Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8

On the Symmetries of Integrability

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.Comment: 23 page

Fourier Transforms

The 21st century ushered in a new era of technology that has been reshaping everyday life, simplifying outdated processes, and even giving rise to entirely new business sectors. Today, contemporary users of products and services expect more and more personalized products and services that can meet their unique needs. In that sense, it is necessary to further develop existing methods, adapt them to new applications, or even discover new methods. This book provides a thorough review of some methods that have an increasing impact on humanity today and that can solve different types of problems even in specific industries. Upgrading with Fourier Transformation gives a different meaning to these methods that support the development of new technologies and have a good projected acceleration in the future

Dispersed reference interferometry for on-machine metrology

The reliance of emerging engineering and scientific applications on nanometre scale surfaces has led to the requirement for embedded metrology instrumentation for on-machine measurement in precision and ultra-precision manufacturing processes. In-situ measurement provides reduction in waste, cost and production cycle time of manufactured components however the presence of environmental noise and difficulties of integration make embedded metrology challenging. In this thesis a review of on-machine methods of optical metrology is provided and commercially available instruments are summarised. The requirement for highly miniaturised methods of distance and surface topography measurement for operation within volume limited manufacturing environments is presented and the necessity for smaller, faster probes with increased range, resolution and robustness outlined. The potential for dispersed reference interferometry (DRI) to exceed the capability of existing single-point remote fibre probing instruments is then introduced. DRI as a technique for single-point distance and surface topography measurement is first evaluated in a bulk optics configuration using a short coherence light source and chromatic dispersion within the reference arm. The resulting spectral interferograms have symmetry positions which are directly related to absolute surface position to yield measurement with a 279 nm axial resolution over a 285 μm axial range. Improvement of this resolution results from introduction of template matching, a signal processing technique which correlates a measured interferogram with a set of pre-calculated template interferograms resulting in a relative method of measurement with an axial resolution of 0.6 nm across the 285 μm axial range. Combination of this high resolution, relative measurement method with low resolution but absolute position data is then explored to improve the robustness of DRI to discontinuous and structured surfaces. Determination of high resolution wraparound order using low resolution absolute data is demonstrated over a 30 μm range, with extension to the full range of DRI expected as future improvements increase the measurement rate. Next a DRI topology is introduced which makes use of miniature common-path probes linked by fibre to a remote interrogation interferometer allowing reduction in size of the on-machine metrology apparatus. Modifications to the DRI are described to enable this common-path remote fibre probing, along with changes to the light source, spectrometer and dispersive element which allow extension of the range of DRI to 800 μm while maintaining nanometre axial resolution. Finally, a further work section offers insights into methods to improve miniaturisation of DRI probes as well as providing discussion of methods of hardware and signal processing optimisation to augment the instrument measurement rate