Finite Extinction Time for Non-Linear Absorption-Diffusion Equations

In this thesis, we develop a numerical method in order to approximate the solutions of one-dimensional, non-linear absorption-diffusion equations. We test our method for accuracy against a linear diffusion equation with a solution that can be written in closed form. We then test various types of diffusion and absorption terms to determine which ones produce extinction in finite time. We also develop a numerical method to computationally solve diffusion-free equations. We compare the numerical solutions of the one-dimensional, non-linear absorption-diffusion equation and the diffusion-free equation and we find that for the cases tested, the numerical absorption-diffusion solutions are always less than the numerical diffusion-free solutions. Furthermore, we find this is true for the cases tested when there is finite and infinite extinction time. We also look at the open problem where we have slow diffusion and weak absorption but, their combined effect is strong. Our results provide some insight into the answer of this problem

An Analytical Model of Nanometer Scale Viscoelastic Properties of Polymer Surfaces Measured Using an Atomic Force Microscope

The United States Air Force and the Department of Defense is increasingly interested in nanomaterials. To study these materials, one needs to measure the mechanics of materials on the nanoscale. Over the past few decades the atomic force microscope (AFM) has been used in various methods to establish local surface properties at the nanoscale. In particular, surface elasticity measurements are crucial to understanding nanoscale surface properties. Problems arise, however, when measuring soft surfaces such as polymers and biological specimens, because these materials have a more complex viscoelastic response. This research focuses on modeling an AFM dynamic nanoindentation experiment intended to characterize near-surface viscoelastic material parameters. The experiment uses an AFM in dynamic contact mode with a polymer surface to gather frequency dependent amplitude and phase data. A three-dimensional, dynamic viscoelastic model of the AFM and surface interaction is developed and then analytically solved in the linear approximation under appropriate physical assumptions based on the physics of the AFM experimental setup. As an illustrative application, the analytical solution is coupled with experimental data from a polystyrene material to ascertain surface material properties at the nanoscale. Our solution allows the direct calculation of the storage and loss modulus from experimental data

The Complexity of Approximately Counting Retractions

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems

Influence of surface states on the conductance spectra for Co adsorbed on Cu(111)

We calculate the conductance spectra of a Co atom adsorbed on Cu(111), considering the Co 3d orbitals within a correlated multiple configurations model interacting through the substrate band with the Co 4s orbital, which is treated in a mean-field-like approximation. By symmetry, only the dz2 orbital couples with the s orbital through the Cu bands, and the interference between both conduction channels introduces a zero-bias anomaly in the conductance spectra. We find that, while the Kondo resonance is mainly determined by the interaction of the Co d orbitals with the bulk states of the Cu(111) surface, a proper description of the contribution given by the coupling with the localized surface states to the Anderson widths is crucial to describe the interference line shape. We find that the coupling of the Co 4s orbital with the Shockley surface states is responsible for two main features observed in the measured conductance spectra, the dip shape around the Fermi energy and the resonance structure at the surface state low band edge.Fil: Tacca, Marcos Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Física del Litoral. Universidad Nacional del Litoral. Instituto de Física del Litoral; Argentina. Universitat Ulm. Faculty Of Natural Sciences; AlemaniaFil: Jacob, T.. Universitat Ulm. Faculty Of Natural Sciences; AlemaniaFil: Goldberg, Edith Catalina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Física del Litoral. Universidad Nacional del Litoral. Instituto de Física del Litoral; Argentin

Unsupervised Distillation of Syntactic Information from Contextualized Word Representations

Contextualized word representations, such as ELMo and BERT, were shown to perform well on various semantic and syntactic tasks. In this work, we tackle the task of unsupervised disentanglement between semantics and structure in neural language representations: we aim to learn a transformation of the contextualized vectors, that discards the lexical semantics, but keeps the structural information. To this end, we automatically generate groups of sentences which are structurally similar but semantically different, and use metric-learning approach to learn a transformation that emphasizes the structural component that is encoded in the vectors. We demonstrate that our transformation clusters vectors in space by structural properties, rather than by lexical semantics. Finally, we demonstrate the utility of our distilled representations by showing that they outperform the original contextualized representations in a few-shot parsing setting.Comment: Accepted in BlackboxNLP@EMNLP202

Multiorbital electronic correlation effects of Co adatoms on graphene: An ionic Hamiltonian approach

In the present work, we propose an ionic Hamiltonian for describing the interaction of graphene with an adsorbed Co atom. In this approach, the electronic correlation effects, related to the many d orbitals involved in the interaction, are taken into account by selecting appropriate electronic configurations of the adsorbed atom. The Hamiltonian parameters are calculated considering the localized and extended features of the atom-surface interacting system. The physical quantities of interest are calculated by using a Green functions formalism, solved by means of the equations of motion method closed up to a second order in the atom-band coupling term. The charge and spin fluctuations in the adsorbed Co atom are inferred from density functional theory calculations and assuming that the lower energy configurations obey Hund's rules. The calculated spectral densities and the occurrence probabilities of the different atomic configurations are analyzed as a function of the Co energy level positions and the surface temperature. In addition, the conductance spectra are calculated by using the Keldysh formalism and compared with existing measurements. We analyze the behavior, under variable bias and gate potentials, of resonancelike features in the conductance spectra which can be related to transitions between atomic configurations of low occurrence probability.Fil: Tacca, Marcos Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Física del Litoral. Universidad Nacional del Litoral. Instituto de Física del Litoral; ArgentinaFil: Jacob, T.. Universitat Ulm. Faculty of Natural Sciences; AlemaniaFil: Goldberg, Edith Catalina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Física del Litoral. Universidad Nacional del Litoral. Instituto de Física del Litoral; Argentin

Quantum control of Rydberg atoms for mesoscopic-scale quantum state and circuit preparation

Individually trapped Rydberg atoms show significant promise as a platform for scalable quantum simulation and for development of programmable quantum computers. In particular, the Rydberg blockade effect can be used to facilitate both fast qubit-qubit interactions and long coherence times via low-lying electronic states encoding the physical qubits. To bring existing Rydberg-atom-based platforms a step closer to fault-tolerant quantum computation, we demonstrate high-fidelity state and circuit preparation in a system of five atoms. We specifically show that quantum control can be used to reliably generate fully connected cluster states and to simulate the error-correction encoding circuit based on the 'Perfect Quantum Error Correcting Code' by Laflamme et al. [Phys. Rev. Lett. 77, 198 (1996)]. Our results make these ideas and their implementation directly accessible to experiments and demonstrate a promising level of noise tolerance with respect to experimental errors. With this approach, we motivate the application of quantum control in small subsystems in combination with the standard gate-based quantum circuits for direct and high-fidelity implementation of few-qubit modules

Counting Homomorphisms to K4K_4-minor-free Graphs, modulo 2

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$