies were 85.77% and 91.92% with F1-scores if 93.10% and 96.11% for online and offline, respectively.
Conclusions
The results show LapTool-Net outperformed state-of-the-art methods significantly, even while using fewer training samples and a shallower architecture. Our context-aware model does not require expert’s domain-specific knowledge, and the simple architecture can potentially improve all existing methods.
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