10 research outputs found
On the spectra of nonsymmetric Laplacian matrices
A Laplacian matrix is a square real matrix with nonpositive off-diagonal
entries and zero row sums. As a matrix associated with a weighted directed
graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized
Laplacian matrix is a Laplacian matrix with the absolute values of the
off-diagonal entries not exceeding 1/n, where n is the order of the matrix. We
study the spectra of Laplacian matrices and relations between Laplacian
matrices and stochastic matrices. We prove that the standardized Laplacian
matrices are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues
of a standardized Laplacian matrix are equal to the in-forest dimension of the
corresponding digraph and one less than the in-forest dimension of the
complementary digraph, respectively. These eigenvalues are semisimple. The
spectrum of a standardized Laplacian matrix belongs to the meet of two closed
disks, one centered at 1/n, another at 1-1/n, each having radius 1-1/n, and two
closed angles, one bounded with two half-lines drawn from 1, another with two
half-lines drawn from 0 through certain points. The imaginary parts of the
eigenvalues are bounded from above by 1/(2n) cot(pi/2n); this maximum converges
to 1/pi as n goes to infinity.
Keywords: Laplacian matrix; Laplacian spectrum of graph; Weighted directed
graph; Forest dimension of digraph; Stochastic matrixComment: 11 page
Coordination in multiagent systems and Laplacian spectra of digraphs
Constructing and studying distributed control systems requires the analysis
of the Laplacian spectra and the forest structure of directed graphs. In this
paper, we present some basic results of this analysis partially obtained by the
present authors. We also discuss the application of these results to
decentralized control and touch upon some problems of spectral graph theory.Comment: 15 pages, 2 figures, 40 references. To appear in Automation and
Remote Control, Vol.70, No.3, 200
Crafting chaos: computational design of contraptions with complex behaviour
The 2010s saw the democratisation of digital fabrication technologies. Although this phenomenon made fabrication more accessible, physical assemblies displaying a complex behaviour are still difficult to design. While many methods support the creation of complex shapes and assemblies, managing a complex behaviour is often assumed to be a tedious aspect of the design process. As a result, the complex parts of the behaviour are either deemed negligible (when possible) or managed directly by the software, without offering much fine-grained user control. This thesis argues that efficient methods can support designers seeking complex behaviours by increasing their level of control over these behaviours. To demonstrate this, I study two types of artistic devices that are particularly challenging to design: drawing machines, and chain reaction contraptions. These artefacts’ complex behaviour can change dramatically even as their components are moved by a small amount. The first case study aims to facilitate the exploration and progressive refinement of complex patterns generated by drawing machines under drawing-level user-defined constraints. The approach was evaluated with a user study, and several machines drawing the expected pattern were fabricated. In the second case study, I propose an algorithm to optimise the layout of complex chain reaction contraptions described by a causal graph of events in order to make them robust to uncertainty. Several machines optimised with this method were successfully assembled and run. This thesis makes the following contributions: (1) support complex behaviour specifications; (2) enable users to easily explore design variations that respect these specifications; and (3) optimise the layout of a physical assembly to maximise the probability of real-life success
Constraint-Based Graphic Statics - A geometrical support for computer-aided structural equilibrium design
This thesis introduces “constraint-based graphic statics”, a geometrical support for computer-aided structural design. This support increases the freedom with which the designer interacts with the plane static equilibriums being shaped. Constraint-based graphic statics takes full advantage of geometry, both its visual expressiveness and its capacity to solve complex problems in simple terms. Accordingly, the approach builds on the two diagrams of classical graphic statics: a form diagram describing the geometry of a strut-and-tie network and a force diagram vectorially representing its inner static quilibrium. Two new devices improve the control of these diagrams: (1) nodes — considered as the only variables — are constrained within Boolean combinations of graphical regions; and (2) the user modifies these diagrams by means of successive operations whose geometric properties do not at any time jeopardise the static equilibrium of the strut-and-tie network. These two devices offer useful features, such as the ability to describe, constrain and modify any static equilibrium using purely geometric grammar, the ability to compute and handle multiple solutions to a problem at the same time, the ability to switch the hierarchy of constraint dependencies, the ability to execute dynamic conditional statements graphically, the ability to compute full interdependency and therefore the ability to remove significantly the limitations of compass-and-straightedge constructions and, finally the ability to propagate some solution domains symbolically. As a result, constraint-based graphic statics encourages the emergence of new structural design approaches that are highly interactive, precognitive and chronology-free: highly interactive because forces and geometries are simultaneously and dynamically steered by the designer; precognitive because the graphical region constraining each points marks out the set of available solutions before they are even explored by the user; and chronology-free because the deductive process undertaken by the designer can be switched whenever desired
Approaches to the Use of Geometry in Architecture: A study of the works of Andrea Palladio, Frank Lloyd Wright, and Frank Gehry
Geometry deals with form, shape, and measurement and is a part of mathematics
where visual thought is dominant. Both design and construction in architecture deal with
visualization, and architects constantly employ geometry. Today, with the advent of
computer software, architects can visualize forms that go beyond our everyday
experience. Some architects claim that the complex forms of their works have
correlations with non-Euclidean geometry, but the space we experience is still
Euclidean. Given this context, I have explored possible correlations that might exist
between mathematical concepts of geometry and the employment of geometry in
architectural design from a historic perspective. The main focus will be to describe the
two phenomena historically, and then investigate any connections that might emerge
from the discussion. While discussing the way geometry has been approached in
architecture, I have focused on the Renaissance, Modern, and Post-modern phases as
they have a distinct style and expression. Andrea Palladio, Frank Lloyd Wright, and Frank Gehry's works will be case studies for the Renaissance, Modern, and Post-modern
phases respectively.
One of the important conclusions of this study is that architects use geometry in a
more subconscious and intuitive manner while designing. Certain approaches to
geometry can be determined by the way an architect deals with form and space. From
the discussions of the works of Palladio, Wright, and Gehry, it can be concluded that
from a two-dimensional simple approach to form and space in architecture, there has
been a development of thinking about complex forms three dimensionally. Similarly, in
mathematics, geometry has developed from a two-dimensional and abstract description
of our surroundings to something that can capture the complex and specific nature of a
phenomena. It is also shown that architects rarely come up with new concepts of
geometry. Significant developments in geometry have always been in the domain of
mathematics. Hence, most correlations between geometry in architecture and geometry
in mathematics develop much later than the introduction of those concepts of geometry
in mathematics. It is also found that the use of Euclidean geometry persists in
architecture and that later concepts like non-Euclidean geometry cannot be used in an
instrumental manner in architecture
Proceedings of the 12th International Conference on Technology in Mathematics Teaching ICTMT 12
Innovation, inclusion, sharing and diversity are some of the words that briefly and suitably characterize the ICTMT series of biennial international conferences – the International Conference
on Technology in Mathematics Teaching. Being the twelfth of a series which began in Birmingham,
UK, in 1993, under the influential enterprise of Professor Bert Waits from Ohio State University,
this conference was held in Portugal for the first time. The 12th International Conference on
Technology in Mathematics Teaching was hosted by the Faculty of Sciences and Technology of the
University of Algarve, in the city of Faro, from 24 to 27 June 2015, and was guided by the original
spirit of its foundation.
The integration of digital technologies in mathematics education across school levels and countries,
from primary to tertiary education, together with the understanding of the phenomena involved in
the teaching and learning of mathematics in technological environments have always been driving
forces in the transformation of pedagogical practices. The possibility of joining at an international
conference a wide diversity of participants, including school mathematics teachers, lecturers,
mathematicians, mathematics educators and researchers, software designers, and curriculum
developers, is one facet that makes this conference rather unique. At the same time, it seeks to foster
the sharing of ideas, experiences, projects and studies while providing opportunities to try-out and
assess tools or didactical proposals during times of hands-on work. The ICTMT 12 had this same
ambition, when embracing and welcoming just over 120 delegates who actively and enthusiastically
contributed to a very packed program of scientific proposals and sessions on various topics
The art and architecture of mathematics education: a study in metaphors
This chapter presents the summary of a talk given at the Eighth European Summer University, held in Oslo in 2018. It attempts to show how art, literature, and history, can paint images of mathematics that are not only useful but relevant to learners as they can support their personal development as well as their appreciation of mathematics as a discipline. To achieve this goal, several metaphors about and of mathematics are explored