The rigidity of countable frameworks in normed spaces

We present a rigorous study of framework rigidity in finite dimensional normed spaces using a wide array of tools to attack these problems, including differential and discrete geometry, matroid theory, convex analysis and graph theory. We shall first focus on giving a good grounding of the area of rigidity theory from a more general view point to allow us to deal with a variety of normed spaces. By observing orbits of placements from the perspective of Lie group actions on smooth manifolds, we obtain upper bounds for the dimension of the space of trivial motions for a framework. Utilising aspects of differential geometry, we prove an extension of Asimow and Roth’s 1978/9 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we establish the independence of all graphs with d + 1 vertices d-dimensional normed space, and also prove they will be flexible if the normed space is non-Euclidean. Next, we prove that a graph has an infinitesimally rigid placement in a nonEuclidean normed plane if and only if it contains a (2, 2)-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph K4 by considering smoothness and strict convexity properties of the unit ball. Finally, we carry our previous results to countably infinite frameworks where this is possible, and otherwise identify when such results cannot be brought forward. We first establish matroidal methods for identifying rigidity and flexibility, and apply these methods to a large class of normed spaces. We characterise a necessary and sufficient condition for countably infinite graphs to have sequentially infinitesimally rigid placements in a general normed plane, and further stengthen the result for a large class normed planes. Finally, we prove that infinitesimal rigidity for countably infinite generic frameworks implies a weaker (but possibly equivalent) form of continuous rigidity, and infinitesimal rigidity for countably infinite algebraically generic frameworks implies continuous rigidity

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Muscle activation mapping of skeletal hand motion: an evolutionary approach.

Creating controlled dynamic character animation consists of mathe- matical modelling of muscles and solving the activation dynamics that form the key to coordination. But biomechanical simulation and control is com- putationally expensive involving complex di erential equations and is not suitable for real-time platforms like games. Performing such computations at every time-step reduces frame rate. Modern games use generic soft- ware packages called physics engines to perform a wide variety of in-game physical e ects. The physics engines are optimized for gaming platforms. Therefore, a physics engine compatible model of anatomical muscles and an alternative control architecture is essential to create biomechanical charac- ters in games. This thesis presents a system that generates muscle activations from captured motion by borrowing principles from biomechanics and neural con- trol. A generic physics engine compliant muscle model primitive is also de- veloped. The muscle model primitive forms the motion actuator and is an integral part of the physical model used in the simulation. This thesis investigates a stochastic solution to create a controller that mimics the neural control system employed in the human body. The control system uses evolutionary neural networks that evolve its weights using genetic algorithms. Examples and guidance often act as templates in muscle training during all stages of human life. Similarly, the neural con- troller attempts to learn muscle coordination through input motion samples. The thesis also explores the objective functions developed that aids in the genetic evolution of the neural network. Character interaction with the game world is still a pre-animated behaviour in most current games. Physically-based procedural hand ani- mation is a step towards autonomous interaction of game characters with the game world. The neural controller and the muscle primitive developed are used to animate a dynamic model of a human hand within a real-time physics engine environment