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We give a complete classification of all nonuniform congruences on orthogonal rook monoids and symplectic rook monoids. We find that there are four kinds of nonuniform congruences on the orthogonal rook monoids ${OR}_n$ for even $n\ne 4$, and we describe each kind of the congruences explicitly in terms of normal subgroups of maximal subgroups. We also find that if $n = 4$, there are six kinds of nonuniform congruences on ${OR}_4$, and we describe these congruences using both $\mathcal{H}$-relations and certain normal subgroups of some maximal subgroups. In contrast, we find that there is only one kind of congruences on the symplectic rook monoids for all even $n\ge 2.$

In this paper, we enumerate lattice paths with certain constraints and apply the corresponding results to develop formulas for calculating the dimensions of submodules of a class of modules for planar upper triangular rook monoids. In particular, we show that the famous Catalan numbers appear as the dimensions of some special modules; we also obtain some combinatorial identities, Comment: arXiv admin note: substantial text overlap with arXiv:1610.04122

In this paper, an overcurrent protection system is designed to ensure the safety of the gyrotrons in the EAST ECRH system. Two overcurrent protection systems were established, a fast one and a slow one. The fast one uses the current transformers as the current sensors. The models of the current transformers and the superconducting magnet were built to analyze the effect of the environmental magnetic field on the current transformers using FI method. The analysis results show that the magnetic induction at the position near the current transformers must less than 0.002 T, i.e., the current transformers should be placed at a distance greater than 2.2 meters from the magnet center to ensure its normal work. The slow one uses the shunt to monitor the currents. An anti-fuse FPGA and a timer is used to realize the signal processing in the fast protection circuit and the slow protection circuit respectively. The response time of the fast protection circuit is less than 100 ns, and the response time of the slow protection circuit is less than 31 {\mu}s., Comment: 8 pages, 11 figures

In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule of V. In particular, we provide a recursive formula for calculating the dimension of the cyclic module generated by v_S, and show that some dimensions are Catalan numbers, giving rise to new combinatorial identities.