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Persistence problem of compact invariant manifold under random perturbation is considered in this dissertation. Under uniformly small random perturbation and the condition of normal hyperbolicity, the original invariant manifold persists and becomes a random invariant manifold. The random counterpart has random local stable and unstable manifolds. They could be invariantly foliated thanks to the normal hyperbolicity. Those underlie an extension of the geometric singular perturbation theory to the random case which means the slow manifold persists and becomes a random manifold so that the local global structure near the slow manifold persists under singular perturbation. A normal form for a random differential equation is obtained and this helps to prove a random version of the exchange lemma.

This dissertation talks about the dynamics of the drop impact in two parts, the impact of the drop on the deep liquid pool with singular jet and sound emission, and the bouncing drop with filaments on the superhydrophoic solid surface. First, we use experiments and simulations to study drop impacts on a deep liquid pool, with a focus on fine vertical jetting and underwater sound emission from entrapped bubbles, during the rebounding of the hemispherical crater. The much larger parametric complexity introduced by the use of two immiscible liquids, compared to that for the same liquid, leads to an extended variety of compound-dimple shapes. The fastest jet occurs from the rebounding of a telescope dimple shape without bubble pinch-off, at around 45 m/s, which leaves a toroidal micro-bubbles from the air-cusp at the base of the dimple. The finest jets have diameter of only 12 µm. A new focusing mechanism for singular jetting from collapsing drop-impact craters is then proposed based on high-resolution numerical simulations. The fastest jet is confined in a converging conical channel with the entrained air sheet providing a free-slip outer boundary condition. Sound can be emitted from the oscillation of the entrapped dimple-bubble, while the tiny bubble from the initial impact is induced to oscillate with the entrapped bubble, triggering the double crest of the acoustic signal. We track the compression of the bubble volume from the high-speed imaging and relate it to the hydrophone signal. In the second part, we investigate the impact of a polymeric drop on a superhydrophobic solid substrate with micropillar structure. The drop spreads on the substrate, wets the tops of the pillars, and rebounds out of the superhydrophobic soild surface. Numerous liquid filaments are stretched from the liquid drop to the attached adjacent pillars, and minuscule threads would be left on the top of the pillars using the inclined superhydrophobic solid surface. The well-organized exposed polymer threads are left on the top of the pillars after solvent evaporation. The thickness of the deposition of filament bundles using the bouncing method are thinner than those formed by drop evaporation or drop rolling from SEM (scanning electron microscope) observation.

The practice of ranking objects, events, and people to determine relevance, importance, or competitive edge is ancient. Recently, the use of rankings has permeated into daily usage, especially in the fields of business and education. When determining the association among those creating the ranks (herein called sources), the traditional assumption is that all sources compare a list of the same items (herein called elements). In the twenty-first century, it is rare that any two sources choose identical elements to rank. Adding to this difficulty, the number of credible sources creating and releasing rankings is increasing. In statistical literature, there is no current methodology that adequately assesses the association among multiple sources. We introduce rank-adapted singular value decomposition (R-A SVD), a new method that uses Kendall's tau as the underlying correlation method. We begin with (P), a matrix of data ranks. The first step is to factor the covariance matrix (K) as follows: K = cov(P) = V D^2 V Here, (V) is an orthonormal basis for the rows that is useful in identifying when sources agree as to the rank order and specifically which sources. D is a diagonal of eigenvalues. By analogy with singular value decomposition (SVD), we define U^* as U^* = PVD^(-1) The diagonal matrix, D, provides the factored eigenvalues in decreasing order. The largest eigenvalue is used to assess the overall association among the sources and is a conservative unbiased method comparable to Kendall's W. Anderson's test determines whether this association is significant and also identifies other significant eigenvalues produced by the covariance matrix.. Using Anderson's test (1963) we identify the a significantly large eigenvalues from D. When one or more eigenvalues is significant, there is evidence that the association among the sources is significant. Focusing on the a corresponding vectors of V specifically identifies which sources agree. In cases where more than one eigenvalue is significant, the $a$ significant vectors of V provide insight into factions. When more than one set of sources is in agreement, each group of agreeing sources is considered a faction. In many cases, more than one set of sources will be in agreement with one another but not necessarily with another set of sources; each group that is in agreement would be considered a faction. Using the a significant vectors of U^* provides different but equally important results. In many cases, the elements that are being ranked can be subdivided into equivalence classes. An equivalence class is defined as subpopulations of ranked elements that are similar to one another but dissimilar from other classes. When these classes exist, U^* provides insight as to how many classes and which elements belong in each class. In summary, the R-A SVD method gives the user the ability to assess whether there is any underlying association among multiple rank sources. It then identifies when sources agree and allows for more useful and careful interpretation when analyzing rank data. / Statistics

Yes / In this paper, a set of singular integral formulations are derived to calculate the surface impedance matrix on the antenna array elements. The proposed mathematical model enables electrodynamic analysis of antenna arrays designed using metamaterial-inspired structures. The formulations allow the determination of the array’s impedance, spatial and polarization characteristics at significantly low computational cost compared to conventional electromagnetic solvers based on method-of-moments (MoM) numerical technique. The accuracy of the surface impedance results obtained from the theoretical formulations are verified using the full wave EM software. It is shown that there is excellent agreement between the proposed formulations and EM software. The accuracy of the theoretical model presented is valid for single layer structures. / RTI2018-095499-B-C31, Funded by Ministerio de Ciencia, Innovación y Universidades, Gobierno de España (MCIU/AEI/FEDER,UE), and innovation programme under grant agreement H2020-MSCA-ITN-2016 SECRET-722424 and the financial support from the UK Engineering and Physical Sciences Research Council (EPSRC) under grant EP/E022936/1

The use of autonomous underwater vehicles (AUVs) for oceanic observation and research is becoming more common. Underwater gliders are a specific class of AUV that do not use conventional propulsion. Instead they change their buoyancy and center of mass location to control attitude and trajectory. The vehicles spend most of their time in long, steady glides, so even minor improvements in glide range can be magnified over multiple dives. This dissertation presents a rigid-body dynamic system for a generic vehicle operating in a moving fluid (ocean current or wind). The model is then reduced to apply to underwater gliders. A reduced-order point-mass model is analyzed for optimal gliding in the presence of a current. Different numerical method solutions are compared while attempting to achieve maximum glide range. The result, although approximate, provides good insight into how the vehicles may be operated more effectively. At the end of each dive, the gliders must change their buoyancy and pitch to transition to a climb. Improper scheduling of the buoyancy and pitch change may cause the vehicle to stall and lose directional stability. Optimal control theory is applied to the buoyancy and angle of attack scheduling of a point-mass model. A rigid-body model is analyzed on a singular arc steady glide. An analytical solution for the control required to stay on the arc is calculated. The model is linearized to calculate possible perturbation directions while remaining on the arc. The nonlinear model is then propagated in forward and reverse time with the perturbations and analyzed. Lastly, one of the numerical solutions is analyzed using the singular arc equations for verification. This work received support from the Office of Naval Research under Grant Number N00014-08-1-0012. / Ph. D.

This thesis presents both a shared memory parallel version of the hybrid classification algorithm IGSCR (iterative guided spectral class rejection) and a novel data reduction technique that can be used in conjuction with pIGSCR (parallel IGSCR). The parallel algorithm is motivated by a demonstrated need for more computing power driven by the increasing size of remote sensing datasets due to higher resolution sensors, larger study regions, and the like. Even with a fast algorithm such as pIGSCR, the reduction of dimension in a dataset is desirable in order to decrease the processing time further and possibly improve overall classification accuracy. pIGSCR was developed to produce fast and portable code using Fortran 95, OpenMP, and the Hierarchical Data Format version 5 (HDF5) and accompanying data access library. The applicability of the faster pIGSCR algorithm is demonstrated by classifying Landsat data covering most of Virginia, USA into forest and non-forest classes with approximately 90 percent accuracy. Parallel results are given using the SGI Altix 3300 shared memory computer and the SGI Altix 3700 with as many as 64 processors reaching speedups of almost 77. This fast algorithm allows an analyst to perform and assess multiple classifications to refine parameters. As an example, pIGSCR was used for a factorial analysis consisting of 42 classifications of a 1.2 gigabyte image to select the number of initial classes (70) and class purity (70%) used for the remaining two images. A feature selection or reduction method may be appropriate for a specific lassification method depending on the properties and training required for the classification method, or an alternative band selection method may be derived based on the classification method itself. This thesis introduces a feature reduction method based on the singular value decomposition (SVD). This feature reduction technique was applied to training data from two multitemporal datasets of Landsat TM/ETM+ imagery acquired over a forested area in Virginia, USA and Rondonia, Brazil. Subsequent parallel iterative guided spectral class rejection (pIGSCR) forest/non-forest classifications were performed to determine the quality of the feature reduction. The classifications of the Virginia data were five times faster using SVD based feature reduction without affecting the classification accuracy. Feature reduction using the SVD was also compared to feature reduction using principal components analysis (PCA). The highest average accuracies for the Virginia dataset (88.34%) and for the Amazon dataset (93.31%) were achieved using the SVD. The results presented here indicate that SVD based feature reduction can produce statistically significantly better classifications than PCA. / Master of Science

The method of singular-perturbation analysis is applied to the determination of range-fuel-time optimal aircraft trajectories. The problem is shown to break down into three sub-problems which are studied separately. In particular, the inner layer containing the altitude path-angle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work. As a step forward in solving the ensuing nonlinear two-point boundary-value problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundary-layer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angle-of-attack from the linear analysis is discussed. Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method. / Master of Science

In his account of the intellectual cognition of singulars, Aquinas claims that the intellect cognizes singulars by way of mental images. Some recent commentators have claimed that Aquinas' appeal to mental images is inadequate to account for the intellectual cognition of singulars because mental images considered in terms of their qualitative character alone have content that is general and are, therefore, insufficient to determine reference to a singular. That is, if Aquinas takes mental images to refer to singulars because those singulars perfectly resemble the mental images, then his account is deficient. In my paper, I argue that the critical interpretation above is predicated on a misunderstanding of Aquinas regarding the intentionality of images. I investigate Aquinas' account of the intentionality of images in order to show that Aquinas understands the reference of mental images to be determined not by their qualitative character alone but also by the causal relation that obtains between the cognizer and a singular. / Master of Arts

With usable spectrum becoming increasingly more congested, the need for robust, adaptive communications to take advantage of spatially-separated signal sources is apparent. Traditional phased array beamforming techniques used for interference removal rely on perfect calibration between elements and precise knowledge of the array configuration; however, if the exact array configuration is not known (unknown or imperfect assumption of element locations, unknown mutual coupling between elements, etc.), these traditional beamforming techniques are not viable, so a blind beamforming approach is required. A novel blind beamforming approach is proposed to address complex narrow-band interference environments where the precise array configuration is unknown. The received signal is decomposed into orthogonal narrow-band partitions using a polyphase filter-bank channelizer, and a rank-reduced version of the received matrix on each sub-channel is computed through reconstruction by retaining a subset of its singular values. The wideband spectrum is synthesized through a near-perfect polyphase reconstruction filter, and a composite wideband spectrum is obtained from the maximum eigenvector of the resulting covariance matrix.The resulting process is shown to suppress numerous interference sources (in special cases even with more than the degrees of freedom of the array), all without any knowledge of the primary signal of interest. Results are validated with both simulation and wireless laboratory over-the-air experimentation. / M.S. / As the number of devices using wireless communications increase, the amount of usable radio frequency spectrum becomes increasingly congested. As a result, the need for robust, adaptive communications to improve spectral efficiency and ensure reliable communication in the presence of interference is apparent. One solution is using beamforming techniques on digital phased array receivers to maximize the energy in a desired direction and steer nulls to remove interference. However, traditional phased array beamforming techniques used for interference removal rely on perfect calibration between antenna elements and precise knowledge of the array configuration. Consequently, if the exact array configuration is not known (unknown or imperfect assumption of element locations, unknown mutual coupling between elements, etc.), these traditional beamforming techniques are not viable, so a beamforming approach with relaxed requirements (blind beamforming) is required. This thesis proposes a novel blind beamforming approach to address complex narrow-band interference in spectrally congested environments where the precise array configuration is unknown. The resulting process is shown to suppress numerous interference sources, all without any knowledge of the primary signal of interest. Results are validated with both simulation and wireless laboratory experimentation conducted with a two-element array, verifying that proposed beamforming approach achieves a similar performance to the theoretical performance bound of receiving packets in AWGN with no interference present.