Compressible flows at small Reynolds numbers

The problem of the slow viscous flow of a gas past a sphere is considered. The fluid cannot be treated incompressible in the limit when the Reynolds number Re, and the Mach number M, tend to zero in such a way that Re ~ o(M^2 ). In this case, the lowest order approximation to the steady Navier-Stokes equations of motion leads to a paradox discovered by Lagerstrom and Chester. This paradox is resolved within the framework of continuum mechanics using the classical slip condition and an iteration scheme that takes into account certain terms in the full Navier-Stokes equations that drop out in the approximation used by the above authors. It is found however that the drag predicted by the theory does not agree with R. A. Millikan's classic experiments on sphere drag.

The whole question of the applicability of the Navier-Stokes theory when the Knudsen number M/Re is not small is examined. A new slip condition is proposed. The idea that the Navier-Stokes equations coupled with this condition may adequately describe small Reynolds number flows when the Knudsen number is not too large is looked at in some detail. First, a general discussion of asymptotic solutions of the equations for all such flows is given. The theory is then applied to several concrete problems of fluid motion. The deductions from this theory appear to interpret and summarize the results of Millikan over a much wider range of Knudsen numbers (almost up to the free molecular or kinetic limit) than hitherto Believed possible by a purely continuum theory. Further experimental tests are suggested and certain interesting applications to the theory of dilute suspensions in gases are noted. Some of the questions raised in the main body of the work are explored further in the appendices.

Plate tectonic constraints on flat subduction and paleomagnetic constraints on rifting

Plate tectonics shapes our dynamic planet through the creation and destruction of lithosphere. This work focuses on increasing our understanding of the processes at convergent and divergent boundaries through geologic and geophysical observations at modern plate boundaries. Recent work had shown that the subducting slab in central Mexico is most likely the flattest on Earth, yet there was no consensus about what caused it to originate. The first chapter of this thesis sets out to systematically test all previously proposed mechanisms for slab flattening on the Mexican case. What we have discovered is that there is only one model for which we can find no contradictory evidence. The lack of applicability of the standard mechanisms used to explain flat subduction in the Mexican example led us to question their applications globally. The second chapter expands the search for a cause of flat subduction, in both space and time. We focus on the historical record of flat slabs in South America and look for a correlation between the shallowing and steepening of slab segments with relation to the inferred thickness of the subducting oceanic crust. Using plate reconstructions and the assumption that a crustal anomaly formed on a spreading ridge will produce two conjugate features, we recreate the history of subduction along the South American margin and find that there is no correlation between the subduction of a bathymetric highs and shallow subduction. These studies have proven that a subducting crustal anomaly is neither a sufficient or necessary condition of flat slab subduction. The final chapter in this thesis looks at the divergent plate boundary in the Gulf of California. Through geologic reconnaissance mapping and an intensive paleomagnetic sampling campaign, we try to constrain the location and orientation of a widespread volcanic marker unit, the Tuff of San Felipe. Although the resolution of the applied magnetic susceptibility technique proved inadequate to contain the direction of the pyroclastic flow with high precision, we have been able to detect the tectonic rotation of coherent blocks as well as rotation within blocks.

Multiplexed DNA-mediated electrochemistry

The aromatic core of double helical DNA possesses the unique and remarkable ability to form a conduit for electrons to travel over exceptionally long molecular distances. This core of π-stacked nucleobases creates an efficient pathway for charge transfer to proceed that is exquisitely sensitive to even subtle perturbations. Ground state electrochemistry of DNA-modified electrodes has been one of the major techniques used both to investigate and to harness the property of DNA-mediated charge transfer. DNA-modified electrodes have been an essential tool for both gaining insights into the fundamental properties of DNA and, due to the exquisite specificity of DNA-mediated charge transfer for the integrity of the π-stack, for use in next generation diagnostic sensing. Here, multiplexed DNA-modified electrodes are used to (i) gain new insights on the electrochemical coupling of metalloproteins to the DNA π-stack with relevance to the fundaments of in vivo DNA-mediated charge transfer and (ii) enhance the overall sensitivity of DNA-mediated reduction for use in the detection of low abundance diagnostic targets.

First, Methylene Blue (MB′) was covalently attached to DNA through a flexible C12 alkyl linker to yield a new redox reporter for DNA electrochemistry measurements with enhanced sensitivity. Tethered, intercalated MB′ was reduced through DNA-mediated charge transport. The redox signal intensity for MB′-dT-C12-DNA was found to be at least 3 fold larger than that of previously used Nile Blue (NB)-dT-DNA, which is coupled to the base stack via direct conjugation. The signal attenuation, due to an intervening mismatch, and therefore the degree of DNA-mediated reduction, does, however, depend on the DNA film morphology and the backfilling agent used to passivate the surface. These results highlight two possible mechanisms for the reduction of MB′ on the DNA-modified electrode that are distinguishable by their kinetics: reduction mediated by the DNA base pair stack and direct surface reduction of MB′ at the electrode. The extent of direct reduction at the surface can be minimized by overall DNA assembly conditions.

Next, a series of intercalation-based DNA-mediated electrochemical reporters were developed, using a flexible alkane linkage to validate and explore their DNA-mediated reduction. The general mechanism for the reduction of distally bound redox active species, covalently tethered to DNA through flexible alkyl linkages, was established to be an intraduplex DNA-mediated pathway. MB, NB, and anthraquinone were covalently tethered to DNA with three different covalent linkages. The extent of electronic coupling of the reporter was shown to correlate with the DNA binding affinity of the redox active species, supporting an intercalative mechanism. These electrochemical signals were shown to be exceptionally sensitive to a single intervening π-stack perturbation, an AC mismatch, in a densely packed DNA monolayer, which further supports that the reduction is DNA-mediated. Finally, this DNA-mediated reduction of MB occurs primarily via intra- rather than inter duplex intercalation, as probed through varying the proximity and integrity of the neighboring duplex DNA. Further gains to electrochemical sensitivity of our DNA-modified devices were then achieved through the application of electrocatalytic signal amplification using these solvent accessible intercalative reporters, MB-dT-C8, and hemoglobin as a novel electron sink. Electrocatalysis offers an excellent means of electrochemical signal amplification, yet in DNA based sensors, its application has been limited due to strict assembly conditions. We describe the use of hemoglobin as a robust and effective electron sink for electrocatalysis in DNA sensing on low density DNA films. Protein shielding of the heme redox center minimizes direct reduction at the electrode surface and permits assays on low density DNA films. Electrocatalysis of MB that is covalently tethered to the DNA by a flexible alkyl linkage allows for efficient interactions with both the base stack and hemoglobin. Consistent suppression of the redox signal upon incorporation of single CA mismatch in the DNA oligomer demonstrates that both the unamplified and the electrocatalytically amplified redox signals are generated through DNA-mediated charge transport. Electrocatalysis with hemoglobin is robust: it is stable to pH and temperature variations. The utility and applicability of electrocatalysis with hemoglobin is demonstrated through restriction enzyme detection, and an enhancement in sensitivity permits femtomole DNA sampling.

Finally, we expanded the application of our multiplexed DNA-modified electrodes to the electrochemical characterization of DNA-bound proteins containing [4Fe-4S] clusters. DNA-modified electrodes have become an essential tool for the characterization of the redox chemistry of DNA repair proteins that contain redox cofactors. Multiplexed analysis of EndonucleaseIII (EndoIII), a DNA repair protein containing a [4Fe-4S] cluster known to be accessible via DNA-mediated charge transport, elucidated subtle differences in the electrochemical behavior as a function of DNA morphology. DNA-bound EndoIII is seen to have two different electron transfer pathways for reduction, either through the DNA base stack or through direct surface reduction. Closely packed DNA films, where the protein has limited surface accessibility, produce electrochemical signals reflecting electron transfer that is DNA-mediated. The electrochemical comparison of EndoIII mutants, including a new family of mutations altering the electrostatics surrounding the [4Fe-4S] cluster, was able to be quantitatively performed. While little change in the midpoint potential was found for this family of mutants, significant variations in the efficiency of DNA-mediated electron transfer were apparent. Based on the stability of these proteins, examined by circular dichroism, we propose that the electron transfer pathway can be perturbed not only by the removal of aromatic residues, but also through changes in solvation near the cluster.

A new high-order Fourier continuation-based elasticity solver for complex three-dimensional geometries

This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.

Nonlinear dynamic transfer functions for certain retinal neuronal systems

The applicability of the white-noise method to the identification of a nonlinear system is investigated. Subsequently, the method is applied to certain vertebrate retinal neuronal systems and nonlinear, dynamic transfer functions are derived which describe quantitatively the information transformations starting with the light-pattern stimulus and culminating in the ganglion response which constitutes the visually-derived input to the brain. The retina of the catfish, Ictalurus punctatus, is used for the experiments.

The Wiener formulation of the white-noise theory is shown to be impractical and difficult to apply to a physical system. A different formulation based on crosscorrelation techniques is shown to be applicable to a wide range of physical systems provided certain considerations are taken into account. These considerations include the time-invariancy of the system, an optimum choice of the white-noise input bandwidth, nonlinearities that allow a representation in terms of a small number of characterizing kernels, the memory of the system and the temporal length of the characterizing experiment. Error analysis of the kernel estimates is made taking into account various sources of error such as noise at the input and output, bandwidth of white-noise input and the truncation of the gaussian by the apparatus.

Nonlinear transfer functions are obtained, as sets of kernels, for several neuronal systems: Light → Receptors, Light → Horizontal, Horizontal → Ganglion, Light → Ganglion and Light → ERG. The derived models can predict, with reasonable accuracy, the system response to any input. Comparison of model and physical system performance showed close agreement for a great number of tests, the most stringent of which is comparison of their responses to a white-noise input. Other tests include step and sine responses and power spectra.

Many functional traits are revealed by these models. Some are: (a) the receptor and horizontal cell systems are nearly linear (small signal) with certain "small" nonlinearities, and become faster (latency-wise and frequency-response-wise) at higher intensity levels, (b) all ganglion systems are nonlinear (half-wave rectification), (c) the receptive field center to ganglion system is slower (latency-wise and frequency-response-wise) than the periphery to ganglion system, (d) the lateral (eccentric) ganglion systems are just as fast (latency and frequency response) as the concentric ones, (e) (bipolar response) = (input from receptors) - (input from horizontal cell), (f) receptive field center and periphery exert an antagonistic influence on the ganglion response, (g) implications about the origin of ERG, and many others.

An analytical solution is obtained for the spatial distribution of potential in the S-space, which fits very well experimental data. Different synaptic mechanisms of excitation for the external and internal horizontal cells are implied.

On the solution of first excursion problems by simulation with applications to probabilistic seismic performance assessment

In a probabilistic assessment of the performance of structures subjected to uncertain environmental loads such as earthquakes, an important problem is to determine the probability that the structural response exceeds some specified limits within a given duration of interest. This problem is known as the first excursion problem, and it has been a challenging problem in the theory of stochastic dynamics and reliability analysis. In spite of the enormous amount of attention the problem has received, there is no procedure available for its general solution, especially for engineering problems of interest where the complexity of the system is large and the failure probability is small.

The application of simulation methods to solving the first excursion problem is investigated in this dissertation, with the objective of assessing the probabilistic performance of structures subjected to uncertain earthquake excitations modeled by stochastic processes. From a simulation perspective, the major difficulty in the first excursion problem comes from the large number of uncertain parameters often encountered in the stochastic description of the excitation. Existing simulation tools are examined, with special regard to their applicability in problems with a large number of uncertain parameters. Two efficient simulation methods are developed to solve the first excursion problem. The first method is developed specifically for linear dynamical systems, and it is found to be extremely efficient compared to existing techniques. The second method is more robust to the type of problem, and it is applicable to general dynamical systems. It is efficient for estimating small failure probabilities because the computational effort grows at a much slower rate with decreasing failure probability than standard Monte Carlo simulation. The simulation methods are applied to assess the probabilistic performance of structures subjected to uncertain earthquake excitation. Failure analysis is also carried out using the samples generated during simulation, which provide insight into the probable scenarios that will occur given that a structure fails.

Dissecting neural circuits for vision in nonhuman primates using fMRI-guided electrophysiology and optogenetics

The visual system is a remarkable platform that evolved to solve difficult computational problems such as detection, recognition, and classification of objects. Of great interest is the face-processing network, a sub-system buried deep in the temporal lobe, dedicated for analyzing specific type of objects (faces). In this thesis, I focus on the problem of face detection by the face-processing network. Insights obtained from years of developing computer-vision algorithms to solve this task have suggested that it may be efficiently and effectively solved by detection and integration of local contrast features. Does the brain use a similar strategy? To answer this question, I embark on a journey that takes me through the development and optimization of dedicated tools for targeting and perturbing deep brain structures. Data collected using MR-guided electrophysiology in early face-processing regions was found to have strong selectivity for contrast features, similar to ones used by artificial systems. While individual cells were tuned for only a small subset of features, the population as a whole encoded the full spectrum of features that are predictive to the presence of a face in an image. Together with additional evidence, my results suggest a possible computational mechanism for face detection in early face processing regions. To move from correlation to causation, I focus on adopting an emergent technology for perturbing brain activity using light: optogenetics. While this technique has the potential to overcome problems associated with the de-facto way of brain stimulation (electrical microstimulation), many open questions remain about its applicability and effectiveness for perturbing the non-human primate (NHP) brain. In a set of experiments, I use viral vectors to deliver genetically encoded optogenetic constructs to the frontal eye field and faceselective regions in NHP and examine their effects side-by-side with electrical microstimulation to assess their effectiveness in perturbing neural activity as well as behavior. Results suggest that cells are robustly and strongly modulated upon light delivery and that such perturbation can modulate and even initiate motor behavior, thus, paving the way for future explorations that may apply these tools to study connectivity and information flow in the face processing network.

Sequence-function relationships in E. coli transcriptional regulation

Understanding how transcriptional regulatory sequence maps to regulatory function remains a difficult problem in regulatory biology. Given a particular DNA sequence for a bacterial promoter region, we would like to be able to say which transcription factors bind there, how strongly they bind, and whether they interact with each other and/or RNA polymerase, with the ultimate objective of integrating knowledge of these parameters into a prediction of gene expression levels. The theoretical framework of statistical thermodynamics provides a useful framework for doing so, enabling us to predict how gene expression levels depend on transcription factor binding energies and concentrations. We used thermodynamic models, coupled with models of the sequence-dependent binding energies of transcription factors and RNAP, to construct a genotype to phenotype map for the level of repression exhibited by the lac promoter, and tested it experimentally using a set of promoter variants from E. coli strains isolated from different natural environments. For this work, we sought to ``reverse engineer'' naturally occurring promoter sequences to understand how variations in promoter sequence affects gene expression. The natural inverse of this approach is to ``forward engineer'' promoter sequences to obtain targeted levels of gene expression. We used a high precision model of RNAP-DNA sequence dependent binding energy, coupled with a thermodynamic model relating binding energy to gene expression, to predictively design and verify a suite of synthetic E. coli promoters whose expression varied over nearly three orders of magnitude.

However, although thermodynamic models enable predictions of mean levels of gene expression, it has become evident that cell-to-cell variability or ``noise'' in gene expression can also play a biologically important role. In order to address this aspect of gene regulation, we developed models based on the chemical master equation framework and used them to explore the noise properties of a number of common E. coli regulatory motifs; these properties included the dependence of the noise on parameters such as transcription factor binding strength and copy number. We then performed experiments in which these parameters were systematically varied and measured the level of variability using mRNA FISH. The results showed a clear dependence of the noise on these parameters, in accord with model predictions.

Finally, one shortcoming of the preceding modeling frameworks is that their applicability is largely limited to systems that are already well-characterized, such as the lac promoter. Motivated by this fact, we used a high throughput promoter mutagenesis assay called Sort-Seq to explore the completely uncharacterized transcriptional regulatory DNA of the E. coli mechanosensitive channel of large conductance (MscL). We identified several candidate transcription factor binding sites, and work is continuing to identify the associated proteins.

Generalizations and extensions of the Fokker-Planck-Kolmogorov equations

The problem of determining probability density functions of general transformations of random processes is considered in this thesis. A method of solution is developed in which partial differential equations satisfied by the unknown density function are derived. These partial differential equations are interpreted as generalized forms of the classical Fokker-Planck-Kolmogorov equations and are shown to imply the classical equations for certain classes of Markov processes. Extensions of the generalized equations which overcome degeneracy occurring in the steady-state case are also obtained.

The equations of Darling and Siegert are derived as special cases of the generalized equations thereby providing unity to two previously existing theories. A technique for treating non-Markov processes by studying closely related Markov processes is proposed and is seen to yield the Darling and Siegert equations directly from the classical Fokker-Planck-Kolmogorov equations.

As illustrations of their applicability, the generalized Fokker-Planck-Kolmogorov equations are presented for certain joint probability density functions associated with the linear filter. These equations are solved for the density of the output of an arbitrary linear filter excited by Markov Gaussian noise and for the density of the output of an RC filter excited by the Poisson square wave. This latter density is also found by using the extensions of the generalized equations mentioned above. Finally, some new approaches for finding the output probability density function of an RC filter-limiter-RC filter system driven by white Gaussian noise are included. The results in this case exhibit the data required for complete solution and clearly illustrate some of the mathematical difficulties inherent to the use of the generalized equations.

Solar System Small-Body Demographics with the Palomar Transient Factory Survey

Observational studies of our solar system's small-body populations (asteroids and comets) offer insight into the history of our planetary system, as these minor planets represent the left-over building blocks from its formation. The Palomar Transient Factory (PTF) survey began in 2009 as the latest wide-field sky-survey program to be conducted on the 1.2-meter Samuel Oschin telescope at Palomar Observatory. Though its main science program has been the discovery of high-energy extragalactic sources (such as supernovae), during its first five years PTF has collected nearly five million observations of over half a million unique solar system small bodies. This thesis begins to analyze this vast data set to address key population-level science topics, including: the detection rates of rare main-belt comets and small near-Earth asteroids, the spin and shape properties of asteroids as inferred from their lightcurves, the applicability of this visible light data to the interpretation of ultraviolet asteroid observations, and a comparison of the physical properties of main-belt and Jovian Trojan asteroids. Future sky-surveys would benefit from application of the analytical techniques presented herein, which include novel modeling methods and unique applications of machine-learning classification. The PTF asteroid small-body data produced in the course of this thesis work should remain a fertile source of solar system science and discovery for years to come.

The isolation and partial biochemical analysis of Sindbis virus proteins

Recently, the amino acid sequences have been reported for several proteins, including the envelope glycoproteins of Sindbis virus, which all probably span the plasma membrane with a common topology: a large N-terminal, extracellular portion, a short region buried in the bilayer, and a short C-terminal intracellular segment. The regions of these proteins buried in the bilayer correspond to portions of the protein sequences which contain a stretch of hydrophobic amino acids and which have other common characteristics, as discussed. Reasons are also described for uncertainty, in some proteins more than others, as to the precise location of some parts of the sequence relative to the membrane.

The signal hypothesis for the transmembrane translocation of proteins is briefly described and its general applicability is reviewed. There are many proteins whose translocation is accurately described by this hypothesis, but some proteins are translocated in a different manner.

The transmembraneous glycoproteins E1 and E2 of Sindbis virus, as well as the only other virion protein, the capsid protein, were purified in amounts sufficient for biochemical analysis using sensitive techniques. The amino acid composition of each protein was determined, and extensive N-terminal sequences were obtained for E1 and E2. By these techniques E1 and E2 are indistinguishable from most water soluble proteins, as they do not contain an obvious excess of hydrophobic amino acids in their N-terminal regions or in the intact molecule.

The capsid protein was found to be blocked, and so its N-terminus could not be sequenced by the usual methods. However, with the use of a special labeling technique, it was possible to incorporate tritiated acetate into the N-terminus of the protein with good specificity, which was useful in the purification of peptides from which the first amino acids in the N-terminal sequence could be identified.

Nanomole amounts of PE2, the intracellular precursor of E2, were purified by an immuno-affinity technique, and its N-terminus was analyzed. Together with other work, these results showed that PE2 is not synthesized with an N-terminal extension, and the signal sequence for translocation is probably the N-terminal amino acid sequence of the protein. This N-terminus was found to be 80-90% blocked, also by Nacetylation, and this acetylation did not affect its function as a signal sequence. The putative signal sequence was also found to contain a glycosylated asparagine residue, but the inhibition of this glycosylation did not lead to the cleavage of the sequence.

Magnetic field and pressure effects in a saturated gas laser amplifier

Theoretical and experimental studies of a gas laser amplifier are presented, assuming the amplifier is operating with a saturating optical frequency signal. The analysis is primarily concerned with the effects of the gas pressure and the presence of an axial magnetic field on the characteristics of the amplifying medium. Semiclassical radiation theory is used, along with a density matrix description of the atomic medium which relates the motion of single atoms to the macroscopic observables. A two-level description of the atom, using phenomenological source rates and decay rates, forms the basis of our analysis of the gas laser medium. Pressure effects are taken into account to a large extent through suitable choices of decay rate parameters.

Two methods for calculating the induced polarization of the atomic medium are used. The first method utilizes a perturbation expansion which is valid for signal intensities which barely reach saturation strength, and it is quite general in applicability. The second method is valid for arbitrarily strong signals, but it yields tractable solutions only for zero magnetic field or for axial magnetic fields large enough such that the Zeeman splitting is much larger than the power broadened homogeneous linewidth of the laser transition. The effects of pressure broadening of the homogeneous spectral linewidth are included in both the weak-signal and strong-signal theories; however the effects of Zeeman sublevel-mixing collisions are taken into account only in the weak-signal theory.

The behavior of a He-Ne gas laser amplifier in the presence of an axial magnetic field has been studied experimentally by measuring gain and Faraday rotation of linearly polarized resonant laser signals for various values of input signal intensity, and by measuring nonlinearity - induced anisotropy for elliptically polarized resonant laser signals of various input intensities. Two high-gain transitions in the 3.39-μ region were used for study: a J = 1 to J = 2 (3s₂ → 3p₄) transition and a J = 1 to J = 1 (3s₂ → 3p₂) transition. The input signals were tuned to the centers of their respective resonant gain lines.

The experimental results agree quite well with corresponding theoretical expressions which have been developed to include the nonlinear effects of saturation strength signals. The experimental results clearly show saturation of Faraday rotation, and for the J = 1 t o J = 1 transition a Faraday rotation reversal and a traveling wave gain dip are seen for small values of axial magnetic field. The nonlinearity induced anisotropy shows a marked dependence on the gas pressure in the amplifier tube for the J = 1 to J = 2 transition; this dependence agrees with the predictions of the general perturbational or weak signal theory when allowances are made for the effects of Zeeman sublevel-mixing collisions. The results provide a method for measuring the upper (neon 3s₂) level quadrupole moment decay rate, the dipole moment decay rates for the 3s₂ → 3p₄ and 3s₂ → 3p₂ transitions, and the effects of various types of collision processes on these decay rates.

Influence of local geology on earthquake ground motion

As a simplified approach for estimating theoretically the influence of local subsoils upon the ground motion during an earthquake, the problem of an idealized layered system subjected to vertically incident plane body waves was studied. Both the technique of steady-state analysis and the technique of transient analysis have been used to analyze the problem.

In the steady-state analysis, a recursion formula has been derived for obtaining the response of a layered system to sinusoidally steady-state input. Several conclusions are drawn concerning the nature of the amplification spectrum of a nonviscous layered system having its layer stiffnesses increasing with depth. Numerical examples are given to demonstrate the effect of layer parameters on the amplification spectrum of a layered system.

In the transient analysis, two modified shear beam models have been established for obtaining approximately the response of a layered system to earthquake-like excitation. The method of continuous modal analysis was adopted for approximate analysis of the models, with energy dissipation in the layers, if any, taken into account. Numerical examples are given to demonstrate the accuracy of the models and the effect of a layered system in modifying the input motion.

Conditions are established, under which the theory is applicable to predict the influence of local subsoils on the ground motion during an earthquake. To demonstrate the applicability of the models to actual cases, three examples of actually recorded earthquake events are examined. It is concluded that significant modification of the incoming seismic waves, as predicted by the theory, is likely to occur in well defined soft subsoils during an earthquake, provided that certain conditions concerning the nature of the incoming seismic waves are satisfied.

Localization of sources of human evoked responses

The evoked response, a signal present in the electro-encephalogram when specific sense modalities are stimulated with brief sensory inputs, has not yet revealed as much about brain function as it apparently promised when first recorded in the late 1940's. One of the problems has been to record the responses at a large number of points on the surface of the head; thus in order to achieve greater spatial resolution than previously attained, a 50-channel recording system was designed to monitor experiments with human visually evoked responses.

Conventional voltage versus time plots of the responses were found inadequate as a means of making qualitative studies of such a large data space. This problem was solved by creating a graphical display of the responses in the form of equipotential maps of the activity at successive instants during the complete response. In order to ascertain the necessary complexity of any models of the responses, factor analytic procedures were used to show that models characterized by only five or six independent parameters could adequately represent the variability in all recording channels.

One type of equivalent source for the responses which meets these specifications is the electrostatic dipole. Two different dipole models were studied: the dipole in a homogeneous sphere and the dipole in a sphere comprised of two spherical shells (of different conductivities) concentric with and enclosing a homogeneous sphere of a third conductivity. These models were used to determine nonlinear least squares fits of dipole parameters to a given potential distribution on the surface of a spherical approximation to the head. Numerous tests of the procedures were conducted with problems having known solutions. After these theoretical studies demonstrated the applicability of the technique, the models were used to determine inverse solutions for the evoked response potentials at various times throughout the responses. It was found that reliable estimates of the location and strength of cortical activity were obtained, and that the two models differed only slightly in their inverse solutions. These techniques enabled information flow in the brain, as indicated by locations and strengths of active sites, to be followed throughout the evoked response.

Tsunamis - a model of their generation and propagation

A general solution is presented for water waves generated by an arbitrary movement of the bed (in space and time) in a two-dimensional fluid domain with a uniform depth. The integral solution which is developed is based on a linearized approximation to the complete (nonlinear) set of governing equations. The general solution is evaluated for the specific case of a uniform upthrust or downthrow of a block section of the bed; two time-displacement histories of the bed movement are considered.

An integral solution (based on a linear theory) is also developed for a three-dimensional fluid domain of uniform depth for a class of bed movements which are axially symmetric. The integral solution is evaluated for the specific case of a block upthrust or downthrow of a section of the bed, circular in planform, with a time-displacement history identical to one of the motions used in the two-dimensional model.

Since the linear solutions are developed from a linearized approximation of the complete nonlinear description of wave behavior, the applicability of these solutions is investigated. Two types of non-linear effects are found which limit the applicability of the linear theory: (1) large nonlinear effects which occur in the region of generation during the bed movement, and (2) the gradual growth of nonlinear effects during wave propagation.

A model of wave behavior, which includes, in an approximate manner, both linear and nonlinear effects is presented for computing wave profiles after the linear theory has become invalid due to the growth of nonlinearities during wave propagation.

An experimental program has been conducted to confirm both the linear model for the two-dimensional fluid domain and the strategy suggested for determining wave profiles during propagation after the linear theory becomes invalid. The effect of a more general time-displacement history of the moving bed than those employed in the theoretical models is also investigated experimentally.

The linear theory is found to accurately approximate the wave behavior in the region of generation whenever the total displacement of the bed is much less than the water depth. Curves are developed and confirmed by the experiments which predict gross features of the lead wave propagating from the region of generation once the values of certain nondimensional parameters (which characterize the generation process) are known. For example, the maximum amplitude of the lead wave propagating from the region of generation has been found to never exceed approximately one-half of the total bed displacement. The gross features of the tsunami resulting from the Alaskan earthquake of 27 March 1964 can be estimated from the results of this study.