<p>This work strengthens and extends an algorithm for computing Betti numbers of symmetric semialgebraic sets developed by Basu and Riener in, <em>Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-Algebraic Sets</em>. We first adapt a construction of Gabrielov and Vorobjov in, <em>Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology,</em> for replacing arbitrary definable sets by compact ones to the symmetric case. The original construction provided maps from the homotopy and homology groups of the replacement set to those of the original; we show that for sets symmetric relative to the action of some finite reflection group <em>G</em>, we may construct these maps to be equivariant. This modification to the construction for compact replacement allows us to extend Basu and Riener's theorem on which submodules appear in the isotypic decomposition of each cohomology space to sets not necessarily closed and bounded. Furthermore, by utilizing this equivariant compact approximation, we may obtain a precise description of the aforementioned decomposition of each cohomology space, and not merely the final dimension of the space, from Basu and Riener's algorithm.</p> <p><br></p> <p>    Though our equivariant compact replacement holds for <em>G</em> any finite reflection group, Basu and Riener's results only consider the case of the action the of symmetric group, sometimes termed type <em>A</em>. As a first step towards generalizing Basu and Riener's work, we examine the next major class of symmetry: the action of the group of signed permutations (known as type <em>B</em>). We focus our attention on Vandermonde varieties, a key object in Basu and Riener's proofs. We show that the intersection of a type <em>B</em> Vandermonde variety with a fundamental region of type <em>B</em> symmetry is topologically regular. We also prove a result about the intersection of a type <em>B</em> Vandermonde variety with the walls of this fundamental region, leading to the elimination of factors in a different decomposition of the homology spaces.</p>

Parallelization is an important part of modern data systems. However, the non-determinism of thread scheduling introduces the difficult problem of considering all different execution orders when constructing an algorithm. Therefore deterministic-by-design concurrent systems are attractive. A new approach called LVars consists of using data which is part of a lattice, with a predefined join operation. Updates to shared data are carried out using the join operation and thus the updates commute. Together with limiting the reads of shared data, this guarantees a deterministic result. The Reactive Async framework follows a similar approach but has several aspects which can cause a non-deterministic result. The goal with this thesis is to explore how we can ammend Reactive Async in order to guarantee a deterministic result. First an exploration into the subtleties of lattice based data combined with concurrency is made. Then a formal model based on a simple object-oriented language is constructed. The constructed small-step operational semantics and type system are shown to guarantee a form of determinism. This shows that LVars-similar system can be implemented in an object-oriented setting. Furthermore the work can act as a basis for future revisions of Reactive Async and similar frameworks. / Parallellisering är en viktig del i moderna datasystem. Flertrådade applikationer innebär dock en svårighet i och med att programmerare måste ta alla exekveringsordningar i beaktning. Därför är beräkningsmodeller vars resultat är garanterat deterministiskt en attraktiv utväg. En ny modell, kallad LVars, använder gitterstrukturer tillsammans med en supremum-operation för att garantera att uppdateringar av delad data kommuterar. Detta tillsammans med begränsningar av läsning av datan garanterar ett deterministiskt resultat. Reactive Async är ett programmeringsramverk som följer en liknande strategi. Det finns dock flera delar i dess konstruktion som i en oförsiktig programmerares händer kan orsaka att ett programs resultat blir icke-deterministiskt. Målet med detta examensarbete är att utforska vilka modifikationer som skulle kunna göras av Reactive Async för att garantera determinism. Först görs en undersökning av de mer svårförståeliga delarna i kombinationen av gitterbaserad data med flertrådad exekvering. Sedan konstrueras en formell beräkningsmodell baserad på ett enkelt objektorienterat språk. Konstruktionens småstegade operationella semantik tillsammans med dess typsystem visas kunna garantera en form av determinism. Detta visar att ett system liknande LVars kan implementeras i ett objektorienterat språk. Därmed skulle detta arbete kunna ligga till grund för framtida versioner av Reactive Async.

The knowledge of the packing behaviour of small organic compounds in crystal lattices is of great importance for industries dealing with solid state materials. The properties of materials depend on how the molecules arrange themselves in a crystalline environment. Crystal structure prediction provides a theoretical approach through the application of computational strategies to seek possible crystal packing arrangements (or polymorphs) a compound may adopt. Based on the chemical diagrams, this thesis investigates polymorphism of several small organic compounds. Plausible crystal packings of those compounds are generated, and their lattice energies are minimised using molecular mechanics and/or quantum mechanics methods. Most of the work presented here is conducted using two software packages commercially available in this field, Polymorph Predictor of Materials Studio 4.0 and GRACE 1.0. In general, the computational techniques implemented in GRACE are very good at reproducing the geometries of the crystal structures corresponding to the experimental observations of the compounds, in addition to describing their solid state energetics correctly. Complementing the CSP results obtained using GRACE with isostructurality offers a route by which new potential polymorphs of the targeted compounds might be crystallised using the existing experimental data. Based on all calculations in this thesis, four new potential polymorphs for four different compounds, which have not yet been determined experimentally, are predicted to exist and may be obtained under the right crystallisation conditions. One polymorph is expected to crystallise under pressure. The remaining three polymorphs might be obtained by using a seeding technique or the utilisation of suitable tailor made additives. / University of Bradford

Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.

Topological states in photonic systems are described by qualitative discrete quantum numbers and feature unique combinations of properties such as robustness to perturbations and dissipationless surface transport of energy and information. Due to the many easily accessible platforms in photonics, topological photonics is one of the spearheads of topological physics that has offered exciting possibilities for investigations of novel topological phenomena and numerous promising technological uses. This master thesis aims at investigating from a theoretical and experimental perspective the properties of topological states of interacting light modes in arrays of coupled waveguides. First, we present a review of recent advances in the recently emerged field of topological physics. Then we review recent work on lattice models that feature topological edge states, and subsequently, we identify the existence of robust corner states on the edge of honeycomb lattices. These states display, similarly to the corner states known in 1D SSH and 2D Kagome lattices, exponential localisation with naturally occurring total destructive interference. Additionally, the corner states share properties with the dynamic transport modes of photonic Floquet topological insulators. However, the origin of the corner localisation is different from the previously studied examples: instead of dimerised coupling strength, the asymmetric and dimerised number of neighbours in the direction of the corner generates the intensity gradient. Motivated by our numerical study, we outline an experimental realisation of the novel honeycomb corner states, by means of waveguide lattices written in glass. The waveguides are 3D written in the bulk of a glass sample, with the lattice pattern orthogonal to the main direction of propagation which acts as a time coordinate in the experiments. Femtosecond direct laser writing enables fabricating truly 3D waveguides. Our experimental preparations cover a study of the effects of fabrication parameters for the relevant structures. We discuss how the localisation and robustness properties of the honeycomb corner states would next be investigated in the prepared experimental setting. / Topologiska tillstånd i fotoniska system beskrivs av särskilda topologiska diskreta kvanttal och har unika kombinationer av egenskaper såsom robusthet mot fluktuationer och förlustfri yttransport av energi och information. Tack vare de många lättillgängliga plattformarna inom fotonik är topologisk fotonik en av spjutspetsarna inom topologisk fysik och har erbjudit unika möjligheter för undersökningar av nya topologiska fenomen och med många lovande tekniska användningsområden. Denna masteruppsats syftar till att undersöka, ur ett teoretiskt och experimentellt perspektiv, egenskaperna hos topologiska tillstånd av interagerande ljusmoder i gitter av kopplade vågledare. Först presenterar vi en översikt av de senaste framstegen inom det nyligen uppkomna området topologisk fysik. Sedan sammanfattar vi det senaste arbetet med gittermodeller som har topologiska kanttillstånd, och därefter identifierar vi förekomsten av robusta hörntillstånd på kanten av bikakegitter. Dessa tillstånd uppvisar, på samma sätt som hörntillstånden som är kända i 1D SSH- och 2D Kagome-gitter, hög kantlokalisering med naturligt förekommande total destruktiv interferens. Dessutom delar hörntillstånden egenskaper med de kända fotoniska Floquet-topologiska isolatorerna som uppvisar topologisk transport. Ursprunget till hörnlokaliseringen skiljer sig dock från de tidigare studerade exemplen: istället för en dimeriserad kopplingsstyrka genererar det asymmetriska och dimeriserade antalet grannar i hörnets riktning intensitetsgradienten. Med motivering i den numeriska studien inleder vi en experimentell realisering av de nya bikakehörntillstånden, med hjälp av vågledargitter skrivna i glas. Vågledarna är 3D-skrivna på insidan av ett glasprov, med gittermönstret ortogonalt mot huvudutbredningsriktningen som fungerar som en tidskoordinat i experimenten. Direkt laserskrivning med femtosekundlaser gör det möjligt att tillverka vågledare i tre dimensioner. Våra experimentella förberedelser omfattar en studie av effekterna av tillverkningsparametrar för de relevanta strukturerna. Vi diskuterar hur lokaliserings- och robusthetsegenskaperna för bikakehörntillstånden sedan skulle undersökas med den experimentella metoden.

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The emergence of metal Additive Manufacturing (AM) over the last two decades has opened venues to mitigate the challenges associated with stochastic open-cell metal foams manufactured through the traditional foaming process. Regular lattices with user-defined unit cell topologies have been reported to exhibit better mechanical properties in comparison to metal foams which extend their applicability to multifunctional heat exchangers subjected to both thermal and mechanical loads. The current study aims at investigating the thermal-hydraulic characteristics of promising novel unit cell topologies realizable through AM technologies. Experimental investigation was conducted on four different topologies, viz (a) Octet, (b) Face-diagonal (FD) cube, (c) Tetrakaidecahedron, and (d) Cube, printed in single-cell thick sandwich type configuration in 420 stainless steel via Binder Jetting technology at same intended porosity. The effective thermal conductivity of the samples was found to be strongly dependent on the lattice porosity, however, no significant dependence on the unit-cell topology was demonstrated. Face-diagonal cube lattice exhibited the highest heat transfer coefficient and pressure drop, and consequently provided the lowest thermal-hydraulic performance. A procedure to incorporate the manufacturing-induced random roughness effects in the samples during numerical modelling is introduced. The numerical simulations were conducted on samples exhibiting the roughness profiles having statistically same mean roughness as the additively manufactured coupons and the results were compared to that obtained from the intended smooth-profiled CAD models that were fed into the printing machines. The analysis showed that inclusion of roughness effects in computational models can significantly improve the thermal performance predictions. Through this study, we demonstrate that additively manufactured ordered lattices exhibit superior thermal transport characteristics and future developmental efforts would require extensive experimentations to characterize their thermal and flow performance as well as local surface quality and AM-induced defect recognition. Experimental findings would also need to be supported by computational efforts where configurations which closely mimic the real AM parts could be modeled. A combined experimental-numerical framework is recommended for advancements in metal additive manufacturing-enabled enhanced heat transfer concepts.

<p dir="ltr">In this work we investigate the existence and experimentally measurable properties of Majorana quasiparticles in a few-body number conserving atomic system.</p>