Discrete Differential Geometry

Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session

Spectral characterizations of complex unit gain graphs

While eigenvalues of graphs are well studied, spectral analysis of complex unit gain graphs is still in its infancy. This thesis considers gain graphs whose gain groups are gradually less and less restricted, with the ultimate goal of classifying gain graphs that are characterized by their spectra. In such cases, the eigenvalues of a gain graph contain sufficient structural information that it might be uniquely (up to certain equivalence relations) constructed when only given its spectrum. First, the first infinite family of directed graphs that is – up to isomorphism – determined by its Hermitian spectrum is obtained. Since the entries of the Hermitian adjacency matrix are complex units, these objects may be thought of as gain graphs with a restricted gain group. It is shown that directed graphs with the desired property are extremely rare. Thereafter, the perspective is generalized to include signs on the edges. By encoding the various edge-vertex incidence relations with sixth roots of unity, the above perspective can again be taken. With an interesting mix of algebraic and combinatorial techniques, all signed directed graphs with degree at most 4 or least multiplicity at most 3 are determined. Subsequently, these characterizations are used to obtain signed directed graphs that are determined by their spectra. Finally, an extensive discussion of complex unit gain graphs in their most general form is offered. After exploring their various notions of symmetry and many interesting ties to complex geometries, gain graphs with exactly two distinct eigenvalues are classified

Resonant Tunnelling Diodes for Millimetre and Sub-Millimetre Wave Mixing Applications

The primary intention of this research work was to evaluate a topology for a sub-harmonic down conversion mixer exploiting the fourth harmonic of a LO signal. Designs were evaluated by simulation at 640GHz and 320GHz with the aim of exploring the potential of a RTD based down-converter at 640GHz, in the 580-750GHz atmospheric window, with an intermediate frequency signal in the range around 2GHz by mixing with the fourth harmonic of a 159.5GHz LO signal. Related design studies were undertaken at 320GHz which gave a simulated single side band (SSB) conversion loss of 5.7dB, and with a LO power requirement of less than -9.5dBm which vindicated the principle, as far as the design stage is concerned, of using RTDs as the non-linear mixing element, where the layer design can be tailored to favour very low pump powers. The other, related, target of the current PhD work was to also explore the potential for high LO drive level mixers and their up-conversion efficiencies using the same novel devices, i.e. RTDs, but with a different layer design, better suited to support high pump powers in this instance. For achieving the latter goal, two different sub-harmonic up-conversion mixers employing a single RTD and using the second harmonic of an LO signal were designed and evaluated at two different frequencies. The first mixer design was aimed at 180 GHz providing -7.5dBm of output power while the second one should work at 110GHz showing output power in the range of -4dBm, and was used to initially evaluate the approach and which could, in principle, be later fabricated and measured. All these down and up-conversion mixers were carefully designed using ADS and HFSS and evaluated using two different technologies, microstrip and Grounded Coplanar Waveguide (GCPW), and both compared with a nearest Schottky diode based approaches, and also their physical mask was produced in anticipation of a later fabrication stage

Binding energies of interstellar molecules on crystalline and amorphous models of water ice by ab-initio calculations

In the denser and colder ($\leq$20 K) regions of the interstellar medium (ISM), near-infrared observations have revealed the presence of sub-micron sized dust grains covered by several layers of H\textsubscript{2}O-dominated ices and dirtied by the presence of other volatile species. Whether a molecule is in the gas or solid-phase depends on its binding energy (BE) on ice surfaces. Thus, BEs are crucial parameters for the astrochemical models that aim to reproduce the observed evolution of the ISM chemistry. In general, BEs can be inferred either from experimental techniques or by theoretical computations. In this work, we present a reliable computational methodology to evaluate the BEs of a large set (21) of astrochemical relevant species. We considered different periodic surface models of both crystalline and amorphous nature to mimic the interstellar water ice mantles. Both models ensure that hydrogen bond cooperativity is fully taken into account at variance with the small ice cluster models. Density functional theory adopting both B3LYP-D3 and M06-2X functionals was used to predict the species/ice structure and their BE. As expected from the complexity of the ice surfaces, we found that each molecule can experience multiple BE values, which depend on its structure and position at the ice surface. A comparison of our computed data with literature data shows agreement in some cases and (large) differences in others. We discuss some astrophysical implications that show the importance of calculating BEs using more realistic interstellar ice surfaces to have reliable values for inclusion in the astrochemical models.Comment: 30 pages (including Appendix), 16 figures (including Appendix). To be published in Astrophysical Journa

Contributions to the Minimum Linear Arrangement Problem

The Minimum Linear Arrangement problem (MinLA) consists in finding an ordering of the nodes of a weighted graph, such that the sum of the weighted edge lengths is minimized. We report on the usefulness of a new model within a branch-and-cut-and-price algorithm for solving MinLA problems to optimality. The key idea is to introduce binary variables d_{ijk}, that are equal to 1 if nodes i and j have distance k in the permutation. We present formulations for complete and for sparse graphs and explain the realization of a branch-and-cut-and-price algorithm. Furthermore, its different settings are discussed and evaluated. To the study of the theoretical aspects concerning the MinLA, we contribute a characterization of a relaxation of the corresponding polyeder

Design of the Synthetic Aperture Microwave Imager-2 for measurement of the edge current density on MAST-U

The Synthetic Aperture Microwave Imager-2 (SAMI-2) is a 2D Doppler backscattering (DBS) diagnostic designed for multiple high quality simultaneous measurements of the edge pitch angle on the Mega-Ampere Spherical Tokamak Upgrade (MAST-U). The specification, design and testing of the SAMI-2 microwave front end, predominantly antennas and mixer circuitry, are described in this thesis. Dual-polarisation sinuous antennas are designed and shown to fulfil their performance criteria over the 20-40 GHz range. A four-channel microwave in-phase and quadrature (IQ) downconverter is designed to mix the radio frequency (RF) signals from each antenna in the SAMI-2 phased array to a lower frequency, for digitisation. The down-converter is tested and found to achieve its performance specifications over the entire frequency range of the antennas. The 2D DBS technique was demonstrated by Synthetic Aperture Microwave Imager (SAMI) on the Mega-Ampere Spherical Tokamak (MAST) to measure the edge magnetic pitch angle. A radial profile of the edge pitch angle enables calculation of the edge current density, a difficult quantity to measure, which is valuable for the validation of models and understanding of pedestal dynamics and edge plasma instabilities, e.g. ELMs. In active probing mode, the SAMI-2 diagnostic is designed to make the first measurements of the edge current density by a DBS diagnostic. In passive mode, SAMI-2 will measure Bernstein wave mode conversion, to inform spherical tokamak microwave heating systems

Quantum Gates for Quantum Dots

Since the mid-20th century it has been understood that a general-purpose quan- tum computer would be able to efficiently solve problems that will forever be out-of-reach for conventional computers. Since then, many quantum algorithms have been developed with applications in a wide range of domains including cryptography, simulations, machine learning and data analysis. While this has resulted in substantial attention being paid to the development of quantum com- puters, the best architectures to use in their fabrication is not yet clear. Semiconductor quantum dot devices are a particularly promising candidate for use in quantum computing architectures, as it is anticipated that once the funda- mental building blocks are implemented, they might be massively scalable using the existing lithography techniques of the semiconductor industry. So far, how- ever, it is not yet clear how best to implement the high-fidelity gates required for general-purpose quantum computation. In this thesis, we present and characterise novel theoretical proposals for fast, simple and high-fidelity two-qubit gates using magnetic (exchange) coupling for specific semiconductor quantum dot qubits; namely, the singlet-triplet and resonant-exchange qubits. These two-qubit operations are simple enough that it is feasible for them to be implemented in experiments of the near future. Success- ful implementations would significantly extend the experimentally demonstrable frontier of semi-conductor quantum dot devices as relevant to their use in uni- versal quantum computing architectures. We also develop simple parameter estimation schemes by which it is possible to substantially mitigate the dominant sources of error for our proposed gates; namely, low-frequency charge and magnetic noise. We develop the techniques in the context of pseudo-static magnetic field gradient fluctuations in singlet- triplet qubits, and demonstrate that these techniques lead to a several orders of magnitude improvement in single-qubit coherence times. With minimal effort this could be ported to other qubit architectures

EM-driven miniaturization of high-frequency structures through constrained optimization

The trends afoot for miniaturization of high-frequency electronic devices require integration of active and passive high-frequency circuit elements within a single system. This high level of accomplishment not only calls for a cutting-edge integration technology but also necessitates accommodation of the corresponding circuit components within a restricted space in applications such as implantable devices, internet of things (IoT), or 5G communication systems. At the same time, size reduction does not remain the only demand. The performance requirements of the abovementioned systems form a conjugate demand to that of the size reduction, yet with a contrasting nature. A compromise can be achieved through constrained numerical optimization, in which two kinds of constrains may exist: equality and inequality ones. Still, the high cost of electromagnetic-based (EM-based) constraint evaluations remains an obstruction. This issue can be partly mitigated by implicit constraint handling using the penalty function approach. Nevertheless, securing its performance requires expensive guess-work-based identification of the optimum setup of the penalty coefficients. An additional challenge lies in allocating the design within or in the vicinity of a thin feasible region corresponding to equality constraints. Furthermore, multimodal nature of constrained miniaturization problems leads to initial design dependency of the optimization results. Regardless of the constraint type and the corresponding treatment techniques, the computational expenses of the optimization-based size reduction persist as a main challenge. This thesis attempts to address the abovementioned issues specifically pertaining to optimization-driven miniaturization of high frequency structures by developing relevant algorithms in a proper sequence. The first proposed approach with automated adjustment of the penalty functions is based on the concept of sufficient constraint violation improvement, thereby eliminating the costly initial trial-and-error stage for the identification of the optimum setup of the penalty factors. Another introduced approach, i.e., correction-based treatment of the equality constraints alleviates the difficulty of allocating the design within a thin feasible region where designs satisfying the equality constraints reside. The next developed technique allows for global size reduction of high-frequency components. This approach not only eliminates the aforementioned multimodality issues, but also accelerates the overall global optimization process by constructing a dimensionality-reduced surrogate model over a pre-identified feasible region as compared to the complete parameter search space. Further to the latter, an optimization framework employing multi-resolution EM-model management has been proposed to address the high cost issue. The said technique provides nearly 50 percent average acceleration of the optimization-based miniaturization process. The proposed technique pivots upon a newly-defined concept of model-fidelity control based on a combination of algorithmic metrics, namely convergence status and constraint violation level. Numerical validation of the abovementioned algorithms has also been provided using an extensive set of high-frequency benchmark structures. To the best of the author´s knowledge, the presented study is the first investigation of this kind in the literature and can be considered a contribution to the state of the art of automated high-frequency design and miniaturization

Robust and Optimal Methods for Geometric Sensor Data Alignment

Geometric sensor data alignment - the problem of finding the rigid transformation that correctly aligns two sets of sensor data without prior knowledge of how the data correspond - is a fundamental task in computer vision and robotics. It is inconvenient then that outliers and non-convexity are inherent to the problem and present significant challenges for alignment algorithms. Outliers are highly prevalent in sets of sensor data, particularly when the sets overlap incompletely. Despite this, many alignment objective functions are not robust to outliers, leading to erroneous alignments. In addition, alignment problems are highly non-convex, a property arising from the objective function and the transformation. While finding a local optimum may not be difficult, finding the global optimum is a hard optimisation problem. These key challenges have not been fully and jointly resolved in the existing literature, and so there is a need for robust and optimal solutions to alignment problems. Hence the objective of this thesis is to develop tractable algorithms for geometric sensor data alignment that are robust to outliers and not susceptible to spurious local optima. This thesis makes several significant contributions to the geometric alignment literature, founded on new insights into robust alignment and the geometry of transformations. Firstly, a novel discriminative sensor data representation is proposed that has better viewpoint invariance than generative models and is time and memory efficient without sacrificing model fidelity. Secondly, a novel local optimisation algorithm is developed for nD-nD geometric alignment under a robust distance measure. It manifests a wider region of convergence and a greater robustness to outliers and sampling artefacts than other local optimisation algorithms. Thirdly, the first optimal solution for 3D-3D geometric alignment with an inherently robust objective function is proposed. It outperforms other geometric alignment algorithms on challenging datasets due to its guaranteed optimality and outlier robustness, and has an efficient parallel implementation. Fourthly, the first optimal solution for 2D-3D geometric alignment with an inherently robust objective function is proposed. It outperforms existing approaches on challenging datasets, reliably finding the global optimum, and has an efficient parallel implementation. Finally, another optimal solution is developed for 2D-3D geometric alignment, using a robust surface alignment measure. Ultimately, robust and optimal methods, such as those in this thesis, are necessary to reliably find accurate solutions to geometric sensor data alignment problems

Small business innovation research. Abstracts of 1988 phase 1 awards

Non-proprietary proposal abstracts of Phase 1 Small Business Innovation Research (SBIR) projects supported by NASA are presented. Projects in the fields of aeronautical propulsion, aerodynamics, acoustics, aircraft systems, materials and structures, teleoperators and robots, computer sciences, information systems, data processing, spacecraft propulsion, bioastronautics, satellite communication, and space processing are covered