Immersions in the stable range

Immersions of a differentiable m-manifold M in a differentiable n-manifold N, 2n > 3m+1, are classified up to regular homotopy by the homotopy classes of fibre maps F: T(M) ----> T(N) such that F(-X)=-F(X) and F(X) is nonzero of X is nonzero.

Smooth regular neighborhoods

If K is a subcomplex of a smooth triangulation of a manifold M and N is a regular neighborhood of K contained in an open neighborhood U of K in M, there is a piecewise regular homeomorphism f of M carrying N onto a smooth submanifold of U, and f reduces to the identity map on K and outside U.

On homotopy spheres of low dimension

Let M be a smooth compact manifold homotopy equivalent to the 4-sphere S^4. Then M x R^1 is homeomorphism co S^4 x R^1.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

Alternating surgeries

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

AIR SIDE HEAT TRANSFER ENHANCEMENT IN HEAT EXCHANGERS UTILIZING INNOVATIVE DESIGNS AND THE ADDITIVE MANUFACTURING TECHNIQUE

Over the last decade, rapid development of additive manufacturing techniques has allowed the fabrication of innovative and complex designs. One field that can benefit from such technology is heat exchanger fabrication, as heat exchanger design has become more and more complex due to the demand for higher performance particularly on the air side of the heat exchanger. By employing the additive manufacturing, a heat exchanger design was successfully realized, which otherwise would have been very difficult to fabricate using conventional fabrication technologies. In this dissertation, additive manufacturing technique was implemented to fabricate an advanced design which focused on a combination of heat transfer surface and fluid distribution system. Although the application selected in this dissertation is focused on power plant dry cooling applications, the results of this study can directly and indirectly benefit other sectors as well, as the air-side is often the limiting side for in liquid or single phase cooling applications. Two heat exchanger designs were studied. One was an advanced metallic heat exchanger based on manifold-microchannel technology and the other was a polymer heat exchanger based on utilization of prime surface technology. Polymer heat exchangers offer several advantages over metals such as antifouling, anticorrosion, lightweight and often less expensive than comparable metallic heat exchangers. A numerical modeling and optimization were performed to calculate a design that yield an optimum performance. The optimization results show that significant performance enhancement is noted compared to the conventional heat exchangers like wavy fins and plain plate fins. Thereafter, both heat exchangers were scaled down and fabricated using additive manufacturing and experimentally tested. The manifold-micro channel design demonstrated that despite some fabrication inaccuracies, compared to a conventional wavy-fin surface, 15% - 50% increase in heat transfer coefficient was possible for the same pressure drop value. In addition, if the fabrication inaccuracy can be eliminated, an even larger performance enhancement is predicted. Since metal based additive manufacturing is still in the developmental stage, it is anticipated that with further refinement of the manufacturing process in future designs, the fabrication accuracy can be improved. For the polymer heat exchanger, by fabricating a very thin wall heat exchanger (150μm), the wall thermal resistance, which usually becomes the limiting side for polymer heat exchanger, was calculated to account for only up to 3% of the total thermal resistance. A comparison of air-side heat transfer coefficient of the polymer heat exchanger with some of the commercially available plain plate fin surface heat exchangers show that polymer heat exchanger performance is equal or superior to plain plate fin surfaces. This shows the promising potential for polymer heat exchangers to compete with conventional metallic heat exchangers when an additive manufacturing-enabled fabrication is utilized. Major contributions of this study are as follows: (1) For the first time demonstrated the potential of additive manufacturing in metal printing of heat exchangers that benefit from a sophisticated design to yield a performance substantially above the respective conventional systems. Such heat exchangers cannot be fabricated with the conventional fabrication techniques. (2) For the first time demonstrated the potential of additive manufacturing to produce polymer heat exchangers that by design minimize the role of thermal conductivity and deliver a thermal performance equal or better that their respective metallic heat exchangers. In addition of other advantages of polymer over metal like antifouling, anticorrosion, and lightweight. Details of the work are documented in respective chapters of this thesis.

Integrated visual perception architecture for robotic clothes perception and manipulation

This thesis proposes a generic visual perception architecture for robotic clothes perception and manipulation. This proposed architecture is fully integrated with a stereo vision system and a dual-arm robot and is able to perform a number of autonomous laundering tasks. Clothes perception and manipulation is a novel research topic in robotics and has experienced rapid development in recent years. Compared to the task of perceiving and manipulating rigid objects, clothes perception and manipulation poses a greater challenge. This can be attributed to two reasons: firstly, deformable clothing requires precise (high-acuity) visual perception and dexterous manipulation; secondly, as clothing approximates a non-rigid 2-manifold in 3-space, that can adopt a quasi-infinite configuration space, the potential variability in the appearance of clothing items makes them difficult to understand, identify uniquely, and interact with by machine. From an applications perspective, and as part of EU CloPeMa project, the integrated visual perception architecture refines a pre-existing clothing manipulation pipeline by completing pre-wash clothes (category) sorting (using single-shot or interactive perception for garment categorisation and manipulation) and post-wash dual-arm flattening. To the best of the author’s knowledge, as investigated in this thesis, the autonomous clothing perception and manipulation solutions presented here were first proposed and reported by the author. All of the reported robot demonstrations in this work follow a perception-manipulation method- ology where visual and tactile feedback (in the form of surface wrinkledness captured by the high accuracy depth sensor i.e. CloPeMa stereo head or the predictive confidence modelled by Gaussian Processing) serve as the halting criteria in the flattening and sorting tasks, respectively. From scientific perspective, the proposed visual perception architecture addresses the above challenges by parsing and grouping 3D clothing configurations hierarchically from low-level curvatures, through mid-level surface shape representations (providing topological descriptions and 3D texture representations), to high-level semantic structures and statistical descriptions. A range of visual features such as Shape Index, Surface Topologies Analysis and Local Binary Patterns have been adapted within this work to parse clothing surfaces and textures and several novel features have been devised, including B-Spline Patches with Locality-Constrained Linear coding, and Topology Spatial Distance to describe and quantify generic landmarks (wrinkles and folds). The essence of this proposed architecture comprises 3D generic surface parsing and interpretation, which is critical to underpinning a number of laundering tasks and has the potential to be extended to other rigid and non-rigid object perception and manipulation tasks. The experimental results presented in this thesis demonstrate that: firstly, the proposed grasp- ing approach achieves on-average 84.7% accuracy; secondly, the proposed flattening approach is able to flatten towels, t-shirts and pants (shorts) within 9 iterations on-average; thirdly, the proposed clothes recognition pipeline can recognise clothes categories from highly wrinkled configurations and advances the state-of-the-art by 36% in terms of classification accuracy, achieving an 83.2% true-positive classification rate when discriminating between five categories of clothes; finally the Gaussian Process based interactive perception approach exhibits a substantial improvement over single-shot perception. Accordingly, this thesis has advanced the state-of-the-art of robot clothes perception and manipulation.

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

Triangular mesh parameterization with trimmed surfaces

Given a 2manifold triangular mesh \(M \subset {\mathbb {R}}^3\), with border, a parameterization of \(M\) is a FACE or trimmed surface \(F=\{S,L_0,\ldots, L_m\}\) -- \(F\) is a connected subset or region of a parametric surface \(S\), bounded by a set of LOOPs \(L_0,\ldots ,L_m\) such that each \(L_i \subset S\) is a closed 1manifold having no intersection with the other \(L_j\) LOOPs -- The parametric surface \(S\) is a statistical fit of the mesh \(M\) -- \(L_0\) is the outermost LOOP bounding \(F\) and \(L_i\) is the LOOP of the ith hole in \(F\) (if any) -- The problem of parameterizing triangular meshes is relevant for reverse engineering, tool path planning, feature detection, redesign, etc -- Stateofart mesh procedures parameterize a rectangular mesh \(M\) -- To improve such procedures, we report here the implementation of an algorithm which parameterizes meshes \(M\) presenting holes and concavities -- We synthesize a parametric surface \(S \subset {\mathbb {R}}^3\) which approximates a superset of the mesh \(M\) -- Then, we compute a set of LOOPs trimming \(S\), and therefore completing the FACE \(F=\ {S,L_0,\ldots ,L_m\}\) -- Our algorithm gives satisfactory results for \(M\) having low Gaussian curvature (i.e., \(M\) being quasi-developable or developable) -- This assumption is a reasonable one, since \(M\) is the product of manifold segmentation preprocessing -- Our algorithm computes: (1) a manifold learning mapping \(\phi : M \rightarrow U \subset {\mathbb {R}}^2\), (2) an inverse mapping \(S: W \subset {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3\), with \ (W\) being a rectangular grid containing and surpassing \(U\) -- To compute \(\phi\) we test IsoMap, Laplacian Eigenmaps and Hessian local linear embedding (best results with HLLE) -- For the back mapping (NURBS) \(S\) the crucial step is to find a control polyhedron \(P\), which is an extrapolation of \(M\) -- We calculate \(P\) by extrapolating radial basis functions that interpolate points inside \(\phi (M)\) -- We successfully test our implementation with several datasets presenting concavities, holes, and are extremely nondevelopable -- Ongoing work is being devoted to manifold segmentation which facilitates mesh parameterization

Reduced models of chemical reaction in chaotic flows

We describe and evaluate two reduced models for nonlinear chemical reactions in a chaotic laminar flow. Each model involves two separate steps to compute the chemical composition at a given location and time. The “manifold tracking model” first tracks backwards in time a segment of the stable manifold of the requisite point. This then provides a sample of the initial conditions appropriate for the second step, which requires solving one-dimensional problems for the reaction in Lagrangian coordinates. By contrast, the first step of the “branching trajectories model” simulates both the advection and diffusion of fluid particles that terminate at the appropriate point; the chemical reaction equations are then solved along each of the branched trajectories in a second step. Results from each model are compared with full numerical simulations of the reaction processes in a chaotic laminar flow.

Mixing, Chemical Reactions, and Combustion in Supersonic Flows

Experiments were conducted at the GALCIT supersonic shear-layer facility to investigate aspects of reacting transverse jets in supersonic crossflow using chemiluminescence and schlieren image-correlation velocimetry. In particular, experiments were designed to examine mixing-delay length dependencies on jet-fluid molar mass, jet diameter, and jet inclination.

The experimental results show that mixing-delay length depends on jet Reynolds number, when appropriately normalized, up to a jet Reynolds number of 500,000. Jet inclination increases the mixing-delay length, but causes less disturbance to the crossflow when compared to normal jet injection. This can be explained, in part, in terms of a control-volume analysis that relates jet inclination to flow conditions downstream of injection.

In the second part of this thesis, a combustion-modeling framework is proposed and developed that is tailored to large-eddy simulations of turbulent combustion in high-speed flows. Scaling arguments place supersonic hydrocarbon combustion in a regime of autoignition-dominated distributed reaction zones (DRZ). The proposed evolution-variable manifold (EVM) framework incorporates an ignition-delay data-driven induction model with a post-ignition manifold that uses a Lagrangian convected 'balloon' reactor model for chemistry tabulation. A large-eddy simulation incorporating the EVM framework captures several important reacting-flow features of a transverse hydrogen jet in heated-air crossflow experiment.

Jointly event extraction and visualization on Twitter via probabilistic modelling

Event extraction from texts aims to detect structured information such as what has happened, to whom, where and when. Event extraction and visualization are typically considered as two different tasks. In this paper, we propose a novel approach based on probabilistic modelling to jointly extract and visualize events from tweets where both tasks benefit from each other. We model each event as a joint distribution over named entities, a date, a location and event-related keywords. Moreover, both tweets and event instances are associated with coordinates in the visualization space. The manifold assumption that the intrinsic geometry of tweets is a low-rank, non-linear manifold within the high-dimensional space is incorporated into the learning framework using a regularization. Experimental results show that the proposed approach can effectively deal with both event extraction and visualization and performs remarkably better than both the state-of-the-art event extraction method and a pipeline approach for event extraction and visualization.

On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space (Formula presented.) and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot (Formula presented.). As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on (Formula presented.), the result of (Formula presented.) surgery in the left-handed trefoil knot (Formula presented.), J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Dios y la dama como referentes en la poesía amorosa española (siglos XV-XVII). A propósito de la hipérbole sacroprofana

In this dissertation I will study the phenomenon of the hipérbole sacroprofana in different Castillian writers of the 15th to the 17th centuries, in order to show in what does this fact consists, and how it has to be sorted out. Thus, I will also show how this tendency unfolds and how the use of either one or another resource leads the hipérbole towards different expressive horizons. Although the hipérbole sacroprofana is usually detected by most of scholars, it has never met proper attention, so that it has turned into a jumble in which the most disparate instances of a lady’s praise. Yet, an accurate analysis of this phenomenon reveals that this is not the case, and that this resource has manifold aspects and varying intentions and expressive ways as well. In this dissertation, where some five thousand hipérboles are analysed, I point out the various kinds of instances, and I classify them according to their expression and their literary-intellectual interweaving. Besides I monitor this fact to explain how hipérbole varies with the passage of times. By so doing a deficiency in the history of literary criticism, which mistook and likened all sorts of hipérboles in poetical texts, is eased. So we can see that, in its origins, hipérbole was confined to using sacred terms in profane poems out of their context (which I call hipérbole léxica o de inserción terminológica). This resource was effectual insofar as the use of words religiously connoted carried along devout attitudes towards poems of earthly love so that the lady and the feeling that the author professed became “adulterated” by Christian attitudes...