Akhet Khufu: Archaeo-astronomical Hints at a Common Project of the Two Main Pyramids of Giza, Egypt

The architectural complexes composed by the two main pyramids of Giza together with their temples are investigated from an interdisciplinary point of view, taking into account their astronomical alignments as well as their relationships with the visible landscape. Combining already known facts together with new clues, the work strongly supports the idea that the two complexes were conceived as parts of a common project.     

Continuity versus Discretization

The threefold interest in architecture, biology and mathematics motivated us to examine and justify new architectural forms. We discuss some notions of rhythm: Euclidean, morphogenetic and morphologic. Contemporary relationships between structure and form are based on the generation of shape by technological processes, thus the resulting objects are restricted to their material expression. Here a phenomenological organisation of form and its continuity with the landscape arise out of the mathematical and architectural creativity. The use of the computer is applied from outside to inside instead from inside to outside; this means that we are dealing with the organisational processes via continuous methods instead of evolutionary processes given by computer simulations, known as genetic algorithms, where the resulting configurations are reduced to a matter of routine. The role of design as an aesthetic component innovates the theoretical framework of structural engineering to establish the architectural environments.     

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.     

Optimum design of long-span suspension bridges considering aeroelastic and kinematic constraints

Cable supported bridges are wind prone structures. Therefore, their aerodynamic behaviour must be studied in depth in order to guarantee their safe performance. In the last decades important achievements have been reached in the study of bridges under wind-induced actions. On the other hand, non-conventional design techniques such as sensitivity analysis or optimum design have not been applied although they have proved their feasibility in the automobile or aeronautic industries. The aim of this research work is to demonstrate how non-conventional design techniques can help designers when dealing with long span bridges considering their aeroelastic behaviour. In that respect, the comprehensive analytical optimum design problem formulation is presented. In the application example the optimum design of the challenging Messina Strait Bridge is carried out. The chosen initial design has been the year 2002 design proposal. Up to a 33% deck material saving has been obtained after finishing the optimization process.     

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.     

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.