On a Stratification Defined by Real Roots of Polynomials

2000 Mathematics Subject Classification: 12D10

A Characterization Theorem for the K-functional Associated with the Algebraic Version of Trigonometric Jackson Integrals

2000 Mathematics Subject Classification: 41A25, 41A36.

A New Characterization of Weighted Peetre K-Functionals (II)

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10. ∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”, Sofia, 2006.

Equivalence Between K-functionals Based on Continuous Linear Transforms

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.

The Lower Estimate for Bernstein Operator

MSC 2010: 41A10, 41A15, 41A25, 41A36

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Electrochemical formation of ordered pore arrays in InP in KCl

Pores are formed electrochemically in n-InP in KCl electrolytes with concentrations of 2 mol dm-3 or greater. The pore morphology is similar to what is seen in other halide-based electrolytes. At low potentials, crystallographically oriented (CO) pores are formed. At higher potentials, current-line oriented (CLO) pores are formed. Crystallographically oriented pore walls are observed for both pore morphologies. When formed at a constant current, potential oscillations are observed which have been correlated to oscillations in the pore width. The CLO pore wall smoothness and overall uniformity increase as KCl concentration is increased. The porous structures formed in KCl compare favourably with those formed in the more acidic or alkaline electrolytes that are typically used to form these structures.

On Provable Algorithms for Determination of Continuous Protein Interdomain Motions from Residual Dipolar Couplings

Dynamics of biomolecules over various spatial and time scales are essential for biological functions such as molecular recognition, catalysis and signaling. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. Unfortunately, these distributions cannot be fully constrained by the limited information from experiments, making the problem an ill-posed one in the terminology of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem needs to be regularized by making assumptions, which inevitably introduce biases into the result.

Here, I present two continuous probability density function approaches to solve an important inverse problem called the RDC trigonometric moment problem. By focusing on interdomain orientations we reduced the problem to determination of a distribution on the 3D rotational space from residual dipolar couplings (RDCs). We derived an analytical equation that relates alignment tensors of adjacent domains, which serves as the foundation of the two methods. In the first approach, the ill-posed nature of the problem was avoided by introducing a continuous distribution model, which enjoys a smoothness assumption. To find the optimal solution for the distribution, we also designed an efficient branch-and-bound algorithm that exploits the mathematical structure of the analytical solutions. The algorithm is guaranteed to find the distribution that best satisfies the analytical relationship. We observed good performance of the method when tested under various levels of experimental noise and when applied to two protein systems. The second approach avoids the use of any model by employing maximum entropy principles. This 'model-free' approach delivers the least biased result which presents our state of knowledge. In this approach, the solution is an exponential function of Lagrange multipliers. To determine the multipliers, a convex objective function is constructed. Consequently, the maximum entropy solution can be found easily by gradient descent methods. Both algorithms can be applied to biomolecular RDC data in general, including data from RNA and DNA molecules.

Open-source virtual bronchoscopy for image guided navigation

This thesis describes the development of an open-source system for virtual bronchoscopy used in combination with electromagnetic instrument tracking. The end application is virtual navigation of the lung for biopsy of early stage cancer nodules. The open-source platform 3D Slicer was used for creating freely available algorithms for virtual bronchscopy. Firstly, the development of an open-source semi-automatic algorithm for prediction of solitary pulmonary nodule malignancy is presented. This approach may help the physician decide whether to proceed with biopsy of the nodule. The user-selected nodule is segmented in order to extract radiological characteristics (i.e., size, location, edge smoothness, calcification presence, cavity wall thickness) which are combined with patient information to calculate likelihood of malignancy. The overall accuracy of the algorithm is shown to be high compared to independent experts' assessment of malignancy. The algorithm is also compared with two different predictors, and our approach is shown to provide the best overall prediction accuracy. The development of an airway segmentation algorithm which extracts the airway tree from surrounding structures on chest Computed Tomography (CT) images is then described. This represents the first fundamental step toward the creation of a virtual bronchoscopy system. Clinical and ex-vivo images are used to evaluate performance of the algorithm. Different CT scan parameters are investigated and parameters for successful airway segmentation are optimized. Slice thickness is the most affecting parameter, while variation of reconstruction kernel and radiation dose is shown to be less critical. Airway segmentation is used to create a 3D rendered model of the airway tree for virtual navigation. Finally, the first open-source virtual bronchoscopy system was combined with electromagnetic tracking of the bronchoscope for the development of a GPS-like system for navigating within the lungs. Tools for pre-procedural planning and for helping with navigation are provided. Registration between the lungs of the patient and the virtually reconstructed airway tree is achieved using a landmark-based approach. In an attempt to reduce difficulties with registration errors, we also implemented a landmark-free registration method based on a balanced airway survey. In-vitro and in-vivo testing showed good accuracy for this registration approach. The centreline of the 3D airway model is extracted and used to compensate for possible registration errors. Tools are provided to select a target for biopsy on the patient CT image, and pathways from the trachea towards the selected targets are automatically created. The pathways guide the physician during navigation, while distance to target information is updated in real-time and presented to the user. During navigation, video from the bronchoscope is streamed and presented to the physician next to the 3D rendered image. The electromagnetic tracking is implemented with 5 DOF sensing that does not provide roll rotation information. An intensity-based image registration approach is implemented to rotate the virtual image according to the bronchoscope's rotations. The virtual bronchoscopy system is shown to be easy to use and accurate in replicating the clinical setting, as demonstrated in the pre-clinical environment of a breathing lung method. Animal studies were performed to evaluate the overall system performance.

Pilot Study on Improving the Efficiency of Transportation Projects Using Laser Scanning, 2003

Laser scanning is a terrestrial laser-imaging system that creates highly accurate three-dimensional images of objects for use in standard computer-aided design software packages. This report describes results of a pilot study to investigate the use of laser scanning for transportation applications in Iowa. After an initial training period on the use of the scanner and Cyclone software, pilot tests were performed on the following projects: intersection and railroad bridge for training purposes; section of highway to determine elevation accuracy and pair of bridges to determine level of detail that can be captured; new concrete pavement to determine smoothness; bridge beams to determine camber for deck-loading calculations; stockpile to determine volume; and borrow pit to determine volume. Results show that it is possible to obtain 2-6 mm precision with the laser scanner as claimed by the manufacturer compared to approximately one-inch precision with aerial photogrammetry using a helicopter. A cost comparison between helicopter photogrammetry and laser scanning showed that laser scanning was approximately 30 percent higher in cost depending on assumptions. Laser scanning can become more competitive to helicopter photogrammetry by elevating the scanner on a boom truck and capturing both sides of a divided roadway at the same time. Two- and three-dimensional drawings were created in MicroStation for one of the scanned highway bridges. It was demonstrated that it is possible to create such drawings within the accuracy of this technology. It was discovered that a significant amount of time is necessary to convert point cloud images into drawings. As this technology matures, this task should become less time consuming. It appears that laser scanning technology does indeed have a place in the Iowa Department of Transportation design and construction toolbox. Based on results from this study, laser scanning can be used cost effectively for preliminary surveys to develop TIN meshes of roadway surfaces. It also appears that this technique can be used quite effectively to measure bridge beam camber in a safer and quicker fashion compared to conventional approaches. Volume calculations are also possible using laser scanning. It seems that measuring quantities of rock could be an area where this technology would be quite beneficial since accuracy is more important with this material compared to soil. Other applications for laser scanning could include developing as-built drawings of historical structures such as the bridges of Madison County. This technology could also be useful where safety is a concern such as accurately measuring the surface of a highway active with traffic or scanning the underside of a bridge damaged by a truck. It is recommended that the Iowa Department of Transportation initially rent the scanner when it is needed and purchase the software. With time, it may be cost justifiable to purchase the scanner as well. Laser scanning consultants can be hired as well but at a higher cost.

Dispersive estimates for the Schrödinger equation

In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.

STATISTICAL METHODS FOR ADVANCED ECONOMETRIC MODELS WITH APPLICATIONS TO VEHICLE HOLDING AND SPEED QUANTILES DISTRIBUTION

This dissertation proposes statistical methods to formulate, estimate and apply complex transportation models. Two main problems are part of the analyses conducted and presented in this dissertation. The first method solves an econometric problem and is concerned with the joint estimation of models that contain both discrete and continuous decision variables. The use of ordered models along with a regression is proposed and their effectiveness is evaluated with respect to unordered models. Procedure to calculate and optimize the log-likelihood functions of both discrete-continuous approaches are derived, and difficulties associated with the estimation of unordered models explained. Numerical approximation methods based on the Genz algortithm are implemented in order to solve the multidimensional integral associated with the unordered modeling structure. The problems deriving from the lack of smoothness of the probit model around the maximum of the log-likelihood function, which makes the optimization and the calculation of standard deviations very difficult, are carefully analyzed. A methodology to perform out-of-sample validation in the context of a joint model is proposed. Comprehensive numerical experiments have been conducted on both simulated and real data. In particular, the discrete-continuous models are estimated and applied to vehicle ownership and use models on data extracted from the 2009 National Household Travel Survey. The second part of this work offers a comprehensive statistical analysis of free-flow speed distribution; the method is applied to data collected on a sample of roads in Italy. A linear mixed model that includes speed quantiles in its predictors is estimated. Results show that there is no road effect in the analysis of free-flow speeds, which is particularly important for model transferability. A very general framework to predict random effects with few observations and incomplete access to model covariates is formulated and applied to predict the distribution of free-flow speed quantiles. The speed distribution of most road sections is successfully predicted; jack-knife estimates are calculated and used to explain why some sections are poorly predicted. Eventually, this work contributes to the literature in transportation modeling by proposing econometric model formulations for discrete-continuous variables, more efficient methods for the calculation of multivariate normal probabilities, and random effects models for free-flow speed estimation that takes into account the survey design. All methods are rigorously validated on both real and simulated data.

Development and characterisation of solution processed vanadium oxide and transparent metal oxide thin films

Metal oxide thin films are important for modern electronic devices ranging from thin film transistors to photovoltaics and functional optical coatings. Solution processed techniques allow for thin films to be rapidly deposited over a range of surfaces without the extensive processing of comparative vapour or physical deposition methods. The production of thin films of vanadium oxide prepared through dip-coating was developed enabling a greater understanding of the thin film formation. Mechanisms of depositing improved large area uniform coverage on a number of technologically relevant substrates were examined. The fundamental mechanism for polymer-assisted deposition in improving thin film surface smoothness and long range order has been delivered. Different methods were employed for adapting the alkoxide based dip-coating technique to produce a variety of amorphous and crystalline vanadium oxide based thin films. Using a wide range of material, spectroscopic and optical measurement techniques the morphology, structure and optoelectronic properties of the thin films were studied. The formation of pinholes on the surface of the thin films, due to dewetting and spinodal effects, was inhibited using the polymer assisted deposition technique. Uniform thin films with sub 50 nm thicknesses were deposited on a variety of substrates controlled through alterations to the solvent-alkoxide dilution ratios and employing polymer assisted deposition techniques. The effects of polymer assisted deposition altered the crystallized VO thin films from a granular surface structure to a polycrystalline structure composed of high density small in-plane grains. The formation of transparent VO based thin film through Si and Na substrate mediated diffusion highlighted new methods for material formation and doping.