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Articles 1 - 9 of 9
Full-Text Articles in Physical Sciences and Mathematics
Jones Polynomial Obstructions For Positivity Of Knots, Lizzie Buchanan
Jones Polynomial Obstructions For Positivity Of Knots, Lizzie Buchanan
Dartmouth College Ph.D Dissertations
The fundamental problem in knot theory is distinguishing one knot from another. We accomplish this by looking at knot invariants. One such invariant is positivity. A knot is positive if it has a diagram in which all crossings are positive. A knot is almost-positive if it does not have a diagram where all crossings are positive, but it does have a diagram in which all but one crossings are positive. Given a knot with an almost-positive diagram, it is in general very hard to determine whether it might also have a positive diagram. This work provides positivity obstructions for three …
Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn
Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn
Dartmouth College Ph.D Dissertations
Recently, there has been an explosion of interest in re-imagining many-body quantum phenomena beyond equilibrium. One such effort has extended the symmetry-protected topological (SPT) phase classification of non-interacting fermions to driven and dissipative settings, uncovering novel topological phenomena that are not known to exist in equilibrium which may have wide-ranging applications in quantum science. Similar physics in non-interacting bosonic systems has remained elusive. Even at equilibrium, an "effective non-Hermiticity" intrinsic to bosonic Hamiltonians poses theoretical challenges. While this non-Hermiticity has been acknowledged, its implications have not been explored in-depth. Beyond this dynamical peculiarity, major roadblocks have arisen in the search …
Spectral Sequences And Khovanov Homology, Zachary J. Winkeler
Spectral Sequences And Khovanov Homology, Zachary J. Winkeler
Dartmouth College Ph.D Dissertations
In this thesis, we will focus on two main topics; the common thread between both will be the existence of spectral sequences relating Khovanov homology to other knot invariants. Our first topic is an invariant MKh(L) for links in thickened disks with multiple punctures. This invariant is different from but inspired by both the Asaeda-Pryzytycki-Sikora (APS) homology and its specialization to links in the solid torus. Our theory will be constructed from a Z^n-filtration on the Khovanov complex, and as a result we will get various spectral sequences relating MKh(L) to Kh(L), AKh(L), and APS(L). Our …
On The Thom Isomorphism For Groupoid-Equivariant Representable K-Theory, Zachary J. Garvey
On The Thom Isomorphism For Groupoid-Equivariant Representable K-Theory, Zachary J. Garvey
Dartmouth College Ph.D Dissertations
This thesis proves a general Thom Isomorphism in groupoid-equivariant KK-theory. Through formalizing a certain pushforward functor, we contextualize the Thom isomorphism to groupoid-equivariant representable K-theory with various support conditions. Additionally, we explicitly verify that a Thom class, determined by pullback of the Bott element via a generalized groupoid homomorphism, coincides with a Thom class defined via equivariant spinor bundles and Clifford multiplication. The tools developed in this thesis are then used to generalize a particularly interesting equivalence of two Thom isomorphisms on TX, for a Riemannian G-manifold X.
A Fast Algorithm For Simulating Multiphase Flows Through Periodic Geometries Of Arbitrary Shape, Gary R. Marple, Alex Barnett, Adrianna Gillman, Shravan Veerapaneni
A Fast Algorithm For Simulating Multiphase Flows Through Periodic Geometries Of Arbitrary Shape, Gary R. Marple, Alex Barnett, Adrianna Gillman, Shravan Veerapaneni
Dartmouth Scholarship
This paper presents a new boundary integral equation (BIE) method for simulating particulate and mul- tiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system—multiple vesicles suspended in a periodic channel of arbitrary shape—to describe the numerical method and test its performance. Rather than relying on the periodic Green’s function as classical BIE methods do, the method combines the free-space Green’s function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a …
Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp
Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp
Dartmouth Scholarship
In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool …
Topological Structures In The Equities Market Network, Gregory Leibon, Scott Pauls, Daniel Rockmore, Robert Savell
Topological Structures In The Equities Market Network, Gregory Leibon, Scott Pauls, Daniel Rockmore, Robert Savell
Dartmouth Scholarship
We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on “partition decoupled null models,” a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an application, we analyze a correlation matrix derived from 4 years of close prices of equities in the New York Stock Exchange (NYSE) and National Association of Securities Dealers Automated Quotation (NASDAQ). In this example, we expose (i) a natural structure composed of 2 interacting partitions of …
Boundary Volume And Length Spectra Of Riemannian Manifolds: What The Middle Degree Hodge Spectrum Doesn't Reveal, Carolyn S. Gordon, Juan P. Rossetti
Boundary Volume And Length Spectra Of Riemannian Manifolds: What The Middle Degree Hodge Spectrum Doesn't Reveal, Carolyn S. Gordon, Juan P. Rossetti
Dartmouth Scholarship
No abstract provided.
Isospectral Deformations Of Closed Riemannian Manifolds With Different Scalar Curvature, Carolyn S. Gordon, Ruth Gornet, Dorothee Schueth, David L. Webb
Isospectral Deformations Of Closed Riemannian Manifolds With Different Scalar Curvature, Carolyn S. Gordon, Ruth Gornet, Dorothee Schueth, David L. Webb
Dartmouth Scholarship
No abstract provided.