Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

Optimization of sequential excavation method for large-section urban subway tunnel: A case study

In response to the requirement for population increase, the cross-section area of the urban subway is enlarged, resulting in that one-time excavation technology cannot be directly applied to the excavation of large-section tunnels. Consequently, how to partition the cross-section of the large-section tunnels and optimize the corresponding construction parameters is of great significance. In this paper, we establish a unified planar partition optimization model based on the four parameters of the number of horizontal layers, the number of transverse partitions, the height of the step, and the width of sections. Moreover, using the dynamic programming principle, we can further obtain the optimal excavation sequence and the construction parameters of the large-section tunnels by solving the planar partition optimization model. Combined with the case of an extra-large cross-section tunnel excavation of Chongqing Metro Central Park East Station, the paper optimizes the excavation method of the tunnel with the aim of the maximum construction efficiency and tunnel stability to obtain the optimal excavation sequence, the optimal construction parameters, and the optimal comprehensive evaluation index. The practice has proved that the optimization model based on the dynamic programming principle can effectively solve the problem of large-section tunnel construction. The case analysis can provide an effective reference for similar large-section tunnel projects.

TOWARDS A SCALE DEPENDENT FRAMEWORK FOR CREATING VARIO-SCALE MAPS

<p><strong>Abstract.</strong> Traditionally, the content for vario-scale maps has been created using a ‘one fits all’ approach equal for all scales. Initially only the delete/merge operation was used to create the vario-scale data using the importance and the compatibility functions defined at class level (and evaluated at instance level) to create the tGAP structure with planar partition as basis. In order to improve the generalization quality other operators and techniques have been added during the past years; e.g. simplify, collapse (change area to line representation), split, attractiveness regions and the introduction of the concept of linear network topology. However, the decision which operation to apply has been hard coded in our software, making it not very flexible. Further, we want to include awareness of the current scale when deciding what generalization operation to apply. For this purpose we propose the scale dependent framework (SDF), which at its core contains the encoding of the generalization knowledge in the SDF conceptual model. This SDF model covers the representation of scale dependent class importance, scale dependent class compatibility values, scale dependent attractiveness regions and last but not least specification of generalization operations that are scale and class dependent. By changing the settings in the SDF configuration and re-running the vario-scale generalization process, we can easily experiment in order to find best settings (for specific map user needs). In this paper we design the SDF conceptual model and explicitly motivate and define the scope of its expressiveness. We further present the improved scale dependent tGAP creation software and present initial results in the form of better created vario-scale map content.</p>

RSK Insertion for Set Partitions and Diagram Algebras

We give combinatorial proofs of two identities from the representation theory of the partition algebra ${\Bbb C} A_k(n)$, $n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$, $f^\lambda$ is the number of standard tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of "vacillating tableaux" of shape $\lambda$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_\lambda (m_k^\lambda)^2$, where $B(2k)$ is the number of set partitions of $\{1, \ldots, 2k\}$. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.