Entangled graphs on surfaces in space

In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation

Sliceforms: Deployable structures from interlocking slices

A sliceform is a volumetric, honeycomb-like structure assembled from an array of cross-sectional planar slices that are interlocked via pairs of complementary slots placed along each intersection. If the slices are thin, these slotted intersections function as revolute joints, and the sliceform is foldable if the geometry of the embedded spatial linkage permits it, for example a lattice sliceform (LS) is bi-directionally flat-foldable. This thesis concerns a study of such sliceforms toward the design of novel deployable structures. A sliceform torus, composed of two sets of inclined slices arranged at regular intervals about a central axis of symmetry, has been discovered to exhibit a surprising and intriguing folding action whereby its incomplete form can be collapsed to a flat-folded stack of coplanar slices. On deployment, the assembly expands smoothly about an arc until the slices have rotated to their design inclination, then, without reaching any apparent physical limit, abruptly ‘locks out’. With a full complement of slices, the outermost intersections can be interlocked to complete and rigidify the ring. The torus is an example of a rotational sliceform (RS), and analysis of these structures proceeds by noting that their structural geometry comprises an array of pyramidal cells that is commensurate to a spherical scissor grid. The conditions for flat-foldability are determined by examination of the intrinsic geometry of each cell; the incompatibility of the slices with apparent rigid-folding revealed by assessment of the extrinsic motion of the slices. Investigation of their compliant kinematics reveals the articulation to be a bistable transition admitted by small transverse deflections of the slices. This structural form is generalised by development of a technique for generating sliceforms along a smooth spatial curve – curve sliceforms (CS). Their synthesis is more involved than for an RS, but a range of sliceform ‘tubes’ are generated and manufactured. Each example retains the flat-foldable, deployable characteristic of an RS, despite the apparent intrinsic rigidity of each constituent skew cell. Examination of the small-scale models indicates that deployable motion is achieved via imperfect action of the slots, and a simple model of the articulation of a single cell is constructed to investigate how this proceeds, verifying that motion is kinematically admissible via local deformations

Exploiting geometry, topology, and optimization for knowledge discovery in big data

2013 Summer.Includes bibliographical references.In this dissertation, we consider several topics that are united by the theme of topological and geometric data analysis. First, we consider an application in landscape ecology using a well-known vector quantization algorithm to characterize and segment the color content of natural imagery. Color information in an image may be viewed naturally as clusters of pixels with similar attributes. The inherent structure and distribution of these clusters serves to quantize the information in the image and provides a basis for classification. A friendly graphical user interface called Biological Landscape Organizer and Semi-supervised Segmenting Machine (BLOSSM) was developed to aid in this classification. We consider four different choices for color space and five different metrics in which to analyze our data, and results are compared. Second, we present a novel topologically driven clustering algorithm that blends Locally Linear Embedding (LLE) and vector quantization by mapping color information to a lower dimensional space, identifying distinct color regions, and classifying pixels together based on both a proximity measure and color content. It is observed that these techniques permit a significant reduction in color resolution while maintaining the visually important features of images. Third, we develop a novel algorithm which we call Sparse LLE that leads to sparse representations in local reconstructions by using a data weighted 1-norm regularization term in the objective function of an optimization problem. It is observed that this new formulation has proven effective at automatically determining an appropriate number of nearest neighbors for each data point. We explore various optimization techniques, namely Primal Dual Interior Point algorithms, to solve this problem, comparing the computational complexity for each. Fourth, we present a novel algorithm that can be used to determine the boundary of a data set, or the vertices of a convex hull encasing a point cloud of data, in any dimension by solving a quadratic optimization problem. In this problem, each point is written as a linear combination of its nearest neighbors where the coefficients of this linear combination are penalized if they do not construct a convex combination, revealing those points that cannot be represented in this way, the vertices of the convex hull containing the data. Finally, we exploit the relatively new tool from topological data analysis, persistent homology, and consider the use of vector bundles to re-embed data in order to improve the topological signal of a data set by embedding points sampled from a projective variety into successive Grassmannians

Small-world interconnection networks for large parallel computer systems

The use of small-world graphs as interconnection networks of multicomputers is proposed and analysed in this work. Small-world interconnection networks are constructed by adding (or modifying) edges to an underlying local graph. Graphs with a rich local structure but with a large diameter are shown to be the most suitable candidates for the underlying graph. Generation models based on random and deterministic wiring processes are proposed and analysed. For the random case basic properties such as degree, diameter, average length and bisection width are analysed, and the results show that a fast transition from a large diameter to a small diameter is experienced when the number of new edges introduced is increased. Random traffic analysis on these networks is undertaken, and it is shown that although the average latency experiences a similar reduction, networks with a small number of shortcuts have a tendency to saturate as most of the traffic flows through a small number of links. An analysis of the congestion of the networks corroborates this result and provides away of estimating the minimum number of shortcuts required to avoid saturation. To overcome these problems deterministic wiring is proposed and analysed. A Linear Feedback Shift Register is used to introduce shortcuts in the LFSR graphs. A simple routing algorithm has been constructed for the LFSR and extended with a greedy local optimisation technique. It has been shown that a small search depth gives good results and is less costly to implement than a full shortest path algorithm. The Hilbert graph on the other hand provides some additional characteristics, such as support for incremental expansion, efficient layout in two dimensional space (using two layers), and a small fixed degree of four. Small-world hypergraphs have also been studied. In particular incomplete hypermeshes have been introduced and analysed and it has been shown that they outperform the complete traditional implementations under a constant pinout argument. Since it has been shown that complete hypermeshes outperform the mesh, the torus, low dimensional m-ary d-cubes (with and without bypass channels), and multi-stage interconnection networks (when realistic decision times are accounted for and with a constant pinout), it follows that incomplete hypermeshes outperform them as well

Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

Discrete Symmetries and their Stringy Origin

Discrete symmetries have proven to be very useful in controlling the phenomenology of theories beyond the standard model. In this work we explore how these symmetries emerge from string compactifications. Our approach is twofold: On the one hand, we consider the heterotic string on orbifold backgrounds. In this case the discrete symmetries can be derived from the orbifold conformal field theory, and it can be shown that they are in close relation with the orbifold geometry. We devote special attention to R-symmetries, which arise from discrete remnants of the Lorentz group in compact space. Further we discuss the physical implications of these symmetries both in the heterotic mini-landscape and in newly constructed models based on the Z2xZ4 orbifold. In both cases we observe that the discrete symmetries favor particular locations in the orbifold where the particles of standard model should live. On the other hand we consider a class of F-theory models exhibiting an SU(5) gauge group, times additional U(1) symmetries. In this case, the smooth compactification background does not permit us to track the discrete symmetries as transparently as in orbifold models. Hence, we follow a different approach and search for discrete subgroups emerging after the U(1)s are broken. We observe that in this approach it is possible to obtain the standard Z2 matter parity of the MSSM