Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

We address the computation of the degrees of minors of a noncommutative symbolic matrix of form $$A[c] := \sum_{k=1}^m A_k t^{c_k} x_k,$$ where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Defining block character

In this paper I propose a clear, efficient, and accurate method for determining if a block of contiguous buildings has an overall character. The work is needed because most contemporary design reviews presuppose the existence of visual character, but existing design principles are often too vague to make the required determination. Clarity is achieved by shifting from vague notions to a definite concept for block character: a design feature will be perceived as part of the overall character of that block if the frequency of the feature is greater than a critical threshold. An experiment suggested that the critical frequency was quite high: over 80%. A case history illustrates how the new concept of visual character could greatly increase the efficiency and accuracy of actual planning decisions.

Configraphics:

This dissertation reports a PhD research on mathematical-computational models, methods, and techniques for analysis, synthesis, and evaluation of spatial configurations in architecture and urban design. Spatial configuration is a technical term that refers to the particular way in which a set of spaces are connected to one another as a network. Spatial configuration affects safety, security, and efficiency of functioning of complex buildings by facilitating certain patterns of movement and/or impeding other patterns. In cities and suburban built environments, spatial configuration affects accessibilities and influences travel behavioural patterns, e.g. choosing walking and cycling for short trips instead of travelling by cars. As such, spatial configuration effectively influences the social, economic, and environmental functioning of cities and complex buildings, by conducting human movement patterns. In this research, graph theory is used to mathematically model spatial configurations in order to provide intuitive ways of studying and designing spatial arrangements for architects and urban designers. The methods and tools presented in this dissertation are applicable in: arranging spatial layouts based on configuration graphs, e.g. by using bubble diagrams to ensure certain spatial requirements and qualities in complex buildings; and analysing the potential effects of decisions on the likely spatial performance of buildings and on mobility patterns in built environments for systematic comparison of designs or plans, e.g. as to their aptitude for pedestrians and cyclists. The dissertation reports two parallel tracks of work on architectural and urban configurations. The core concept of the architectural configuration track is the ‘bubble diagram’ and the core concept of the urban configuration track is the ‘easiest paths’ for walking and cycling. Walking and cycling have been chosen as the foci of this theme as they involve active physical, cognitive, and social encounter of people with built environments, all of which are influenced by spatial configuration. The methodologies presented in this dissertation have been implemented in design toolkits and made publicly available as freeware applications