Mathematics

We present here a novel approach to the analysis of common knowledge based on Category Theory. We formalize knowledge hierarchies as presheaves over a category of agent sequences. The category of these presheaves constitutes a topos. We define an unfolding monad on the resulting topos, and use a Knaster–Tarski theorem to obtain common knowledge as a greatest fixed point under natural uniformity and exchangeability conditions on agent sequences.

When the resources for experimentation are limited, experimenters usually turn to the class of 2-level fractional factorial designs. Resolution III fractional factorial designs are the smallest available designs, but they alias main effects and 2-factor interactions. The class of Resolution IV designs avoids this and provides clear estimates of the main effects, assuming that 3-factor and higher-order interactions are not active. Meanwhile, some two-factor interactions remain aliased with each other. Resolution V designs have no aliasing of main effects and two-factor interactions, assuming that all higher-order interactions are inactive. However, they are often too large for situations with six or more factors of interest. For example, with six factors, the only design capable of estimating all main effects and 2-factor interactions has 32 runs. Consequently, resource restrictions often require experimenters to use smaller designs of lower resolution, typically Resolution IV. The aliasing of effects often requires additional follow-up experimentation to de-alias all active effects. However, there are situations in which follow-up experiments are impossible to perform due to the unavailability of certain test resources. An alternative to using a 16-run Resolution IV design and a follow-up experiment is to use a design with more than 16 runs as the initial experiment. We investigated a strategy for initially augmenting a class of 16-run Resolution IV designs with either 4 or 8 runs. We use a simulation study to show that this augmentation strategy improves the ability to estimate active factors when standard analysis methods are employed. The analysis methods used in this study are Stepwise, LASSO, and Dantzig.

The rapid proliferation of electric vehicles (EVs) has introduced significant challenges to the efficient operation of hydrogen-containing integrated energy systems (H-IESs). To cope with these challenges, this paper develops a bi-level optimal scheduling strategy for H-IESs that simultaneously incorporates a ladder-type carbon emission trading mechanism, demand response, and the operational characteristics of EVs. A demand response model is formulated by considering the coupling characteristics of electric and thermal loads. Price-based incentive signals are further designed to coordinate the interactions between the H-IES operator and EV users, enabling flexible resources to actively participate in system scheduling. In the proposed bi-level framework, the upper-level problem aims to minimize the total operating cost of the H-IES, while the lower-level problem seeks to reduce the charging cost of EV users. The resulting bi-level optimization problem is reformulated and solved using the Karush–Kuhn–Tucker (KKT) conditions. Case study results demonstrate that, compared with the single-level benchmark, the proposed bi-level strategy reduces the total operating cost by 34.79% and lowers the EV charging cost by 4.50%.