Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.Comment: 13 page

Uncertainty quantification of coal seam gas production prediction using Polynomial Chaos

A surrogate model approximates a computationally expensive solver. Polynomial Chaos is a method to construct surrogate models by summing combinations of carefully chosen polynomials. The polynomials are chosen to respect the probability distributions of the uncertain input variables (parameters); this allows for both uncertainty quantification and global sensitivity analysis. In this paper we apply these techniques to a commercial solver for the estimation of peak gas rate and cumulative gas extraction from a coal seam gas well. The polynomial expansion is shown to honour the underlying geophysics with low error when compared to a much more complex and computationally slower commercial solver. We make use of advanced numerical integration techniques to achieve this accuracy using relatively small amounts of training data

Further biembeddings of twofold triple systems

We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ξ 16 or 28 (mod 48). The biembeddings come from index 1 current graphs lifted under a group ℤn/4 × 4

Distributive and anti-distributive Mendelsohn triple systems

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively

Identifying flaws in the security of critical sets in latin squares via triangulations

In this paper we answer a question in theoretical cryptography by reducing it to a seemingly unrelated geometrical problem. Drápal (1991) showed that a given partition of an equilateral triangle of side n into smaller, integer-sided equilateral triangles gives rise to, under certain conditions, a latin trade within the latin square based on the addition table for the integers (mod n). We apply this result in the study of flaws within certain theoretical cryptographic schemes based on critical sets in latin squares. We classify exactly where the flaws occur for an infinite family of critical sets. Using Drápal's result, this classification is achieved via a study of the existence of triangulations of convex regions that contain prescribed triangles

High-speed TIRF and 2D super-resolution structured illumination microscopy with a large field of view based on fiber optic components

Super-resolved structured illumination microscopy (SR-SIM) is among the most flexible, fast, and least perturbing fluorescence microscopy techniques capable of surpassing the optical diffraction limit. Current custom-built instruments are easily able to deliver two-fold resolution enhancement at video-rate frame rates, but the cost of the instruments is still relatively high, and the physical size of the instruments based on the implementation of their optics is still rather large. Here, we present our latest results towards realizing a new generation of compact, cost-efficient, and high-speed SR-SIM instruments. Tight integration of the fiber-based structured illumination microscope capable of multi-color 2D- and TIRF-SIM imaging, allows us to demonstrate SR-SIM with a field of view of up to 150 × 150 µm2 and imaging rates of up to 44 Hz while maintaining highest spatiotemporal resolution of less than 100 nm. We discuss the overall integration of optics, electronics, and software that allowed us to achieve this, and then present the fiberSIM imaging capabilities by visualizing the intracellular structure of rat liver sinusoidal endothelial cells, in particular by resolving the structure of their trans-cellular nanopores called fenestrations

On the upper embedding of Steiner triple systems and Latin squares

It is proved that for any prescribed orientation of the triples of either a Steiner triple system or a Latin square of odd order, there exists an embedding in an orientable surface with the triples forming triangular faces and one extra large face