Geometry and Topology in Memory and Navigation

Okinawa Institute of Science and Technology Graduate UniversityDoctor of PhilosophyGeometry and topology offer rich mathematical worlds and perspectives with which to study and improve our understanding of cognitive function. Here I present the following examples: (1) a functional role for inhibitory diversity in associative memories with graph- ical relationships; (2) improved memory capacity in an associative memory model with setwise connectivity, with implications for glial and dendritic function; (3) safe and effi- cient group navigation among conspecifics using purely local geometric information; and　(4) enhancing geometric and topological methods to probe the relations between neural activity and behaviour. In each work, tools and insights from geometry and topology are used in essential ways to gain improved insights or performance. This thesis contributes to our knowledge of the potential computational affordances of biological mechanisms (such as inhibition and setwise connectivity), while also demonstrating new geometric and topological methods and perspectives with which to deepen our understanding of cognitive tasks and their neural representations.doctoral thesi

Virtual Element based formulations for computational materials micro-mechanics and homogenization

In this thesis, a computational framework for microstructural modelling of transverse behaviour of heterogeneous materials is presented. The context of this research is part of the broad and active field of Computational Micromechanics, which has emerged as an effective tool both to understand the influence of complex microstructure on the macro-mechanical response of engineering materials and to tailor-design innovative materials for specific applications through a proper modification of their microstructure. While the classical continuum approximation does not account for microstructural details within the material, computational micromechanics allows detailed modelling of a heterogeneous material's internal structural arrangement by treating each constituent as a continuum. Such an approach requires modelling a certain material microstructure by considering most of the microstructure's morphological features. The most common numerical technique used in computational micromechanics analysis is the Finite Element Method (FEM). Its use has been driven by the development of mesh generation programs, which lead to the quasi-automatic discretisation of the artificial microstructure domain and the possibility of implementing appropriate constitutive equations for the different phases and their interfaces. In FEM's applications to computational micromechanics, the phase arrangements are discretised using continuum elements. The mesh is created so that element boundaries and, wherever required, special interface elements are located at all interfaces between material's constituents. This approach can be effective in modelling many microstructures, and it is readily available in commercial codes. However, the need to accurately resolve the kinematic and stress fields related to complex material behaviours may lead to very large models that may need prohibitive processing time despite the increasing modern computers' performance. When rather complex microstructure's morphologies are considered, the quasi-automatic discretisation process stated before might fail to generate high-quality meshes. Time-consuming mesh regularisation techniques, both automatic and operator-driven, may be needed to obtain accurate numeric results. Indeed, the preparation of high-quality meshes is today one of the steps requiring more attention, and time, from the analyst. In this respect, the development of computational techniques to deal with complex and evolving geometries and meshes with accuracy, effectiveness, and robustness attracts relevant interest. The computational framework presented in this thesis is based on the Virtual Element Method (VEM), a recently developed numerical technique that has proven to provide robust numerical results even with highly-distorted mesh. These peculiar features have been exploited to analyse two-dimensional representations of heterogeneous materials' microstructures. Ad-hoc polygonal multi-domain meshing strategies have been developed and tested to exploit the discretisation freedom that VEM allows. To further simplify the preprocessing stage of the analysis and reduce the total computational cost, a novel hybrid formulation for analysing multi-domain problems has been developed by combining the Virtual Element Method with the well-known Boundary Element Method (BEM). The hybrid approach has been used to study both composite material's transverse behaviour in the presence of inclusions with complex geometries and damage and crack propagation in the matrix phase. Numerical results are presented that demonstrate the potential of the developed framework

Conception de nouveaux couplages en circuits supraconducteurs

Les circuits supraconducteurs se sont imposés dans le domaine de l'informatique quantique. Une preuve nette de leur performance est la récente démonstration de la suprématie quantique par Google. Des processeurs quantiques à grande échelle sont déjà construits à l'heure actuelle. Toutefois, des améliorations de l'architecture supraconductrice, tant du point de vue des qubits que des portes logiques, sont toujours nécessaires. En effet, la propagation d'erreurs dans le processeur est toujours trop importante pour être enrayée, par exemple, avec des codes correcteurs. Des solutions pour réduire les erreurs sont, par conséquent, primordiales. La propagation d'erreurs dans un processeur quantique résulte ultimement du couplage entre les qubits qui est toutefois nécessaire pour réaliser des portes logiques. La réduction des erreurs dans le processeur passera donc en partie par l'optimisation de ces couplages. Dans cette thèse, je propose de nouvelles façons de coupler des modes en circuits supraconducteurs et présente trois projets réalisés pendant mon doctorat. Premièrement, afin de mieux concevoir les processeurs quantiques, il est nécessaire de pouvoir estimer avec le plus de précision possible la vitesse et la fidélité des portes logiques ainsi que l'amplitude des erreurs logiques. J'introduis une théorie de perturbation qui permet d'analyser les effets du couplage et du pilotage forts des qubits dans un processeur quantique. Cette théorie est basée à la fois sur la théorie de Floquet, la théorie de Schrödinger et la théorie des graphes. Deuxièmement, les coupleurs à deux qubits permettent de mitiger les erreurs dans les processeurs à plusieurs qubits. Cependant, la plupart des coupleurs y arrivent en ajustant de manière précise leurs paramètres de circuit et ne se concentrent généralement que sur l'élimination de certaines interactions parasitaires. J'introduis un coupleur supraconducteur à deux qubits qui soulage ces limitations en supprimant toutes les interactions entre les qubits avec un rapport marche-arrêt exponentiellement grand et qui ne nécessite pas de calibration précise. Il s'agit d'un coupleur à deux modes: un mode ``bus'' est connecté à un mode d'un résonateur non linéaire ancillaire. Le pilotage linéaire du mode ancillaire entraine un déplacement du champ dans le résonateur qui dépend de l'état propre du bus. Cela engendre une élimination exponentielle des interactions réelles et virtuelles entre les qubits avec l'amplitude du pilotage. Finalement, je propose une nouvelle interaction de type charge-flux entre deux modes dans une architecture hybride supraconductrice semi-conductrice. L'interaction devient non réciproque en présence d'un champ magnétique externe statique. J'utilise cette propriété pour concevoir un gyrateur passif

Modular Understanding: A Taxonomy and Toolkit for Designing Modularity in Audio Software and Hardware

Modular synthesis is a continually evolving practice. Currently, an eectivetaxonomy for analyzing modular synthesizer design does not exist, which isa signicant barrier for pedagogy and documentation. In this dissertation,I will dene new taxonomies for modular control, patching strategies, andpanel design. I will also analyze how these taxonomies can be used to in-uence the design of musical applications outside of hardware, such as mycompany Unltered Audio's software products. Finally, I will present EuroReakt, my collection of over 140 module designs for the Reaktor Blocks formatand walk through the design process of each

New two-equation turbulence model for aerodynamics applications

Two-equation turbulence modelling for computational fluid dynamics and especially for analyses of high-lift aerodynamics applications is studied in depth in this thesis. Linear Boussinesq-type modelling is abandoned and a more sophisticated explicit algebraic Reynolds stress modelling (EARSM) approach is chosen as a constitutive relation between the turbulent stress tensor and the mean-velocity gradient and turbulent scales. The proposed techniques to extend the EARSM method for significantly curved flows are critically discussed and assessed. The main focus of this study is on development of a new scale-determining two-equation model to be used with the EARSM as a constitutive model. This new k – ω model is especially designed for the requirements typical in high-lift aerodynamics. In the model development, attention is especially paid to the model sensitivity to pressure gradients, model behaviour at the turbulent/laminar edges, and to calibration of the model coefficients for appropriate flow phenomena. The model development is based on both theoretical studies and numerical experimenting. A systematic study is carried out to find the most suitable operational second scale-variable for this model. According to this study, ω itself was chosen. The developed model is finally assessed and validated for a set of realistic flow problems including high-lift aerofoil flows.reviewe

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Safety and Reliability - Safe Societies in a Changing World

The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management - mathematical methods in reliability and safety - risk assessment - risk management - system reliability - uncertainty analysis - digitalization and big data - prognostics and system health management - occupational safety - accident and incident modeling - maintenance modeling and applications - simulation for safety and reliability analysis - dynamic risk and barrier management - organizational factors and safety culture - human factors and human reliability - resilience engineering - structural reliability - natural hazards - security - economic analysis in risk managemen

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout