Hull operators in bounded posets

Hull operators and interval operators in (L,M)-fuzzy convex spaces

The arity of convex spaces is a numerical feature which shows the ability of finite subsets spanning to the whole space via the hull operators. This paper gives it a formal and strict definition by introducing the truncation of convex spaces. The relations that between the arity of quotient spaces and the original spaces, that between the arity of subspaces and superspaces, that between the arity of product spaces and factors spaces, and that between the arity of disjoint sums and term spaces, are systematically studied. A mistake of a formula in [M. Van De Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993] is corrected. It is shown that a convex space is Alexandrov iff its arity is 1. The convex structures with arity ≤n are equivalent to structured sets with n-restricted hull operators.

In this paper, we introduce (quasi-)consistent spaces and (quasi-)adjacent spaces to characterize convexity spaces. Firstly, we show that convexity spaces can be characterized by quasi-consistent spaces. They can be induced by each other. In particular, each convexity space can be quasi-consistentizable. Every quasi-consistency $\mathcal{U}$ can induce two hull operators and thus determine different convexities $\mathcal{C}^{\mathcal{U}}$ and $\mathcal{C}_{\mathcal{U}}$. And $\mathcal{C}^{\mathcal{U}}=\mathcal{C}_{\mathcal{U}}$ holds when $\mathcal{U}$ is a consistency. Secondly, we use quasi-adjacent spaces to characterize convexity spaces. Each convexity space can be quasi-adjacentizable. In both of characterizations of convexity, remotehood systems play an important role in inducing convexity. Finally, we show there exists a close relation between a quasi-consistency and a quasi-adjacency. Furthermore, there exists a one-to-one correspondence between a quasi-adjacency and a fully ordered quasi-consistency. And we deeply study the relationships among these structures.

Unmanned Sea Surface Vehicle (USSV) motion simulation in time domain is an important component of USSV design and operation. This capability is needed for hull design, operator training, controller synthesis and testing. Many applications such as simulators for operator training require real-time performance. Traditional approaches based on strip theory, although fast, abstract hulls into slender bodies, making the simulation results insensitive to changes in geometry. Computational fluid dynamics (CFD) based approaches are accurate but very slow. Potential flow based approaches are sensitive to hull geometry and take lesser amount of time than CFD based approaches. The motion simulation using potential flow theory involves following four main operations: (1) computation of dynamic pressure head due to fluid flow around the hull under the ocean wave, (2) computation of wet surface, (3) computing surface integral of dynamic pressure head over wet surface, and (4) solving the rigid body dynamics equation. First three operations depend upon the boat geometry complexity and need to be performed at each time step, making the simulation run very slow. In this paper, we investigate the problem of model simplification for real-time simulation of USSV model in time domain using potential flow theory, with arbitrary geometry under ocean waves with inviscid and irrotational flow. Using clustering based simplification scheme and parallel computing we obtained real time simulation performance.

Exact Set-valued Estimation using Constrained Convex Generators for uncertain Linear Systems

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

Flatbed scanners are commonly used for root analysis, but typical manual segmentation methods are time-consuming and prone to errors, especially in large-scale, multi-plant studies. Furthermore, the complex nature of root structures combined with noisy backgrounds in images complicates automated analysis. Addressing these challenges, this article introduces RhizoNet, a deep learning-based workflow to semantically segment plant root scans. Utilizing a sophisticated Residual U-Net architecture, RhizoNet enhances prediction accuracy and employs a convex hull operation for delineation of the primary root component. Its main objective is to accurately segment root biomass and monitor its growth over time. RhizoNet processes color scans of plants grown in a hydroponic system known as EcoFAB, subjected to specific nutritional treatments. The root detection model using RhizoNet demonstrates strong generalization in the validation tests of all experiments despite variable treatments. The main contributions are the standardization of root segmentation and phenotyping, systematic and accelerated analysis of thousands of images, significantly aiding in the precise assessment of root growth dynamics under varying plant conditions, and offering a path toward self-driving labs.

A new channel coding approach was proposed in [1] for random multiple access communication over the discrete-time memoryless channel. The coding approach allows users to choose their communication rates independently without sharing the rate information among each other or with the receiver. The receiver will either decode the message or report a collision depending on whether reliable message recovery is possible. It was shown that, asymptotically as the codeword length goes to infinity, the set of communication rates supporting reliable message recovery can be characterized by an achievable region which equals Shannon's information rate region possibly without a convex hull operation. In this paper, we derive achievable bounds on error probabilities, including the decoding error probability and the collision miss detection probability, of random multiple access systems with a finite codeword length. Achievable error exponents are obtained by taking the codeword length to infinity.

Some Modifications of Hull Operators in Archimedean Lattice-Ordered Groups with Weak Unit