arXiv subjects

George Wilson

Publications and source records attributed to George Wilson.

10 recordsLinked to original sources

arXiv2018

An adaptive self-organizing fuzzy logic controller in a serious game for motor impairment rehabilitation

Rehabilitation robotics combined with video game technology provides a means of assisting in the rehabilitation of patients with neuromuscular disorders by performing various facilitation movements. The current work presents ReHabGame, a serious game using a fusion of implemented technologies that can be easily used by patients and therapists to assess and enhance sensorimotor performance and also increase the activities in the daily lives of patients. The game allows a player to control avatar movements through a Kinect Xbox, Myo armband and rudder foot pedal, and involves a series of reach-grasp-collect tasks whose difficulty levels are learnt by a fuzzy interface. The orientation, angular velocity, head and spine tilts and other data generated by the player are monitored and saved, whilst the task completion is calculated by solving an inverse kinematics algorithm which orientates the upper limb joints of the avatar. The different values in upper body quantities of movement provide fuzzy input from which crisp output is determined and used to generate an appropriate subsequent rehabilitation game level. The system can thus provide personalised, autonomously-learnt rehabilitation programmes for patients with neuromuscular disorders with superior predictions to guide the development of improved clinical protocols compared to traditional theraputic activities.

cs.HC↗

arXiv2018

Development of Rehabilitation System (ReHabgame) through Monte-Carlo Tree Search Algorithm

Computational Intelligence (CI) in computer games plays an important role that could simulate various aspects of real-life problems. CI in real-time decision-making games can provide a platform for the examination of tree search algorithms. In this paper, we present a rehabilitation serious game (ReHabgame) in which the Monte-Carlo Tree Search (MCTS) algorithm is utilized. The game is designed to combat the physical impairment of post-stroke/brain injury casualties in order to improve upper limb movement. Through the process of ReHabgame the player chooses paths via upper limb according to his/her movement ability to reach virtual goal objects. The system adjusts the difficulty level of the game based on the player's quality of activity through MCTS. It learns from the movements made by a player and generates further subsequent objects for collection. The system collects orientation, muscle and joint activity data and utilizes them to make decisions. Players data are collected through Kinect Xbox One and Myo Armband. The results show the effectiveness of the MCTS in the ReHabgame that progresses from highly achievable paths to the less achievable ones, thus configuring and personalizing the rehabilitation process.

cs.HC↗

arXiv2015

Notes on the vector adelic Grassmannian

These are notes about the connection between the Gibbons-Hermsen system and the multi-component KP hierarchy: they were intended as a first sketch of a paper on that subject, but since it seems unlikely that I will ever write such a paper I have decided to make them available in their present form. The notes were compiled in 2009, and have not been revised since then.

math.SG↗

arXiv2010

Picard groups of differential operators on curves

Let X be a complex affine curve (not isomorphic to the affine line), and let Pic(D) be the group of autoequivalences of the category of D(X)-modules. Cannings and Holland have shown that Pic(D) fits into an exact sequence in which the other groups can be considered known: we show that this sequence is split.

math.QA↗

arXiv2010

Equivariant maps between Calogero-Moser spaces

This is a footnote to earlier joint work with Yu. Berest, which constructed a bijection between the space of ideal classes of the Weyl algebra and a union of Calogero-Moser varieties. A key property of this bijection is that it is equivariant with respect to the action of the automorphism group of the Weyl algebra: the main result of the present note is that it is uniquely determined by that property.

math.QA↗

arXiv2008

Differential operators on an affine curve: ideal classes and Picard groups

Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration γ: R \to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of γinto Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define γin the framework of the Grassmannian description of R due to Cannings and Holland.

math.QA↗

arXiv2006

Mad Subalgebras of Rings of Differential Operators on Curves

We study the maximal abelian ad-nilpotent (mad) subalgebras of the domains D Morita equivalent to the first Weyl algebra. We give a complete description both of the individual mad subalgebras and of the space of all such. A surprising consequence is that this last space is independent of D. Our results generalize some classic theorems of Dixmier about the Weyl algebra.

math.QA↗

arXiv2003

Differential Isomorphism and Equivalence of Algebraic Varieties

In this survey article we discuss the question: to what extent is an algebraic variety determined by its ring of differential operators? In the case of affine curves, this question leads to a variety of mathematical notions such as the Weyl algebra, Calogero-Moser spaces and the adelic Grassmannian. We give a fairly detailed overview of this material.

math.AG↗

arXiv2001

Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh)

Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map $θ: R \to C$ by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that $θ$ is inverse to a bijection $ω: C \to R$ constructed in \cite{BW} by a completely different method. The main step in the proof is to show that $θ$ is equivariant with respect to natural actions of the group G=Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.

math.AG↗

arXiv2001

Automorphisms and Ideals of the Weyl Algebra

Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \cite{W1}. We also make some comments on the group $G$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.

math.QA↗