The Other Mathematical Bridge

This paper contextualises, describes and discusses a student project which takes a particular exploratory approach to using mathematical surface definition as a language and vehicle for co-rational design co-authorship for architecture and engineering. The project has two authors, one from an architectural and one from an engineering educational background. It investigates the metaphorical and operational role of mathematics in the design process and outcomes.     

Two- and Three-Dimensional Constructions Based on Leonardo Grids

In 1989 I made a drawing of a net on a cube, consisting of 12 lines/elements. They were connected in a way that, a couple of months later, I recognised them in 899v in Leonardo’s Codex Atlanticus. I don’t know which moment impressed me the most: my own discovery of a very simple and powerful connecting system or the discovery of the Leonardo drawings, which implied that my own discovery was in fact a rediscovery. What we see in Leonardo’s drawings are some examples of roof constructions built with a lot of straight elements. These drawings can be ‘translated’ into the following definition: On each element we define four points at some distance of each other – two points somewhere in the middle and two points closer to the ends. To make constructions with these elements we need only connect a middle point of one element to an end point of another one in a regular over-under pattern. Out of the simple definition of the elements, I designed many different patterns for my so-called “+ - - +” structures: domes, spheres, cylinders and other models were made.     

Gaining insight through case-based explanation

Traditional explanation strategies in machine learning have been dominated by rule and decision tree based approaches. Case-based explanations represent an alternative approach which has inherent advantages in terms of transparency and user acceptability. Case-based explanations are based on a strategy of presenting similar past examples in support of and as justification for recommendations made. The traditional approach to such explanations, of simply supplying the nearest neighbour as an explanation, has been found to have shortcomings. Cases should be selected based on their utility in forming useful explanations. However, the relevance of the explanation case may not be clear to the end user as it is retrieved using domain knowledge which they themselves may not have. In this paper the focus is on a knowledge-light approach to case-based explanations that works by selecting cases based on explanation utility and offering insights into the effects of feature-value differences. In this paper we examine to two such a knowledge-light frameworks for case-based explanation. We look at explanation oriented retrieval (EOR) a strategy which explicitly models explanation utility and also at the knowledge-light explanation framework (KLEF) that uses local logistic regression to support case-based explanation.

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     

Inverse NP Problems

The inverse problem relative to a verifier V of proofs of membership for a NP language is the problem of deciding, given a set π of proofs, whether or not there exists a string x having exactly π as its set of proofs. In this paper, we study the complexity of inverse problems. We develop a new notion of reduction which allows one to compare the complexity of inverse problems. Using this notion, we classify as coNP-complete the inverse problems for the “natural” verifiers of many NP-complete problems. We also show that the inverse complexity of a verifier for a language L cannot be predicted solely from the complexity of L, but rather, is highly dependent upon the choice of verifier used to accept L. In this context, a verifier with a Σ₂ ^p -complete inverse problem is exhibited, giving a new and natural example of a Σ₂ ^p -complete problem.      

Functional diagnosis of the nonlinear dynamic delay systems

Consideration was given to the problem of functional diagnosis of the nonlinear dynamic delay systems. Its solution implies design of redundancy relations whose verification underlies conclusions about system operability and the kind of system defect. For the systems obeying differential equations with a polynomial right-hand side, a method was proposed for designing the redundancy relations. It admits that some or all constant coefficients of the polynomials are unknown.     

Analysis of the oral interface in the interactive servicing systems. II

A classification of the dialogue control algorithms at oral interaction of the client with the computer-aided servicing system was worked out. The most typical algorithms were described, analyzed, and compared in terms of duration using the formulas derived in the first part of the paper.     

Design of discrete control system of flexible spacecraft maintaining robust stability of elastic oscillations

For the flexible spacecraft with a nonlinear orientation control system using flywheel engines, an approach to making its elastic oscillations robust stable was considered. It relies on a purposeful variation of the boundaries of the stability domains in the parameter space of the spacecraft and controller with the aim of maximizing the number of robust stable elastic modes of the flexible spacecraft. To make its control stable in the large, consideration was given to the possibility of expanding the basic algorithm by adding the sum of the components using the estimated coordinates of the remaining series of low-frequency unstable modes. Presented was an example of mathematical modeling of the proposed spacecraft orientation system which corroborated operability of the approach making the multifrequency spacecraft stable to the elastic oscillations.     

Output control of linear systems with unmodeled dynamics

Recently, an output control law for linear parametrically uncertain plants was proposed by one of the authors of this paper. Here, the feasibility of this approach is substantiated in the presence of unmodeled asymptotically stable dynamics. A theorem is proved which formulates the feasibility conditions for the control law for systems with unmodeled dynamics.     

Tracking in linear MIMO systems under external perturbations

Consideration was given to tracking the given trajectories of the output variables of the linear multiple-input multiple-output dynamic systems under uncontrollable external perturbations. A procedure to rearrange the mathematical model of a control plant in a consistent block form of controllability and observability of the output (measurable) variables with regard for the external perturbations was developed within the framework of the block approach. This form underlies the decomposition procedures of feedback design enabling one to track the given trajectories invariantly to the external perturbations.     

Linear control design under phase constraints

A new design method is proposed for linear stabilizing controllers which satisfy the given constraints on the phase variables and control input and minimize the upper bound on the maximum deviation of the plant output. The method is based on the apparatus of linear matrix inequalities; the theory is exemplified via the linear shock absorber design.     

Detection and estimation of the motion velocity vector of a inhomogeneous spatially extended object by image. I

In the paper, we study maximally plausible algorithms for detection and estimation of the motion velocity vector of image of a inhomogeneous spatially extended object for an applicative model of interaction of the useful image and background. Asymptotic expressions for characteristics of synthesized algorithms are obtained by using a method of local additive approximation of logarithm of likelihood ratio functional.     

Transformations of control systems for the investigation of pulse modes

It is considered a scheme of transformation of control system with unbounded velocity hodograph to a system of reduced order (called the derived system) via integral of the limit system which describes behavior of the original one under pulse (practically sufficiently large) control inputs. There are formulated the conditions (connected with controllability of the limit system on its integral manifold) when some solution of the derived system will be a generalized solution of the original one that is it can be approximated by a sequence of regular solutions. Methodical and applied examples are given.     

Guaranteed results in games with forbidden situations

The best guaranteed results in games where “others’” control actions influence not only the utility function but also the admissibility set of each participant are analyzed and compared. The research is conducted without a formal reduction to classical games on direct products of rigid admissibility sets by means of penalized extension of the utility function. By the example of the generalized Cournot duopoly, it is shown that, for a chosen participant, his right to be the first to use common scarce resources might be more important than awareness of previous choices of his competitors. This means the possibility of violating the fundamental inequality of the classical game theory which states that the max-min of the utility function does not exceed its min-max, established for games on direct products of sets of admissible control actions.     

Control of motion of the autonomous underwater vehicle at trajectory survey of the oceanic physical fields

Consideration was given to the control of motion of the autonomous underwater vehicle performing automatic measurements and mapping of the oceanic physical fields. In the general case, it is required to set up a control maintaining motion along the field isolines relying on the trajectory measurements of the level and field gradient components and identification and contouring of the anomalies by the characteristic points and generalized reference points. A practical example based on the experience of bathymetrical and hydrophysical measurements in the Arctic region using an autonomous underwater vehicle was discussed.     

Method to determine an optimal control in the minimum of resource consumption for the nonlinear stationary systems

An iterative method to solve the nonlinear problems of minimization of resource consumption was proposed. It generalizes to a more general class of nonlinear systems the method of resource minimization for the nonlinear systems with the control-linear right-hand side separated in state and control.