CUPOLETS: Chaotic unstable periodic orbits theory and applications

Recent theoretical work suggests that periodic orbits of chaotic systems are a rich source of qualitative information about the dynamical system. The presence of unstable periodic orbits located densely on the attractor is a typical characteristic of chaotic systems. This abundance of unstable periodic orbits can be utilized in a wide variety of theoretical and practical applications [19]. In particular, chaotic communication techniques and methods of controlling chaos depend on this property of chaotic attractors [12, 13]. In the first part of this thesis, a control scheme for stabilizing the unstable periodic orbits of chaotic systems is presented and the properties of these orbits are investigated. The technique allows for creation of thousands of periodic orbits. These approximated chaotic unstable periodic orbits are called cupolets (C&barbelow;haotic U&barbelow;nstable P&barbelow;eriodic O&barbelow;rbit- lets). We show that these orbits can be passed through a phase transformation to a compact cupolet state that possesses a wavelet-like structure and can be used to construct adaptive bases. The cupolet transformation can be regarded as an alternative to Fourier and wavelet transformations. In fact, this new framework provides a continuum between Fourier and wavelet transformations and can be used in variety of applications such as data and music compression, as well as image and video processing. The key point in this method is that all of these different dynamical behaviors are easily accessible via small controls. This technique is implemented in order to produce cupolets which are essentially approximate periodic orbits of the chaotic system. The orbits are produced with small perturbations which in turn suggests that these orbits might not be very far away from true periodic orbits. The controls can be considered as external numerical errors that happen at some points along the computer generated orbits. This raises the question of shadowability of these orbits. It is very interesting to know if there exists a true orbit of the system with a slightly different initial condition that stays close to the computer generated orbit. This true orbit, if it exists, is called a shadow and the computer generated orbit is then said to be shadowable by a true orbit. We will present two general purpose shadowing theorems for periodic and nonperiodic orbits of ordinary differential equations. The theorems provide a way to establish the existence of true periodic and non-periodic orbits near the approximated ones. Both theorems are suitable for computations and the shadowing distances, i.e., the distance between the true orbits and approximated orbits are given by quantities computable form the vector field of the differential equation

Virtual Reality-Based Home Visualization and Interaction

The disclosure describes virtual reality (VR) techniques to create a virtual home for visualizing an appearance and functionality of a home setup. Per the techniques, a home can be visualized in virtual reality before completion or at any later time. The techniques, which can be implemented in a smart home application, enable users to preview furniture, appliances, etc. as placed within their virtual home. The user can virtually place furniture in their homes, examine different furniture placement styles and make an informed decision before making a purchase. The techniques also enable testing smart home device automation behaviors in the virtual home. The use of virtual reality offers an immersive experience by allowing users to interact with a digital environment in three dimensions

INVARIANT OPERATOR RANGES AND SIMILARITY DOMINANCE IN BANACH AND VON NEUMANN ALGEBRAS

\begin{Abstractpage} \setlength{\baselineskip}{1.5\baselineskip} { Suppose $\mathcal{M}$ is a von Neumann algebra. An \textbf{operator range in }$\mathcal{M}$ is the range of an operator in $\mathcal{M}$. When $\mathcal{M}=B\left( H\right)$, the algebra of operators on a Hilbert space $H$, R. Douglas and C. Foia\c{s} proved that if $S,T\in B\left( H\right)$, and $T$ is not algebraic, and if $S$ leaves invariant every $T$-invariant operator range, then $S=f\left( T\right)$ for some entire function $f$. In the first part of this thesis, we prove versions of this result when $B\left( H\right)$ is replaced with a factor von Neumann algebra $\mathcal{M}$ and $T$ is normal. Then using the direct integral theory, we extend our result to an arbitrary von Neumann algebra. In the second part of the thesis, we investigate the notion of \textbf{similarity dominance.} Suppose $\mathcal{A}$ is a unital Banach algebra and $S,T\in \mathcal{A}$. We say that $T$ sim-dominates $S$ provided, for every $R\u3e0$,% $$\sup \left( \left \{ \left \Vert A^{-1}SA\right \Vert :A\in \mathcal{A},\text{ }A\text{ invertible, }\left \Vert A^{-1}TA\right \Vert \leq R\right \} \right) \u3c\infty \text{.}$$ When $\mathcal{A}$ is the algebra $B\left( H\right)$, J. B. Conway and D. Hadwin proved that $T$ sim-dominates $S$ implies $S=\varphi \left( T\right)$ for some entire function $\varphi$. We prove this for a large class of operators in a type III factor von Neumann algebra. We also prove, for any unital Banach algebra $\mathcal{A}$, if $T$ sim-dominates $S$, then $S$ is in the approximate double commutant of $T$ in $\mathcal{A}$. Moreover, we prove that sim-domination is preserved under approximate similarity. } \end{Abstractpage

Effects of Degree of Particle Melt and Crystallinity in SLS Nylon-12 Parts

Differential Scanning Calorimetry (DSC) traces for SLS Nylon-12 parts display two distinct melt peaks, which have been related to the presence of both melted and crystallised regions, and un-melted particle cores within the part. The relative proportions of each region are defined by the term ‘Degree of Particle Melt’ (DPM), and have a large effect on the mechanical properties of a part. This paper demonstrates that the % crystallinity of SLS Nylon-12 parts is dependent on the DPM. Crucially, research has also shown that the trends for some tensile properties (notably Tensile Strength and Young’s Modulus) change once full melting is complete.Mechanical Engineerin

Bluetooth-based Home Intrusion Detection Using Smart Home Devices

Security cameras and motion detectors used for home security can be expensive and may not cover the entirety of a home or property. Also, even when an intruder is detected by the camera, there is not much that can be done after the intruder leaves. This disclosure describes the use of in-home smart devices to detect unrecognized devices via Bluetooth and automatically detect an intrusion based on such detection. The described techniques can detect intruders and can also help locate the intruders after they leave the home. The techniques can improve home security without requiring additional hardware

Registration and Control of Smart Home Devices via Augmented Reality

Smart home devices can be controlled via a smartphone or other device. However, when a user has several smart home devices in their home, identifying a particular device and accessing its controls can be cumbersome. This disclosure describes the use of augmented reality (AR) to detect and register smart home devices, and to automatically surface device control user interface for devices within a user’s view. Per the techniques, object recognition is used to detect the make and model of devices within a user’s camera view to enable the user to easily configure the device. Mesh network protocols are used to locate devices within the local network, relative to each other and with respect to the user’s smartphone or other device used for control. The user can simply select a device within the camera viewport and access the user interface to control the device

Visual Debugger for Internet-of-Things (IoT) Ecosystem

When a smart home device does not work as expected, the device vendor has to investigate many interacting parts in the ecosystem in order to determine a root cause for the issue. With a high number of interacting parts involved in debugging, the debugging process can be time-consuming and tedious. This disclosure describes a visual debugger that is optimized for IoT ecosystems. The visual debugger unifies logs generated by IoT devices to display the evolution of device states with time. Network requests and responses are indicated visually by clickable arrows indicating the directions of network traffic. An event on a device that triggers a cascade of actions on other devices is displayed using an action path that indicates the movement of the action through the network. A log file framework enables components of the IoT network originating from different manufacturers to be analyzed and displayed in a systematic and unified manner. IoT network developers and third-party device vendors can efficiently triage bugs, determine the root causes of failures, and improve performance and interoperability

Portable Smart Home Gateway for Easy Setup of Smart Home Devices

Smart home devices need to be set up to access home WiFi networks and registered to a user account before the device can be used in a home. The multi-step setup procedure can be different for different devices and can take considerable time and user effort. This disclosure describes a portable smart home gateway that simplifies the process of adding new smart home devices to a home network. The smart home gateway can connect to a home WiFi network and to other smart devices via a home networking protocol. New devices can be connected to the smart home gateway via USB (or other suitable connector). Upon connection, the smart home gateway can route requests from the device to the WiFi network and can automate the setup process

Investigation into crystallinity and degree of particle melt in selective laser sintering

Selective Laser Sintering (SLS) is a manufacturing process which has emerged from numerous other technologies as the leading process considered viable for Rapid Manufacturing. SLS of polymers has found use in a wide range of industries ranging from aerospace to medicine. The ability to easily manufacture previously difficult or impossible to produce parts, without tooling, has proved invaluable. This industry backed manufacturing engineering PhD investigation examined material properties of Nylon-12 parts produced by the SLS process. Crystallinity relates to and determines mechanical properties in traditionally processed polymers. The nature of crystallinity in SLS processed Nylon-12 has been examined in this study and shown to be fundamentally different to that of traditionally processed polymers. Rather than depending primarily upon crystallisation factors such as cooling rate, it was shown to depend on the degree of particle melt (DPM). DPM was shown to be quantifiable by DSC measurements (by the degree of crystallinity and Core Peak Height) and distinct relationships between the quantified DPM and mechanical properties were found. Additionally, this study showed for the first time that parts remain above the crystallisation onset temperature even after build completion. This work has expanded the knowledge base of SLS by shedding light on critical aspects of the process. The ability to quantify the degree to which particles melt (DPM) provides a new level of understanding into the causes of changes in mechanical properties with changes in process parameters. This new understanding can lead to improved process modelling and could aid in the development of new processes and materials. DPM could be implemented into new quality control methods and the knowledge of post build crystallisation shows that post build cool down is an aspect of the build process that requires control for improved consistency of properties

Tailoring the Mechanical Properties of Selective Laser Sintered Parts

The ~£1 million IMCRC-funded integrated project ‘Personalised Sports Footwear: From Elite to High Street’ is investigating the use of Rapid Manufacturing to produce personalised sports shoes, with the aim of enhancing performance, reducing injury, and providing improved functionality. Research has identified that, for sprinting, performance benefits can be achieved by tuning the bending stiffness of a shoe to the characteristics of an individual athlete. This paper presents research to date on several novel methods of influencing the mechanical properties of Selective Laser Sintered shoe soles, with a particular focus on stiffness.Mechanical Engineerin