Dissertation (Ph.D.)--Worcester Polytechnic Institute. / Keywords: Periodic Solutions, Under-Determined Systems, Continuation, Nonlinear Eigenvalue, Inexact Newton Methods, Newton's Method, Trust Region Methods Includes bibliographical references (p.93-95).

Motivated by collaborative recycling efforts for non-profit agencies, we study a variant of the periodic location routing problem, in which one decides the set of open depots from the customer set, the capacity of open depots, and the visit frequency to nodes, in an effort to design networks for collaborative pickup activities. We formulate this problem, highlighting the challenges introduced by these decisions. We examine the relative dfficulty introduced with each decision through exact solutions and a heuristic approach which can incorporate extensions of model constraints and solve larger instances. The work is motivated by a project with a network of hunger relief agencies (e.g., food pantries, soup kitchens and shelters) focusing on collaborative approaches to address their cardboard recycling challenges collectively. We present a case study based on data from the network. In this novel setting, we evaluate collaboration in terms of participation levels and cost impact. These insights can be generalized to other networks of organizations that may consider pooling resources.

This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two.

Hyperkalemic periodic paralysis (HyperKPP) is an autosomal dominant human skeletal muscle channelopathy that causes periods of myotonic discharge and periodic paralysis due to defective Nav1.4 sodium channels. Patients are asymptomatic at birth, attacks become short and frequent during childhood, and more severe during adolescence. Since the Nav1.4 content in the cell membrane is relatively constant during childhood, it was hypothesized that some symptoms start with the defective Nav1.4 channels, while other symptoms start after some changes occur in gene expression affecting other membrane channel content and/or activity. To test the hypothesis, the contractile characteristics of EDL and soleus muscles from HyperKPP mice from the age of 0.5 to 12 months were tested in vitro. For both EDL and soleus, contractile defects, including low force generation, instability and large unstimulated force were observed by two weeks of age. With aging, the defects did not worsen, but muscles actually showed some improvement. Considering that Nav1.4 protein content reaches maximum at three weeks of age, the data suggests that HyperKPP symptoms are solely due to the defective Nav1.4 channels.

<p> This thesis describes a proposed design of a special-purpose digital instrument that will obtain the first 32 coefficients of the Walsh-Fourier series of a low-fundamental frequency periodic voltage. The mathematics are developed for applying Walsh functions to obtain a Walsh-Fourier series in the same manner as sinusoidal waves are used to obtain a Fourier series of a periodic wave. It is shown how Walsh-Fourier coefficients are employed to obtain a Fourier series. Some familiar waveforms are shown as examples. The mathematical concepts are applied to the design of the instrument, of which two major portions have been constructed using integrated circuits. The Walsh-Fourier coefficients are available at the end of the second cycle of the input. The upper fundamental frequency limit of the instrument is approximately 60 Hz. There is no low-frequency limit.</p> / Thesis / Master of Engineering (MEngr)

No description available.

<p> Hybrid task-lists are sets of periodic and asynchronous processes. To verifiably schedule hybrid tasks-lists with hard and soft real-time requirements, Xu and Lam proposed Integrated Pre-Run-Time scheduling (IPRTS) [13], a two phase method that first makes use of pre-run-time scheduling techniques, converting some asynchronous tasks with hard deadlines to periodic tasks and reserving processor capacity for the remaining hard deadline asynchronous tasks. These remaining asynchronous tasks are scheduled by a novel run-time scheduler that enforces arbitrary exclusion relations between any combination of periodic and asynchronous processes. The technique has two significant drawbacks: (i) a custom run-time scheduler is required that is not available on existing Real-Time Operating Systems (RTOS) and (ii) in many circumstances the reservation of processor capacity is overly pessimistic, causing the failure of the method for many simple task lists. To overcome these drawbacks, this thesis narrows the set of task-lists considered to those where the asynchronous tasks exclude periodic tasks and periodic processes do not exclude asynchronous tasks. A high priority polling server is then used to handle all hard asynchronous tasks. In cases where the method succeeds, it is easily implementable on any RTOS that has priority based scheduling with phased release times, and inherits the error handling and soft real-time process scheduling capabilities of the RTOS. A set of software tools which partially automates the technique, including an open source implementation of the Xu-Parnas pre-run-time scheduling algorithm [14], has been developed and applied to the examples in the thesis.</p> / Thesis / Master of Applied Science (MASc)

This research investigates the harmonic and narrowband disturbance rejection problem for linear time-periodic (LTP) systems. The consequence of disturbances on LTP systems is similar to their linear time-invariant (LTI) counterparts, but is complicated by the interaction of the disturbance and plant acting at different frequencies, which manifests itself in the modulation of the disturbance signal. The result, for an m-periodic plant and disturbance containing a single tone, is that the output contains m tones. Using various disturbance rejection architectures, harmonic and narrowband disturbance rejection is investigated for linear time-periodic plants. Included are classical and multivariable feedback controllers, fixed-gain feedforward designs using finite impulse response (FIR) filters and H-infinity synthesis tools, and adaptive feedforward controllers. The objective of time-periodic, narrowband, disturbance rejection seeks to place a zero in the controlled system's disturbance path and align the zero direction, defined by the null space of the controlled system at the disturbance frequency, with the disturbance. In this research, constraints on controlled system infinity-norms specify nominal performance and robust stability objectives. Periodic controllers are found using existing LTI H-infinity control theory, and causality is satisfied using two techniques which can be added easily to H-infinity solvers: loop-shifting and Q-parameterization. The resulting controllers are high-gain, narrowband-pass, periodic filters; the closed-loop sensitivity has a zero at the disturbance frequency, and the disturbance is in the sensitivity's null space. It is also shown that classical designs do not achieve the same performance levels as periodic controllers. Similar developments are made using the feedforward disturbance rejection architecture. Objectives are given which minimize the weighted infinity-norm of the controlled system. Such feedforward controllers achieve perfect disturbance rejection. A multivariable equivalent of the tapped-delay line is used in the description of periodic FIR filters. In addition, periodic FIR filters are made adaptive using an algorithm similar to filtered-X least mean square (LMS) but modified for periodic systems. / Ph. D.

We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra. / Master of Science / In quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.

In this thesis, the dynamical system is used as a function on afinite group, to show how states change. We investigate the'numberof cycles' and 'length of cycle' under finite groups. Using grouptheory, fixed point, periodic points and some examples, formulas tofind 'number of cycles' and 'length of cycle' are derived. Theexamples used are on finite cyclic group Z_6 with respectto binary operation '+'. Generalization using finite groups ismade. At the end, I compared the dynamical system over finite cyclic groups with the finite non-cyclic groups and then prove the general formulas to find 'number of cycles' and 'length of cycle' for both cyclic and non-cyclic groups.