Department of Mathematics and Computer Studies
https://dspace.mic.ul.ie/handle/10395/34
Wed, 17 Apr 2024 10:14:46 GMT2024-04-17T10:14:46ZWhich graphs are rigid in lpd?
https://dspace.mic.ul.ie/handle/10395/3037
Which graphs are rigid in lpd?
We present three results which support the conjecture that a graph is minimally rigid in d-dimensional ℓp-space, where p∈(1,∞) and p≠2, if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from ℓdp to ℓd+1p. We then prove that every (d, d)-sparse graph with minimum degree at most d+1 and maximum degree at most d+2 is independent in ℓdp. Finally, we prove that every triangulation of the projective plane is minimally rigid in ℓ3p. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.
Sat, 13 Mar 2021 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/30372021-03-13T00:00:00ZThe stability space of the derived category of holomorphic triples and further investigations
https://dspace.mic.ul.ie/handle/10395/2978
The stability space of the derived category of holomorphic triples and further investigations
In this thesis we give a complete description of the Bridgeland stability space for the bounded derived category of holomorphic triples over a smooth projective curve of genus one as a connected, four dimensional complex manifold.
We will then prove a number of helpful facts that characterise the bounded derived category of holomorphic triples and will subsequently generalise some of the results on the stability space of the bounded derived category of holomorphic triples to that of holomorphic chains.
Wed, 14 Apr 2021 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/29782021-04-14T00:00:00ZConstructing isostatic frameworks for the l1 and l infinity plane (Pre-published)
https://dspace.mic.ul.ie/handle/10395/2964
Constructing isostatic frameworks for the l1 and l infinity plane (Pre-published)
We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph G = (V,E) has a partition into two spanning trees T1 and T2 then there is a map p : V → R2, p(v) = (p(v)1,p(v)2), such that |p(u)i −p(v)i| > |p(u)3−i−p(v)3−i| for every edge uv in Ti (i = 1,2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the `1 or `∞-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.
Fri, 12 Jun 2020 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/29642020-06-12T00:00:00ZGraph rigidity for unitarily invariant matrix norms (Pre-published)
https://dspace.mic.ul.ie/handle/10395/2963
Graph rigidity for unitarily invariant matrix norms (Pre-published)
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of -sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.
Sun, 15 Nov 2020 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/29632020-11-15T00:00:00Z