Physical Sciences and Mathematics Commons^™

Advances In Graph-Cut Optimization: Multi-Surface Models, Label Costs, And Hierarchical Costs, Andrew T. Delong

Electronic Thesis and Dissertation Repository

Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of "low-level" inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing …

Noncommutative Complex Geometry Of Quantum Projective Spaces, Ali Moatadelro

Electronic Thesis and Dissertation Repository

In this thesis, we study complex structures of quantum projective
spaces that was initiated in [19] for the quantum projective line,
$\mathbb{C}P^1_q$. In Chapters 2 and 3, which are the main parts of this thesis, we generalize the the results of [19] to the spaces $\mathbb{C}P^{2}_q$ and $\mathbb{C}P^{\ell}_q$. We consider a natural holomorphic structure on the quantum projective space already presented in [11,9],
and define holomorphic structures on its canonical quantum line bundles.
The space of holomorphic sections of these line bundles then will determine
the quantum homogeneous coordinate ring of the quantum projective space as the space of twisted …

Gkm Theory Of Rationally Smooth Group Embeddings, Richard P. Gonzales

Electronic Thesis and Dissertation Repository

This thesis is concerned with the study of rationally smooth group embeddings. We prove that the equivariant cohomology of any of these compactifications
can be described, via GKM-theory, as certain ring of piecewise polynomial functions.
Moreover, building on previous work of Renner, we show that the embeddings under consideration come equipped with both a canonical decomposition into rational cells and a filtration by equivariantly formal closed subvarieties.

The techniques developed in this monograph supply a method for constructing free
module generators on the equivariant cohomology of Q-filtrable GKM-varieties.
Our findings extend the earlier work of Arabia and Guillemin-Kogan on equivariant …

Arrangements Of Submanifolds And The Tangent Bundle Complement, Priyavrat Deshpande

Electronic Thesis and Dissertation Repository

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\A$ of locally flat codimension $1$ submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) $\FA$ and the \emph{poset of intersections} $L(\A)$. We also associate two topological spaces to $\A$. First, the complement of the union of submanifolds in $X$ which we call the \emph{set of chambers} and denote by $\Ch$. Second, the complement of union of tangent bundles of these …

Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

Electronic Thesis and Dissertation Repository

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some …

Descent Systems, Eulerian Polynomials And Toric Varieties, Letitia Mihaela Golubitsky

Electronic Thesis and Dissertation Repository

It is well-known that the Eulerian polynomials, which count permutations in S_n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the Sn -orbit for a generic weight in the weight lattice of Sn . Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections …