Finite and infinitesimal rigidity with polyhedral norms

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R^d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R^d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R^2.Comment: 26 page

Controversial Orthodoxy: The Efficient Capital Markets Hypothesis And Loss Causation

Since the Supreme Court’s landmark holding in Basic, Inc. v. Levinson, courts have incorporated the efficient capital markets hypothesis as an analytical tool in securities fraud cases. Nevertheless, recent turmoil in the financial markets and a growing chorus of scholarship challenging traditional notions of market efficiency have caused some courts to reconsider the role of the efficient capital markets hypothesis in securities fraud litigation. This Note analyzes a question that has split the circuits and marks the intersection of market efficiency and securities fraud: how quickly must an equity security depreciate in price following the publication of a corrective disclosure for a plaintiff to plead and prove loss causation? Part I introduces the efficient capital markets hypothesis, securities fraud actions, and the ways in which courts have traditionally employed concepts of market efficiency into their analyses. Part II analyzes the circuit split regarding the speed with which the market must incorporate information into price for a plaintiff to properly plead and prove loss causation. Finally, Part III argues that courts should resist the temptation to draw bright-line rules in the context of loss causation and should engage each case on its facts by analyzing the efficiency of the relevant market during each event giving rise to the fraud and economic loss

Viability of endophytic fungus in different perennial ryegrass (Lolium perenne) varieties kept in different storage conditions : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science at Massey University, Manawatū, New Zealand

Epichloë endophytes form symbiotic relationships with cool-season grasses of the Pooideae family and are known to synthesise a range of bio-protective alkaloids. These alkaloids can provide the grass host with benefits for greater survival including; deterrence of herbivorous pests, increased persistence, better livestock health and protection from abiotic stressors. The commercialisation of novel endophytes is on the increase, and it is important to ensure the survival of the endophyte is maintained so their benefits can be realised. This study examined the effects of different storage conditions on the viability of three commercial novel endophytes (AR1, AR37 and NEA2/6) and one pre-commercial novel endophyte (815). The different storage conditions were the top of a warehouse, the bottom of a warehouse and a temperature and humidity controlled cool store to simulate current commercial seed storage environments. The viability of different endophytes decreases independently of grass seed germination (p = NS) however there are many factors influencing the endophyte survival. Over the one year storage period there were significant interactions between endophyte x ploidy (host), endophyte x location and endophyte x ploidy (host) x location. The pre-commercial endophyte, 815, had the largest reduction in viable endophyte when stored outside of the controlled cool store dropping 70 percentage points at the top of the warehouse, compared with AR37 (12 percentage points), AR1 (16 percentage points), and NEA2/6 (46 percentage points) (p<.001, LSD = 15.9). In the cool store there was no significant decrease in any of the treatments. As more novel endophyte/grass combinations are released for commercial sale it is important to test each for compatibility and performance post-storage. The results of this study recommend controlled low-temperature, low-humidity storage to maintain endophyte viability. Keywords: Endophyte, Epichloë, perennial ryegrass, Lolium perenne, storag

Maxwell-Laman counts for bar-joint frameworks in normed spaces

The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks where the unit ball is a quadrilateral.Comment: 17 page

Symmetric isostatic frameworks with ℓ1\ell^1 or ℓ∞\ell^\infty distance constraints

Combinatorial characterisations of minimal rigidity are obtained for symmetric 2-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.Comment: 20 pages. Main theorem extended. Construction schemes refined. New titl

The rigidity of infinite graphs

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit of an inclusion tower of finite graphs for which the inclusions satisfy a relative rigidity property. For d>2 a countable graph which is rigid for generic placements in R^d may fail the stronger property of sequential rigidity, while for d=2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.Comment: 51 page