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We give a Bishop-style constructive analysis of the statement that a continuous homomorphism from the real line onto a compact metric abelian group is periodic; constructive versions of this statement and its contrapositive are given. It is shown that the existence of a minimal period in general is not derivable, but the minimal period is derivable under a simple geometric condition when the group is contained in two dimensional Euclidean space. A number of results about one-one and injective mappings are proved en route to our main theorems. A few Brouwerian examples show that some of our results are the best possible in a constructive framework.

We give a Bishop-style constructive analysis of the statement that a continuous homomorphism from the real line onto a compact metric abelian group is periodic; constructive versions of this statement and its contrapositive are given. It is shown that the existence of a minimal period in general is not derivable, but the minimal period is derivable under a simple geometric condition when the group is contained in two dimensional Euclidean space. A number of results about one-one and injective mappings are proved en route to our main theorems. A few Brouwerian examples show that some of our results are the best possible in a constructive framework.

Thesis (Ph. D.)--Carleton University, 2001. / Includes bibliographical references (p. 118-133). Also available in electronic format on the Internet.

Thesis (Ph. D.)--University of Florida, 1982. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 148-153).

<p> Periodic arrays of Au patterns (dots and lines) were produced via electron beam lithography (EBL). GaAs mesas were produced by using the Au structures as a mask and wet etching the GaAs (lll)B substrates, leaving Au resting above GaAs pillars. Annealing experiments at typical nanowire growth temperatures (550°C) were performed on both mesa-supported samples and a control sample without mesas, and were later characterized by scanning electron microscopy (SEM). From SEM images, a model is proposed to describe the evolution of the Au seed particle during exposure to typical growth conditions. The Au particle is subject to not only a melting process but is also modified by a volume increase due to incorporating Ga atoms and a subsequent crystal structure change. Palpable discrepancies between the mesa-supported and control samples were observed after annealing experiments, suggesting the mesas were effective in confining the migration of the Au. NW s were then grown via gas source molecular beam epitaxy (GS-MBE). Discemable variation amongst the results was evident when a comparison between annealed samples and the grown counterpart was made. The inconsistency is ascribed to the NW growth process beginning only after supersaturation at the growth interface. This saturation took place only after 2-D film growth on the substrate surpassed the height of the mesas rendering the structures less functional. </p> / Thesis / Master of Applied Science (MASc)

abstract: In this work, we focused on the stability and reducibility of quasi-periodic systems. We examined the quasi-periodic linear Mathieu equation of the form x &#776;+(ä+&#1013;[cost+cosùt])x=0 The stability of solutions of Mathieu's equation as a function of parameter values (ä,&#1013;) had been analyzed in this work. We used the Floquet type theory to generate stability diagrams which were used to determine the bounded regions of stability in the ä-ù plane for fixed &#1013;. In the case of reducibility, we first applied the Lyapunov- Floquet (LF) transformation and modal transformation, which converted the linear part of the system into the Jordan form. Very importantly, quasi-periodic near-identity transformation was applied to reduce the system equations to a constant coefficient system by solving homological equations via harmonic balance. In this process we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system to a constant one. / Dissertation/Thesis / M.S.Tech Engineering 2012

Periodically correlated time series generally exist in several fields including hydrology, climatology, economics and finance, and are commonly modelled using periodic autoregressive (PAR) model. For a time series with stochastic periodic trend, for which a unit root is expected, a periodically integrated autoregressive PIAR model with periodic and/or seasonal unit root has been shown to be a satisfactory model. The existing theory used the multivariate methodology to study PIAR models. However, this theory is convoluted, majority of it only developed for quarterly time series and its generalisation to time series with larger number of periods is quite cumbersome. This thesis studies the existing theory and highlights its restrictions and flaws. It provides a coherent presentation of the steps for analysing PAR and PIAR models for different number of periods. It presents the different unit roots representations and compares the performance of different unit root tests available in literature. The restrictions of existing studies gave us the impetus to develop a unified theory that gives a clear understanding of the integration and unit roots in the periodic models. This theory is based on the spectral information of the multi-companion matrix of the periodic models. It is more general than the existing theory, since it can be applied to any number of periods whereas the existing methods are developed for quarterly time series. Using the multi-companion method, we specify and estimate the periodic models without the need to extract complicated restrictions on the model parameters corresponding to the unit roots, as required by NLS method. The multi-companion estimation method performed well and its performance is equivalent to the NLS estimation method that has been used in the literature. Analysing integrated multivariate models is a problematic issue in time series. The multi-companion theory provides a more general approach than the error correction method that is commonly used to analyse such time series. A modified state state representation for the seasonal periodically integrated autoregressive (SPIAR) model with periodic and seasonal unit roots is presented. Also an alternative state space representations from which the state space representations of PAR, PIAR and the seasonal periodic autoregressive (SPAR) models can be directly obtained is proposed. The seasons of the parameters in these representations have been clearly specified, which guarantees correct estimated parameters. Kalman filter have been used to estimate the parameters of these models and better estimation results are obtained when the initial values were estimated rather than when they were given.

Experimental investigations of structurally-dictated surface waves supported by periodically textured metallic substrates with different symmetries, are the primary focus of the work presented in this thesis. The electromagnetic response of three near perfectly conducting substrates perforated with arrays of holes with different geometries,together with a low-profile high-impedance structure are characterised. Experimental measurement techniques are employed to record the transmission, and reflection from the structures under investigation, together with phase-resolved measurements to directly obtain the dispersion of the surface waves supported by these structures. From these measurements information about the nature of the surface modes supported by the structures under investigation can be observed. A study of diffractively coupled surface waves supported by a close-packed array of square cross-section, close-ended holes in the limit where the wavelength of incident radiation and periodicity of the hole array are comparable, is presented. An additional grating, which has a periodicity comparable to the hole array is used to control the strength of diffractive coupling to the mode. Using a free-space measurement technique,information about the dispersion of the modes supported by the structure is obtained by recording the azimuthal-dependent reflection from the structure. It is found that the relative positions of the hole array and `coupling-in' grating is significant, a key issue not addressed in the literature when investigating grating-coupling to surface modes. Good agreement with numerical predictions is demonstrated. Structurally-dictated surface waves on a metallic substrate pierced by a close-packed array of deep, rectangular holes is characterised. In this arrangement, the fundamental resonance in the holes in the orthogonal directions is different and the frequency therefore to which the dispersion of the surface waves supported by the structure is limited, varies with sample orientation. The anisotropic dispersion, resulting from an ellipsoid of limiting frequencies, is directly mapped using a phase-resolved measurement technique. Furthermore by exploiting the anisotropy of the unit cell, a family of higher order surface waves associated with the quantisation of the electromagnetic fields within the holes is explored in this chapter. Once again good agreement with numerical predictions is shown.The `enhanced transmission' recorded through a `zigzag' hole array, attributed to the excitation of diffractively coupled surface waves, is explored. Due to the specific symmetry of the unit cell of the zigzag hole array it is shown that coupling to these surface waves can be achieved with both transverse magnetic and transverse electric polarised incident radiation. Further, incident radiation can directly couple to the surface modes supported by the zigzag hole array, via scattering from its inherent in-plane periodicity. The observed polarisation-selective excitation of individual surface wave bands, agrees well with numerical predictions and is shown to be a direct consequence of the reduced symmetry of the system. Finally, the dispersion of the modes supported by an ultra-thin, high-impedance surface in the form of a Sievenpiper `mushroom' structure, with rectangular geometry is directly recorded. The behaviour of the Sievenpiper structure is rather complex and to aid understanding of the electromagnetic response of the structure, the results are compared with the modes supported by a simpler patch array structure. The anisotropy arising from the rectangular geometry is characterised and an in depth discussion of the origin of the modes presented.

We present a hybrid optimization algorithm for mixed-integer linear programming, embedding both heuristic and exact components. In order to validate it we use the periodic vehicle routing problem (PVRP) as a case study. This problem consists of determining a set of minimum cost routes for each day of a given planning horizon, with the constraints that each customer must be visited a required number of times (chosen among a set of valid day combinations), must receive every time the required quantity of product, and that the number of routes per day (each respecting the capacity of the vehicle) does not exceed the total number of available vehicles. This is a generalization of the well-known vehicle routing problem (VRP). Our algorithm is based on the linear programming (LP) relaxation of a set-covering-like integer linear programming formulation of the problem, with additional constraints. The LP-relaxation is solved by column generation, where columns are generated heuristically by an iterated local search algorithm. The whole solution method takes advantage of the LP-solution and applies techniques of fixing and releasing of the columns as a local search, making use of a tabu list to avoid cycling. We show the results of the proposed algorithm on benchmark instances from the literature and compare them to the state-of-the-art algorithms, showing the effectiveness of our approach in producing good quality solutions. In addition, we report the results on realistic instances of the PVRP introduced in Pacheco et al. (2011) [24] and on benchmark instances of the periodic traveling salesman problem (PTSP), showing the efficacy of the proposed algorithm on these as well. Finally, we report the new best known solutions found for all the tested problems. (authors' abstract)