This paper studies the inverse source problem for the Helmholtz equation with a point source in a two dimensional domain. Given complete boundary data and appropriate discretization Tikhonov regularization is established to be an effective method at finding the point source. Furthermore, it was found that Tikhonov regularization can locate point sources even given significant noise, as well as incomplete boundary data in complicated domains.

In this thesis we investigate theoretically various types of sculptured thin film (STF) envisioned as platforms for optical sensing. A STF consists of an array of parallel nanowires which can be grown on a substrate using vapour deposition techniques. Typically, each nanowire has a diameter in the range from ~ 10-300 nmwhile the film thickness is ~<1μm. Through careful control of the fabrication process, both the optical properties and the porosity of the STF can be tailored to order. These abilities make STFs promising for optical sensing applications, wherein it is envisaged that the material to be sensed infiltrates the void region in between the parallel nanowires and hence changes the optical properties of the STF. Various homogenization formalisms can be used to estimate the constitutive parameters of the infiltrated STFs. In this thesis two different homogenization formalisms were used: the Bruggeman formalism (extended and non–extended versions) and the strong-permittivityfluctuation theory (SPFT). These were used in investigations of the following optical–sensing scenarios: (i) Electromagnetic radiation emitted by a dipole source inside an infiltrated chiral STF. The effects of using the extended Bruggeman homogenization formalism, which takes into account the nonzero size of the component particles,were studied. (ii) Surface–plasmon– polariton waves on ametal–coated, infiltrated columnar thin film. The influences of using the extended SPFT formalism, which takes into account the nonzero size of the component particles and their statistical distributions, were explored. (iii) A metal-coated infiltrated chiral STF which supports both surface-plasmon-polariton waves and the circular Bragg phenomenon. The possibility of using in parallel both surface-plasmon-polariton waves and the circular Bragg phenomenon was investigated using the non–extended Bruggeman formalism. Our numerical studies revealed that the design performance parameters of the infiltrated STF are bode well for these optical–sensing scenarios. The use of inverse Bruggeman formalism was also investigated: this was found to be problematic in certain constitutive parameter regimes, but not those for optical–sensing scenarios considered in this thesis.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

While the bulk of topography on Earth is generated and maintained by variations in the thickness and density of crust and lithosphere, a significant time-variable contribution is expected as a result of convective flow in the underlying mantle. For over three decades, this dynamic topography has been calculated numerically from inferred density structure and radial viscosity profiles. Resulting models predict ±2 km of long wavelength (i.e., ~ 20,000 km) dynamic topography with minor contributions at wavelengths shorter than ~ 5,000 km. Recently, observational studies have revealed that, at the longest wavelengths, dynamic topography variation is ~ 30% that predicted, with ±1 km amplitudes recovered at shorter wavelengths. Here, the existing database of water-loaded basement depths is streamlined, revised and augmented. By fitting increasingly sophisticated thermal models to a combined database of these oceanic basement depths and corrected heat flow measurements, the average thermal structure of oceanic lithosphere is constrained. Significantly, optimal models are consistent with invariable geochemical and seismological constraints whilst yielding similar values of mantle potential temperature and plate thickness, irrespective of whether heat flow, subsidence or both are fit. After recalculating residual depth anomalies relative to optimal age-depth subsidence and combining them with continental constraints from gravity anomalies, a global spherical harmonic representation is generated. Although, long wavelength dynamic topography increases by ~ 40% in the revised observation-based model, spectral analysis confirms that a fundamental discrepancy between observations and predictions remains. Significantly, residual depth anomalies reveal a ~4,000 km-scale eastward tilt across the Indian Peninsula. This asymmetry extends onshore from the high-elevation Western Ghats in the west to the Krishna-Godavari floodplains in the east. Calibrated inverse modelling of drainage networks suggest that the tilt of the peninsula grew principally in Neogene times with vertical motions linked to asthenospheric temperature anomalies. Uplift rates of up to 0.1 mm a⁻¹ place important constraints on the spatio-temporal evolution of dynamic topography and suggest that rates of transient vertical motion exceed those predicted by many modelling studies. Most numerical models excise the upper ~ 300 km of Earth's mantle and are unable to reconstruct the wavelength and rate of uplift observed across Peninsular India. By contrast, through conversion of upper mantle shear wave velocities to density using a calibrated anelastic parameterisation, it is shown that shorter wavelength (i.e., ≤ 5,000 km) dynamic topography, can mostly be explained by ±150°C asthenospheric temperature anomalies. Inclusion of anelastically corrected density structure in whole-mantle instantaneous flow models also serves to reduce discrepancy between predictions and observations of dynamic topography at long wavelengths. Residual mismatch between observations and predictions is further improved if the basal 300-600 km of large low shear wave velocity regions in the deep mantle are geochemically distinct and negatively buoyant. Finally, inverse modelling of geoid, dynamic topography, gravity and core-mantle boundary topography observations using adapted density structure suggests that geodynamic constraints can be acceptably fit using plausible radial viscosity profiles, contradicting a long-standing assertion that modest long wavelength dynamic topography is incompatible with geoid observations.

This study examines why students have difficulty with inverse functions (inverse functions is the process of doing and undoing operations) and what we can do to support their learning. This was a quasi-experimental design in a math classroom in an urban comprehensive high school in California. After two weeks of instruction one group of students was taught the traditional way of inverse functions and another group was taught conceptually. About (N=80) mathematics students in the sampling were assessed before and after the study. Students were given a test to measure their learning of inverse functions and a questionnaire to measure their perspectives on the unit of study of inverse functions. Qualitative and quantitative methods were used to analyze the data. The results will be discussed hoping that in this study students taught conceptually would perform better than the controlled. Also, this study will be useful for teachers and educators to recognize that conceptual teaching yields better results than direct instruction of rote instruction

Im König-Ludwig-Haus wurde im Zeitraum von 2001 bis 2006 bei 65 Patienten nach fehlgeschlagener Versorgung einer proximalen Humerusfraktur eine inverse Prothese implantiert. Davon konnten 46 Patienten nachuntersucht werden. Vorbehandelt waren die Patienten zu 39,1% (n=18) konservativ, zu 26,1% (n=12) mit winkelstabiler Platte, zu 4,3% (n=2) mit proximalem Humerusnagel, zu 10,9% (n=5) mit Drähten, zu 6,6% (n=3) mit Drähten und Schrauben und zu 8,7% (n=4) mit Schrauben. Bei 4,3% (n=2) wurde eine Humeruskopfresektion durchgeführt. Das durchschnittliche Alter der Patienten lag bei 73,9 Jahren. 28,2% (n=13) waren männlich, 71,1% (n=33) waren weiblich. Im Mittel erreichten die Patienten postoperativ einen Constant Score von 45 Punkten. Bereinigt nach Alter und Geschlecht ergibt das 64 % des altersentsprechenden Normwertes. Minimalinvasiv mit Schrauben bzw. Schrauben und Draht versorgte Patienten zeigten die besten Ergebnisse mit durchschnittlich 55,3 bzw. 56,7 Punkten. 9 Patienten gaben an mit dem Ergebnis der TEP-Implantation sehr zufrieden zu sein, 37 waren zufrieden. Lediglich zwei Patienten waren unzufrieden mit dem postoperativen Ergebnis, wobei diese nicht dazu bereit waren, weitere Auskünfte zu geben. Auch aus den Krankenakten waren die Gründe hierfür nicht ersichtlich. Neben zwei periprothetischen Frakturen nach erneutem Sturzereignis trat lediglich eine Abszedierung nach Schweissdrüsenabszess als Komplikation auf. Die inverse Schulterprothese stellt somit eine gute Möglichkeit der Versorgung nach fehlgeschlagenen Voroperationen bei proximaler Humerusfraktur dar. Damit ist zwar eine Verbesserung der Funktion und hohe Zufriedenheit unter den Patienten zu erzielen, jedoch sind die funktionellen Ergebnisse deutlich schlechter als bei primärer TEP-Implantation bei Defektarthropathie. Mögliche Gründe hierfür waren operationsbedingte Morbidität und Deltaatrophien aufgrund des Operationszugangs bei osteosynthetischer Versorgung / In the period from 2001 to 2006 in König-Ludwig-Haus reverse total shoulder arthroplasty in 65 patients after failed supply a proximal humerus fracture was implanted. Of these 46 patients could be followed up. 39.1% (n = 18) of these were pertreated conservative, 26.1% (n = 12) with plat, 4.3% (n = 2) with a nail, 10.9% (n = 5) with wires, 6.6% (n = 3) with wires and screws and 8.7% (n = 4) with screws. The average age of the patients was 73.9 years. 28.2% (n = 13) were male, 71.1% (n = 33) were female. On average, the patients achieved an postoperative Constant score of 45 points. Adjusted for age and sex is 64% of age-matched normal range. Minimally invasive supplied with screws or bolts and wire patients showed the best results with an average of 55.3 and 56.7 points. 9 patients reported being very satisfied with the results of the implantation, 37 were satisfied. Only two patients were dissatisfied with the postoperative results, which were not prepared to provide further information. The reverse shoulder prosthesis is thus a good way of care after failed previous surgery with proximal humerus fractures represent thus an improvement of function and high satisfaction among patients is to achieve, but the functional results clearly worse than in primary TEP implantation in arthropathy.

Consider the Sturm-Liouville system : 8 > > > > > < > > > > > : − y00 + q(x)y = y y(0) cos + y0(0) sin = 0 y(1) cos + y0(1) sin = 0 , where q 2 L 1 (0, 1) and , 2 [0, £¾). Let 0 < x(n)1 < x(n)2 < ... < x(n)n − 1 < 1 be the nodal points of n-th eigenfunction in (0,1). The inverse nodal problem involves the determination of the parameters (q, , ) in the system by the knowledge of the nodal points . This problem was first proposed and studied by McLaughlin. Hald-McLaughlin gave a reconstruc- tion formula of q(x) when q 2 C 1 . In 1999, Law-Shen-Yang improved a result of X. F. Yang to show that the same formula converges to q pointwisely for a.e. x 2 (0, 1), when q 2 L 1 . We found that there are some mistakes in the proof of the asymptotic formulas for sn and l(n)j in Law-Shen-Yang¡¦s paper. So, in this thesis, we correct the mistakes and prove the reconstruction formula for q 2 L 1 again. Fortunately, the mistakes do not affect this result.Furthermore, we show that this reconstruction formula converges to q in L 1 (0, 1) . Our method is similar to that in the proof of pointwise convergence.

In general, to compute the modular inverse is very important in information security, many encrypt/decrypt and signature algorithms always need to use it. In 2007, Liu, Horng, and Liu proposed a variation on Euclidean algorithm, which can calculate the modular inverses as simple as calculate GCDs. This paper analyzes another type of left-shift binary GCD algorithm, which is suitable for the variation and that needs the fewer bit-operations than LSBGCD, which is analyzed by Shallit, and Sorenson.

This thesis reports on research undertaken in the area of microwave tomography (MWT) where the goal is to find the dielectric profile of an object of interest using microwave measurements collected outside the object. The main focus of this research is on the development of inversion algorithms which solve the electromagnetic inverse scattering problem associated with MWT. Various regularization techniques for the Gauss-Newton inversion algorithm are studied and classified. It is shown that these regularization techniques can be viewed from within a single consistent framework after applying some modifications. Within the framework of the two-dimensional MWT problem, the inversion of transverse magnetic and transverse electric data sets are considered and compared in terms of computational complexity, image quality and convergence rate. A new solution to the contrast source inversion formulation of the microwave tomography problem for the case where the MWT chamber consists of a circular conductive enclosure is introduced. This solution is based on expressing the unknowns of the problem as truncated eigenfunction expansions corresponding to the Helmholtz operator for a homogeneous background medium with appropriate boundary conditions imposed at the chamber walls. The MWT problem is also formulated for MWT chambers made of conducting cylinders of arbitrary shapes. It is shown that collecting microwave scattered-field data inside MWT setups with different boundary conditions can provide a robust set of useful information for the reconstruction of the dielectric profile. This leads to a novel MWT setup wherein a rotatable conductive triangular enclosure is used to generate scattered-field data. Antenna arrays, with as few as only four elements, that are fixed with respect to the object of interest can provide sufficient data to give good reconstructions, if the triangular enclosure is rotated a sufficient number of times. Preliminary results of using the algorithms presented herein on data collected using two different MWT prototypes currently under development at the University of Manitoba are reported. Using the open-region MWT prototype, a resolution study using the Gauss-Newton inversion method was performed using various cylindrical targets.

This thesis reports on research undertaken in the area of microwave tomography (MWT) where the goal is to find the dielectric profile of an object of interest using microwave measurements collected outside the object. The main focus of this research is on the development of inversion algorithms which solve the electromagnetic inverse scattering problem associated with MWT. Various regularization techniques for the Gauss-Newton inversion algorithm are studied and classified. It is shown that these regularization techniques can be viewed from within a single consistent framework after applying some modifications. Within the framework of the two-dimensional MWT problem, the inversion of transverse magnetic and transverse electric data sets are considered and compared in terms of computational complexity, image quality and convergence rate. A new solution to the contrast source inversion formulation of the microwave tomography problem for the case where the MWT chamber consists of a circular conductive enclosure is introduced. This solution is based on expressing the unknowns of the problem as truncated eigenfunction expansions corresponding to the Helmholtz operator for a homogeneous background medium with appropriate boundary conditions imposed at the chamber walls. The MWT problem is also formulated for MWT chambers made of conducting cylinders of arbitrary shapes. It is shown that collecting microwave scattered-field data inside MWT setups with different boundary conditions can provide a robust set of useful information for the reconstruction of the dielectric profile. This leads to a novel MWT setup wherein a rotatable conductive triangular enclosure is used to generate scattered-field data. Antenna arrays, with as few as only four elements, that are fixed with respect to the object of interest can provide sufficient data to give good reconstructions, if the triangular enclosure is rotated a sufficient number of times. Preliminary results of using the algorithms presented herein on data collected using two different MWT prototypes currently under development at the University of Manitoba are reported. Using the open-region MWT prototype, a resolution study using the Gauss-Newton inversion method was performed using various cylindrical targets.