Nexus VII: Relationships Between Architecture and Mathematics

Sylvie Duvernoy reports on the seventh international, interdisciplinary Nexus conference for architecture and mathematics. Point Loma Nazarene University, San Diego, USA 23-25 June 2008     

Mathematics for/from Society: The Role of the Module in Modernizing Japanese Architectural Production

This paper presents an examination of the process of the development of module in the works and theories of Japanese architect Ikebe Kiyoshi (1920-79). Ikebe based his idea of module on the belief that “Beauty is Mathematics.” He applied his ideas of module in various ways from the 1940s to the 1970s. Analyzing his ideas and works against their historical background, the social and creative meanings of the idea of module and of mathematics in architecture will be re-examined. This allows us to see how Ikebe developed his ideas of module from a characteristic mathematical approach, and how he developed his idea of mathematical logic into his creative theories based on the flexible nature of people’s lifestyles and social conditions. Going beyond the cultural and social differences and the limitations of Le Corbusier’s Modulor, the idea of module as the method for organizing human space in a harmonious manner was reframed in Ikebe’s works, and was developed in a more flexible mathematical way.     

The Holy Alignment: Geodesic and Astronomical Fundamentals for Calculating the Adjustment of Medieval Naves

It is commonly believed that the longitudinal axes of churches extend exactly in an east-west direction. However, thorough investigations have shown that this is not always correct; rather, both southern and northern deviations of up to about 25° can occur. The angular deviation between the church axis and true east is called the Holy Alignment. This present study presents the possibility that the nave is oriented towards the direction of the sun-rising point on the name day of the patron saint of the church. If several saints share the patronage, the Holy Alignment equals the algebraic sum of the angular distances for each saint. The orientation of a nave can be analyzed by means of common mathematical relations used in geodesy, astronomy and gnomonics. In order to perform such an analysis, it is necessary to know the history of the patronage of the church; the Gauss-Krüger coordinates of the ground plan; and characteristic astronomical quantities at the time the church was built. A calculated example with the saints Andrew, James and Philip for the year 980 illustrates the analysis. An appendix deals with the influence of atmospheric refraction on the apparent altitude of the sun near the horizon.     

Controller synthesis with guaranteed closed-loop phase constraints

In this paper, we present an analysis and synthesis approach for guaranteeing that the phase of a single-input, single-output closed-loop transfer function is contained in the interval [−α,α] for a given α>0 at all frequencies. Specifically, we first derive a sufficient condition involving a frequency domain inequality for guaranteeing a given phase constraint. Next, we use the Kalman–Yakubovich–Popov theorem to derive an equivalent time domain condition. In the case where , we show that frequency and time domain sufficient conditions specialize to the positivity theorem. Furthermore, using linear matrix inequalities, we develop a controller synthesis approach for guaranteeing a phase constraint on the closed-loop transfer function. Finally, we extend this synthesis approach to address mixed gain and phase constraints on the closed-loop transfer function.     

Controller certification

A modern problem from aerospace control involves the certification of a large set of potential controllers with either a single plant or a fleet of potential plant systems, with both plants and controllers being MIMO and, for the moment, linear. Experiments on a limited number of controller/plant pairs should establish the stability and a certain level of margin of the complete set. We consider this certification problem for a set of controllers and provide algorithms for selecting an efficient subset for testing. This is done for a finite set of candidate controllers and, at least for SISO plants, for compact infinite set. In doing this, the ν-gap metric will be the main tool. Computational examples are given, including one of certification of an aircraft engine controller. The overarching aim is to introduce truly MIMO margin calculations and to understand their efficacy in certifying stability over a set of controllers and in replacing legacy single-loop gain and phase margin calculations.     

Application of Cayley-Klein parameters in the problem of strapdown navigation system stochastic state vector synthesis for a self-contained space vehicle

The state vector and observer equations are obtained for a strapdown (gimballess) navigation system with three accelerometers, three angular rate sensors, and a barometric altimeter. These equations are based on the nonlinear filtration methods that ensure required estimation accuracy under arbitrary space maneuver of a vehicle.     

Mathematical models of formation of rational organizational hierarchies

A survey of Russian and foreign papers devoted to one of the burning trends of the theory of control of social and economic systems, i.e., mathematical models of formational of rational organization hierarchies is made. The following lines of investigations are emphasized: multilevel symmetrical hierarchies, hierarchies of knowledge, multilevel hierarchies of data processing, hierarchies and command theory, hierarchies of decision-making, hierarchies and contracts theory, and general model of optimal hierarchies search.     

Estimating parameters of the synchronous twofold-stochastic flow of events

Consideration was given to estimation of the parameters of the synchronous twofold-stochastic flow of events which makes up a mathematical model of the information flow of demands circulating in the queuing systems and networks. Two variants were studied. For the first variant, the problem of optimal estimation of the parameters of a synchronous flow of events with a finite arbitrary number of states was solved. The second variant differs in that the event flow operates in the environment where part of events is lost during the so-called dead time. Consideration was given to a synchronous flow with two states. The problem of estimating the length of the dead time in the conditions of continued dead time was solved. The results of numerical estimation based on the simulation model of the synchronous flow were presented for both variants.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

Adaptive suboptimal robust control of the first-order plant

For a linear first-order plant, consideration was given to minimization of the deviation of its output from the specified value in an uncertain environment. Both the parameters of the plant itself and the upper bounds of the external disturbances and operator perturbations in output and control were assumed to be unknown. The suboptimal adaptive control was constructed on the basis of an approximate solution of the problem of optimal identification from the measurement data where the performance of the problem of control was used as an identification criterion.