Matching Items (73)
Description
Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions. This dissertation studies the global behavior of finite energy solutions to the $d$-dimensional focusing NLS equation, $i partial _t u+Delta u+ |u|^{p-1}u=0, $ with initial data $u_0in H^1,; x in Rn$; the nonlinearity power $p$ and the dimension $d$ are chosen so that the scaling index $s=frac{d}{2}-frac{2}{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.
ContributorsGuevara, Cristi Darley (Author) / Roudenko, Svetlana (Thesis advisor) / Castillo_Chavez, Carlos (Committee member) / Jones, Donald (Committee member) / Mahalov, Alex (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2011
Description
In this thesis I introduce a new direction to computing using nonlinear chaotic dynamics. The main idea is rich dynamics of a chaotic system enables us to (1) build better computers that have a flexible instruction set, and (2) carry out computation that conventional computers are not good at it. Here I start from the theory, explaining how one can build a computing logic block using a chaotic system, and then I introduce a new theoretical analysis for chaos computing. Specifically, I demonstrate how unstable periodic orbits and a model based on them explains and predicts how and how well a chaotic system can do computation. Furthermore, since unstable periodic orbits and their stability measures in terms of eigenvalues are extractable from experimental times series, I develop a time series technique for modeling and predicting chaos computing from a given time series of a chaotic system. After building a theoretical framework for chaos computing I proceed to architecture of these chaos-computing blocks to build a sophisticated computing system out of them. I describe how one can arrange and organize these chaos-based blocks to build a computer. I propose a brand new computer architecture using chaos computing, which shifts the limits of conventional computers by introducing flexible instruction set. Our new chaos based computer has a flexible instruction set, meaning that the user can load its desired instruction set to the computer to reconfigure the computer to be an implementation for the desired instruction set. Apart from direct application of chaos theory in generic computation, the application of chaos theory to speech processing is explained and a novel application for chaos theory in speech coding and synthesizing is introduced. More specifically it is demonstrated how a chaotic system can model the natural turbulent flow of the air in the human speech production system and how chaotic orbits can be used to excite a vocal tract model. Also as another approach to build computing system based on nonlinear system, the idea of Logical Stochastic Resonance is studied and adapted to an autoregulatory gene network in the bacteriophage λ.
ContributorsKia, Behnam (Author) / Ditto, William (Thesis advisor) / Huang, Liang (Committee member) / Lai, Ying-Cheng (Committee member) / Helms Tillery, Stephen (Committee member) / Arizona State University (Publisher)
Created2011
Description
This thesis outlines the development of a vector retrieval technique, based on data assimilation, for a coherent Doppler LIDAR (Light Detection and Ranging). A detailed analysis of the Optimal Interpolation (OI) technique for vector retrieval is presented. Through several modifications to the OI technique, it is shown that the modified technique results in significant improvement in velocity retrieval accuracy. These modifications include changes to innovation covariance portioning, covariance binning, and analysis increment calculation. It is observed that the modified technique is able to make retrievals with better accuracy, preserves local information better, and compares well with tower measurements. In order to study the error of representativeness and vector retrieval error, a lidar simulator was constructed. Using the lidar simulator a thorough sensitivity analysis of the lidar measurement process and vector retrieval is carried out. The error of representativeness as a function of scales of motion and sensitivity of vector retrieval to look angle is quantified. Using the modified OI technique, study of nocturnal flow in Owens' Valley, CA was carried out to identify and understand uncharacteristic events on the night of March 27th 2006. Observations from 1030 UTC to 1230 UTC (0230 hr local time to 0430 hr local time) on March 27 2006 are presented. Lidar observations show complex and uncharacteristic flows such as sudden bursts of westerly cross-valley wind mixing with the dominant up-valley wind. Model results from Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS®) and other in-situ instrumentations are used to corroborate and complement these observations. The modified OI technique is used to identify uncharacteristic and extreme flow events at a wind development site. Estimates of turbulence and shear from this technique are compared to tower measurements. A formulation for equivalent wind speed in the presence of variations in wind speed and direction, combined with shear is developed and used to determine wind energy content in presence of turbulence.
ContributorsChoukulkar, Aditya (Author) / Calhoun, Ronald (Thesis advisor) / Mahalov, Alex (Committee member) / Kostelich, Eric (Committee member) / Huang, Huei-Ping (Committee member) / Phelan, Patrick (Committee member) / Arizona State University (Publisher)
Created2013
Description
In this thesis, we present the study of several physical properties of relativistic mat- ters under extreme conditions. We start by deriving the rate of the nonleptonic weak processes and the bulk viscosity in several spin-one color superconducting phases of quark matter. We also calculate the bulk viscosity in the nonlinear and anharmonic regime in the normal phase of strange quark matter. We point out several qualitative effects due to the anharmonicity, although quantitatively they appear to be relatively small. In the corresponding study, we take into account the interplay between the non- leptonic and semileptonic weak processes. The results can be important in order to relate accessible observables of compact stars to their internal composition. We also use quantum field theoretical methods to study the transport properties in monolayer graphene in a strong magnetic field. The corresponding quasi-relativistic system re- veals an anomalous quantum Hall effect, whose features are directly connected with the spontaneous flavor symmetry breaking. We study the microscopic origin of Fara- day rotation and magneto-optical transmission in graphene and show that their main features are in agreement with the experimental data.
ContributorsWang, Xinyang, Ph.D (Author) / Shovkovy, Igor (Thesis advisor) / Belitsky, Andrei (Committee member) / Easson, Damien (Committee member) / Peng, Xihong (Committee member) / Vachaspati, Tanmay (Committee member) / Arizona State University (Publisher)
Created2013
Description
Numerical simulations are very helpful in understanding the physics of the formation of structure and galaxies. However, it is sometimes difficult to interpret model data with respect to observations, partly due to the difficulties and background noise inherent to observation. The goal, here, is to attempt to bridge this gap between simulation and observation by rendering the model output in image format which is then processed by tools commonly used in observational astronomy. Images are synthesized in various filters by folding the output of cosmological simulations of gasdynamics with star-formation and dark matter with the Bruzual- Charlot stellar population synthesis models. A variation of the Virgo-Gadget numerical simulation code is used with the hybrid gas and stellar formation models of Springel and Hernquist (2003). Outputs taken at various redshifts are stacked to create a synthetic view of the simulated star clusters. Source Extractor (SExtractor) is used to find groupings of stellar populations which are considered as galaxies or galaxy building blocks and photometry used to estimate the rest frame luminosities and distribution functions. With further refinements, this is expected to provide support for missions such as JWST, as well as to probe what additional physics are needed to model the data. The results show good agreement in many respects with observed properties of the galaxy luminosity function (LF) over a wide range of high redshifts. In particular, the slope (alpha) when fitted to the standard Schechter function shows excellent agreement both in value and evolution with redshift, when compared with observation. Discrepancies of other properties with observation are seen to be a result of limitations of the simulation and additional feedback mechanisms which are needed.
ContributorsMorgan, Robert (Author) / Windhorst, Rogier A (Thesis advisor) / Scannapieco, Evan (Committee member) / Rhoads, James (Committee member) / Gardner, Carl (Committee member) / Belitsky, Andrei (Committee member) / Arizona State University (Publisher)
Created2012
Description
Understanding the temperature structure of protoplanetary disks (PPDs) is paramount to modeling disk evolution and future planet formation. PPDs around T Tauri stars have two primary heating sources, protostellar irradiation, which depends on the flaring of the disk, and accretional heating as viscous coupling between annuli dissipate energy. I have written a "1.5-D" radiative transfer code to calculate disk temperatures assuming hydrostatic and radiative equilibrium. The model solves for the temperature at all locations simultaneously using Rybicki's method, converges rapidly at high optical depth, and retains full frequency dependence. The likely cause of accretional heating in PPDs is the magnetorotational instability (MRI), which acts where gas ionization is sufficiently high for gas to couple to the magnetic field. This will occur in surface layers of the disk, leaving the interior portions of the disk inactive ("dead zone"). I calculate temperatures in PPDs undergoing such "layered accretion." Since the accretional heating is concentrated far from the midplane, temperatures in the disk's interior are lower than in PPDs modeled with vertically uniform accretion. The method is used to study for the first time disks evolving via the magnetorotational instability, which operates primarily in surface layers. I find that temperatures in layered accretion disks do not significantly differ from those of "passive disks," where no accretional heating exists. Emergent spectra are insensitive to active layer thickness, making it difficult to observationally identify disks undergoing layered vs. uniform accretion. I also calculate the ionization chemistry in PPDs, using an ionization network including multiple charge states of dust grains. Combined with a criterion for the onset of the MRI, I calculate where the MRI can be initiated and the extent of dead zones in PPDs. After accounting for feedback between temperature and active layer thickness, I find the surface density of the actively accreting layers falls rapidly with distance from the protostar, leading to a net outward flow of mass from ~0.1 to 3 AU. The clearing out of the innermost zones is possibly consistent with the observed behavior of recently discovered "transition disks."
ContributorsLesniak, Michael V., III (Author) / Desch, Steven J. (Thesis advisor) / Scannapieco, Evan (Committee member) / Timmes, Francis (Committee member) / Starrfield, Sumner (Committee member) / Belitsky, Andrei (Committee member) / Arizona State University (Publisher)
Created2012
Description
This thesis deals with the first measurements done with a cold neutron beam at the Spallation Neutron Source at Oak Ridge National Laboratory. The experimental technique consisted of capturing polarized cold neutrons by nuclei to measure parity-violation in the angular distribution of the gamma rays following neutron capture. The measurements presented here for the nuclei Chlorine ( 35Cl) and Aluminum ( 27Al ) are part of a program with the ultimate goal of measuring the asymmetry in the angular distribution of gamma rays emitted in the capture of neutrons on protons, with a precision better than 10-8, in order to extract the weak hadronic coupling constant due to pion exchange interaction with isospin change equal with one ( hπ 1). Based on theoretical calculations asymmetry in the angular distribution of the gamma rays from neutron capture on protons has an estimated size of 5·10-8. This implies that the Al parity violation asymmetry and its uncertainty have to be known with a precision smaller than 4·10-8. The proton target is liquid Hydrogen (H2) contained in an Aluminum vessel. Results are presented for parity violation and parity-conserving asymmetries in Chlorine and Aluminum. The systematic and statistical uncertainties in the calculation of the parity-violating and parity-conserving asymmetries are discussed.
ContributorsBalascuta, Septimiu (Author) / Alarcon, Ricardo (Thesis advisor) / Belitsky, Andrei (Committee member) / Doak, Bruce (Committee member) / Comfort, Joseph (Committee member) / Schmidt, Kevin (Committee member) / Arizona State University (Publisher)
Created2012
Description
Complex dynamical systems consisting interacting dynamical units are ubiquitous in nature and society. Predicting and reconstructing nonlinear dynamics of units and the complex interacting networks among them serves the base for the understanding of a variety of collective dynamical phenomena. I present a general method to address the two outstanding problems as a whole based solely on time-series measurements. The method is implemented by incorporating compressive sensing approach that enables an accurate reconstruction of complex dynamical systems in terms of both nodal equations that determines the self-dynamics of units and detailed coupling patterns among units. The representative advantages of the approach are (i) the sparse data requirement which allows for a successful reconstruction from limited measurements, and (ii) general applicability to identical and nonidentical nodal dynamics, and to networks with arbitrary interacting structure, strength and sizes. Another two challenging problem of significant interest in nonlinear dynamics: (i) predicting catastrophes in nonlinear dynamical systems in advance of their occurrences and (ii) predicting the future state for time-varying nonlinear dynamical systems, can be formulated and solved in the framework of compressive sensing using only limited measurements. Once the network structure can be inferred, the dynamics behavior on them can be investigated, for example optimize information spreading dynamics, suppress cascading dynamics and traffic congestion, enhance synchronization, game dynamics, etc. The results can yield insights to control strategies design in the real-world social and natural systems. Since 2004, there has been a tremendous amount of interest in graphene. The most amazing feature of graphene is that there exists linear energy-momentum relationship when energy is low. The quasi-particles inside the system can be treated as chiral, massless Dirac fermions obeying relativistic quantum mechanics. Therefore, the graphene provides one perfect test bed to investigate relativistic quantum phenomena, such as relativistic quantum chaotic scattering and abnormal electron paths induced by klein tunneling. This phenomenon has profound implications to the development of graphene based devices that require stable electronic properties.
ContributorsYang, Rui (Author) / Lai, Ying-Cheng (Thesis advisor) / Duman, Tolga M. (Committee member) / Akis, Richard (Committee member) / Huang, Liang (Committee member) / Arizona State University (Publisher)
Created2012
Description
What can classical chaos do to quantum systems is a fundamental issue highly relevant to a number of branches in physics. The field of quantum chaos has been active for three decades, where the focus was on non-relativistic quantumsystems described by the Schr¨odinger equation. By developing an efficient method to solve the Dirac equation in the setting where relativistic particles can tunnel between two symmetric cavities through a potential barrier, chaotic cavities are found to suppress the spread in the tunneling rate. Tunneling rate for any given energy assumes a wide range that increases with the energy for integrable classical dynamics. However, for chaotic underlying dynamics, the spread is greatly reduced. A remarkable feature, which is a consequence of Klein tunneling, arise only in relativistc quantum systems that substantial tunneling exists even for particle energy approaching zero. Similar results are found in graphene tunneling devices, implying high relevance of relativistic quantum chaos to the development of such devices. Wave propagation through random media occurs in many physical systems, where interesting phenomena such as branched, fracal-like wave patterns can arise. The generic origin of these wave structures is currently a matter of active debate. It is of fundamental interest to develop a minimal, paradigmaticmodel that can generate robust branched wave structures. In so doing, a general observation in all situations where branched structures emerge is non-Gaussian statistics of wave intensity with an algebraic tail in the probability density function. Thus, a universal algebraic wave-intensity distribution becomes the criterion for the validity of any minimal model of branched wave patterns. Coexistence of competing species in spatially extended ecosystems is key to biodiversity in nature. Understanding the dynamical mechanisms of coexistence is a fundamental problem of continuous interest not only in evolutionary biology but also in nonlinear science. A continuous model is proposed for cyclically competing species and the effect of the interplay between the interaction range and mobility on coexistence is investigated. A transition from coexistence to extinction is uncovered with a non-monotonic behavior in the coexistence probability and switches between spiral and plane-wave patterns arise. Strong mobility can either promote or hamper coexistence, while absent in lattice-based models, can be explained in terms of nonlinear partial differential equations.
ContributorsNi, Xuan (Author) / Lai, Ying-Cheng (Thesis advisor) / Huang, Liang (Committee member) / Yu, Hongbin (Committee member) / Akis, Richard (Committee member) / Arizona State University (Publisher)
Created2012
Description
It is possible in a properly controlled environment, such as industrial metrology, to make significant headway into the non-industrial constraints on image-based position measurement using the techniques of image registration and achieve repeatable feature measurements on the order of 0.3% of a pixel, or about an order of magnitude improvement on conventional real-world performance. These measurements are then used as inputs for a model optimal, model agnostic, smoothing for calibration of a laser scribe and online tracking of velocimeter using video input. Using appropriate smooth interpolation to increase effective sample density can reduce uncertainty and improve estimates. Use of the proper negative offset of the template function has the result of creating a convolution with higher local curvature than either template of target function which allows improved center-finding. Using the Akaike Information Criterion with a smoothing spline function it is possible to perform a model-optimal smooth on scalar measurements without knowing the underlying model and to determine the function describing the uncertainty in that optimal smooth. An example of empiric derivation of the parameters for a rudimentary Kalman Filter from this is then provided, and tested. Using the techniques of Exploratory Data Analysis and the "Formulize" genetic algorithm tool to convert the spline models into more accessible analytic forms resulted in stable, properly generalized, KF with performance and simplicity that exceeds "textbook" implementations thereof. Validation of the measurement includes that, in analytic case, it led to arbitrary precision in measurement of feature; in reasonable test case using the methods proposed, a reasonable and consistent maximum error of around 0.3% the length of a pixel was achieved and in practice using pixels that were 700nm in size feature position was located to within ± 2 nm. Robust applicability is demonstrated by the measurement of indicator position for a King model 2-32-G-042 rotameter.
ContributorsMunroe, Michael R (Author) / Phelan, Patrick (Thesis advisor) / Kostelich, Eric (Committee member) / Mahalov, Alex (Committee member) / Arizona State University (Publisher)
Created2012