12023 research outputs found
Sort by
Empiricism and Mathematical Knowledge
No full textEmpiricism asserts that sensory perceptions and experiences are the basis of all
knowledge, i.e. essentially everything we know has either been directly perceived or deduced
from perceived knowledge. This would seem to conflict with the wealth of mathematical
knowledge that we have accrued as a civilization. Mathematics consists primarily of facts
concerning ideal objects, such as topological spaces, groups, and categories, which we have never
encountered in physical reality, and cannot reasonably expect to. Then from an empiricist
perspective, it seems impossible for us to know so much about these objects, when there is no
sensory experience on which we can base our knowledge. I first assert that mathematical
knowledge is not knowledge in the sense used by empiricists, and thus its existence does not
conflict with their views. However, I argue that first-hand mathematical knowledge, specifically
one's belief in the validity of a proof, in some sense has an empirical component
Nanofabricated Neural Probe System for Dense 3-D Recordings of Brain Activity
Get PDFComputations in brain circuits involve the coordinated activation of large populations of neurons distributed across brain areas. However, monitoring neuronal activity in the brain of intact animals with high temporal and spatial resolution has remained a technological challenge. Here we address this challenge by developing dense, three-dimensional (3-D) electrode array system for electrophysiology. The front-end of the system is composed of nanofabricated neural probes with ultrathin shanks that are engineered to minimize tissue damage. The probes are connected via flexible cables to custom PCBs that multiplex the electrophysiological signals. This system architecture decouples the front-end both mechanically and thermally from the PCB which carries all active electronics for signal conditioning and multiplexing. This system was validated in vivo with hippocampal recordings from head-fixed mice. The culmination of these efforts was a 3-D array with 1024 sites packed within 0.6 mm3 of tissue that yielded the densest electrophysiological recordings to date
Dynamics of Chemically Active Suspensions
Get PDFChemically active particles may swim by self-diffusiophoresis in a concentration gradient of chemical solutes they created themselves by patterned surface catalytic reactions. Those particles can also interact via normal diffusiophoresis in the same solute concentration field. The interaction can be attractive or repulsive. This 'field-driven' nature of the system makes its dynamics different from a thermodynamic system and is analyzed with a new simulation method. Simulations show that attractive active particles exhibit coexistence of dense and dilute regions, but it is different from a liquid-gas phase equilibrium. To explain the behavior, a continuum mechanics theory is developed based on the minimal Active Brownian Particles (ABP) model. In the continuum description, the surface force is found to be the swim stress, which can be anisotropic. The body force includes the average swim force as an internal contribution and an 'activity-gradient' force contribution. Further, behaviors of active matter at the sub-continuum scale are also analyzed. The continuum mechanics theory is shown to accurately describe the behaviors of chemically active particles. Particle clustering is explained with a linear stability analysis, and the steady state is explained with a sedimentation-like mechanical force balance
Nonvanishing of L-Functions for GL(n)
Get PDFIn this thesis I study two different approaches towards proving average results on values of L-functions, with an interest toward establishing new results on automorphic L-functions, especially concerning the nonvanishing of L-functions of degree > 2 at the center of the critical strip (and at other points of the complex plane), and their applications, particularly to p-adic L-functions. In the first problem, I evaluate a twisted average of L-values using the approximate functional equation in order to prove a result on the determination of isobaric representations of GL(3, AQ) by certain L-values of p-power twists. I also provide an application to the adjoint p-adic L-function of an elliptic curve. More specifically, I show that if E is an elliptic curve over Q with semistable reduction at some fixed prime p, then the adjoint p-adic L-function of E evaluated at any infinite set of integers relatively prime to p completely determines up to a quadratic twist the isogeny class of E.
For the second problem, which is part of a long project, I present some results towards proving an average result for the degree 4 L-function on GSp(4)/Q at the center using the relative trace formula. More specifically, I consider a suitable relative trace formula such that the spectral side is an average of central L-values of genus 2 holomorphic Siegel eigenforms of weight k and level N twisted by some fixed character. I then work towards computing the corresponding geometric side.</p
Understanding Micro- and Macro-Mechanics of Soil Liquefaction: A Necessary Step for Field-Scale Assessment
Get PDFLiquefaction is a devastating instability associated with saturated, loose, and cohesionless soils. It poses a significant risk to distributed infrastructure systems that are vital for the security, economy, safety, health, and welfare of societies. In order to make our cities resilient to the effects of liquefaction, it is important to be able to identify areas that are most susceptible. Some of the prevalent methodologies employed to identify susceptible areas include conventional slope stability analysis and the use of so-called liquefaction charts. However, these methodologies have some limitations, which motivate our research objectives. In this dissertation, we investigate the mechanics of origin of liquefaction in a laboratory test using grain-scale simulations, which helps (i) understand why certain soils liquefy under certain conditions, and (ii) identify a necessary precursor for onset of flow liquefaction. Furthermore, we investigate the mechanics of liquefaction charts using a continuum plasticity model; this can help in modeling the surface hazards of liquefaction following an earthquake. Finally, we also investigate the microscopic definition of soil shear wave velocity, a soil property that is used as an index to quantify liquefaction resistance of soil. We show that anisotropy in fabric, or grain arrangement can be correlated with anisotropy in shear wave velocity. This has the potential to quantify the effects of sample disturbance when a soil specimen is extracted from the field. In conclusion, by developing a more fundamental understanding of soil liquefaction, this dissertation takes necessary steps for a more physical assessment of liquefaction susceptibility at the field-scale
Functional, Clustered, Feedforward, and Mesoscale Brain Networks
Get PDFThe brain is a network spanning multiple scales from subcellular to macroscopic. In this thesis I present four projects studying brain networks at different levels of abstraction. The first involves determining a functional connectivity network based on neural spike trains and using a graph theoretical method to cluster groups of neurons into putative cell assemblies. In the second project I model neural networks at a microscopic level. Using diferent clustered wiring schemes, I show that almost identical spatiotemporal activity patterns can be observed, demonstrating that there is a broad neuro-architectural basis to attain structured spatiotemporal dynamics. Remarkably, irrespective of the precise topological mechanism, this behavior can be predicted by examining the spectral properties of the synaptic weight matrix. The third project introduces, via two circuit architectures, a new paradigm for feedforward processing in which inhibitory neurons have the complex and pivotal role in governing information flow in cortical network models. Finally, I analyze axonal projections in sleep deprived mice using data collected as part of the Allen Institute's Mesoscopic Connectivity Atlas. After normalizing for experimental variability, the results indicate there is no single explanatory difference in the mesoscale network between control and sleep deprived mice. Using machine learning techniques, however, animal classification could be done at levels significantly above chance. This reveals that intricate changes in connectivity do occur due to chronic sleep deprivation
Packaging and Deployment of Large Planar Spacecraft Structures
Get PDFThis thesis presents a set of novel methods to biaxially package planar structures by folding and wrapping. The structure is divided into strips connected by folds that can slip during wrapping to accommodate material thickness. These packaging schemes are highly efficient, with theoretical packaging efficiencies approaching 100%. Packaging tests on meter-scale physical models have demonstrated packaging efficiencies of up to 83%. These methods avoid permanent deformation of the structure, allowing an initially flat structure to be deployed to a flat state.
Also presented are structural architectures and deployment schemes that are compatible with these packaging methods. These structural architectures use either in-plane pretension -- suitable for membrane structures -- or out-of-plane bending stiffness to resist loading. Physical models are constructed to realize these structural architectures. The deployment of these types of structures is shown to be controllable and repeatable by conducting experiments on lab-scale models.
These packaging methods, structural architectures, and deployment schemes are applicable to a variety of spacecraft structures such as solar power arrays, solar sails, antenna arrays, and drag sails; they have the potential to enable larger variants of these structures while reducing the packaging volume required. In this thesis, these methods are applied to the preliminary structural design of a space solar power satellite. This deployable spacecraft, measuring 60 m x 60 m, can be packaged into a cylinder measuring 1.5 m in height and 1 m in diameter. It can be deployed to a flat configuration, where it acts as a stiff lightweight support framework for multifunctional tiles that collect sunlight, generate electric power, and transmit it to a ground station on Earth.</p
Disorder Driven Transitions in Non-Equilibrium Quantum Systems
Get PDFThis thesis presents studies of the role of disorder in non-equilibrium quantum systems. The quantum states relevant to dynamics in these systems are very different from the ground state of the Hamiltonian. Two distinct systems are studied, (i) periodically driven Hamiltonians in two dimensions, and (ii) electrons in a one-dimensional lattice with power-law decaying hopping amplitudes. In the first system, the novel phases that are induced from the interplay of periodic driving, topology and disorder are studied. In the second system, the Anderson transition in all the eigenstates of the Hamiltonian are studied, as a function of the power-law exponent of the hopping amplitude.
In periodically driven systems the study focuses on the effect of disorder in the nature of the topology of the steady states. First, we investigate the robustness to disorder of Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are generated by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator.
Interestingly, the effects of disorder are not necessarily adverse, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet Topological Anderson Insulator (FTAI). Such a state would be a dynamical realization of the topological Anderson insulator. We identify the conditions on the driving field necessary for observing such a transition. We realize such a disorder induced topological Floquet spectrum in the driven honeycomb lattice and quantum well models.
Finally, we show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition.
The thesis also present the study of disordered systems using Wegner's Flow equations. The Flow Equation Method was proposed as a technique for studying excited states in an interacting system in one dimension. We apply this method to a one-dimensional tight binding problem with power-law decaying hoppings. This model presents a transition as a function of the exponent of the decay. It is shown that the the entire phase diagram, i.e. the delocalized, critical and localized phases in these systems can be studied using this technique. Based on this technique, we develop a strong-bond renormalization group that procedure where we solve the Flow Equations iteratively. This renormalization group approach
provides a new framework to study the transition in this system.</p
Optomechanics with Superfluid Helium-4
Get PDFWe demonstrate the utility of superfluid helium-4 as an extremely low loss optomechanical element. We form an optomechanical system with a cylindrical niobium superconducting TE011 resonator whose 40 cm3 inner cylindrical cavity is filled with 4He. [1] Coupling is realized via the variations in permittivity resulting from the density profile of the acoustic modes. Acoustic losses in helium-4 below 500 mK are governed by the intrinsic nonlinearity of sound, leading to an attenuation which drops as T 4, indicating the possibility of quality factors (Q) over 1010 at 10 mK. In our lowest loss mode, we demonstrate this T 4 law down to 50 mK, realizing an acoustic Q of 1.35·108 at 8.1 kHz. When coupled with a low phase noise microwave source, we expect this system to be utilized as a probe of macroscopic quantized motion, for precision measurements to search for fundamental physical length scales, and as a continuous gravitational wave detector. Our estimates suggest that a resonant superfluid acoustic system could exceed the sensitivity of current broad-band detectors for narrow-band sources such as pulsars [2]
Robustness, Adaptation, and Learning in Optimal Control
Get PDFRecent technological advances have opened the door to a wide variety of dynamic control applications, which are enabled by increasing computational power in ever smaller devices. These advances are backed by reliable optimization algorithms that allow specification, synthesis, and embedded implementation of sophisticated learning-based controllers. However, as control systems become more pervasive, dynamic, and complex, the control algorithms governing them become more complex to design and analyze. In many cases, optimal control policies are practically impossible to determine unless the state dimension is small, or the dynamics are simple. Thus, in order to make implementation progress, the control designer must specialize to suboptimal architectures and approximate control. The major engineering challenge in the upcoming decades will be how to cope with the complexity of designing implementable control architectures for these smart systems while certifying their safety, robustness, and performance.
This thesis tackles the design and verification complexity by carefully employing tractable lower and upper bounds on the Lyapunov function, while making connections to robust control, formal synthesis, and machine learning. Specifically, optimization-based upper bounds are used to specify robust controllers, while lower bounds are used to obtain performance bounds and to synthesize approximately optimal policies. Implementation of these bounds depends critically on carrying out learning and optimization in the loop. Examples in aerospace, formal methods, hybrid systems, and networked adaptive systems are given, and novel sources of identifiability and persistence of excitation are discussed.</p