Bifurcation theory of nonlinear boundary value problems

The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.

A singularly perturbed linear two-point boundary-value problem

We consider the following singularly perturbed linear two-point boundary-value problem:

Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)

By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)

Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.

A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.

Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).

Neural routing circuits for forming invariant representations of visual objects

This thesis presents a biologically plausible model of an attentional mechanism for forming position- and scale-invariant representations of objects in the visual world. The model relies on a set of control neurons to dynamically modify the synaptic strengths of intra-cortical connections so that information from a windowed region of primary visual cortex (Vl) is selectively routed to higher cortical areas. Local spatial relationships (i.e., topography) within the attentional window are preserved as information is routed through the cortex, thus enabling attended objects to be represented in higher cortical areas within an object-centered reference frame that is position and scale invariant. The representation in V1 is modeled as a multiscale stack of sample nodes with progressively lower resolution at higher eccentricities. Large changes in the size of the attentional window are accomplished by switching between different levels of the multiscale stack, while positional shifts and small changes in scale are accomplished by translating and rescaling the window within a single level of the stack. The control signals for setting the position and size of the attentional window are hypothesized to originate from neurons in the pulvinar and in the deep layers of visual cortex. The dynamics of these control neurons are governed by simple differential equations that can be realized by neurobiologically plausible circuits. In pre-attentive mode, the control neurons receive their input from a low-level "saliency map" representing potentially interesting regions of a scene. During the pattern recognition phase, control neurons are driven by the interaction between top-down (memory) and bottom-up (retinal input) sources. The model respects key neurophysiological, neuroanatomical, and psychophysical data relating to attention, and it makes a variety of experimentally testable predictions.

Coding for information storage

Storage systems are widely used and have played a crucial rule in both consumer and industrial products, for example, personal computers, data centers, and embedded systems. However, such system suffers from issues of cost, restricted-lifetime, and reliability with the emergence of new systems and devices, such as distributed storage and flash memory, respectively. Information theory, on the other hand, provides fundamental bounds and solutions to fully utilize resources such as data density, information I/O and network bandwidth. This thesis bridges these two topics, and proposes to solve challenges in data storage using a variety of coding techniques, so that storage becomes faster, more affordable, and more reliable.

We consider the system level and study the integration of RAID schemes and distributed storage. Erasure-correcting codes are the basis of the ubiquitous RAID schemes for storage systems, where disks correspond to symbols in the code and are located in a (distributed) network. Specifically, RAID schemes are based on MDS (maximum distance separable) array codes that enable optimal storage and efficient encoding and decoding algorithms. With r redundancy symbols an MDS code can sustain r erasures. For example, consider an MDS code that can correct two erasures. It is clear that when two symbols are erased, one needs to access and transmit all the remaining information to rebuild the erasures. However, an interesting and practical question is: What is the smallest fraction of information that one needs to access and transmit in order to correct a single erasure? In Part I we will show that the lower bound of 1/2 is achievable and that the result can be generalized to codes with arbitrary number of parities and optimal rebuilding.

We consider the device level and study coding and modulation techniques for emerging non-volatile memories such as flash memory. In particular, rank modulation is a novel data representation scheme proposed by Jiang et al. for multi-level flash memory cells, in which a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. It eliminates the need for discrete cell levels, as well as overshoot errors, when programming cells. In order to decrease the decoding complexity, we propose two variations of this scheme in Part II: bounded rank modulation where only small sliding windows of cells are sorted to generated permutations, and partial rank modulation where only part of the n cells are used to represent data. We study limits on the capacity of bounded rank modulation and propose encoding and decoding algorithms. We show that overlaps between windows will increase capacity. We present Gray codes spanning all possible partial-rank states and using only ``push-to-the-top'' operations. These Gray codes turn out to solve an open combinatorial problem called universal cycle, which is a sequence of integers generating all possible partial permutations.

On erasure coding for distributed storage and streaming communications

The work presented in this thesis revolves around erasure correction coding, as applied to distributed data storage and real-time streaming communications.

First, we examine the problem of allocating a given storage budget over a set of nodes for maximum reliability. The objective is to find an allocation of the budget that maximizes the probability of successful recovery by a data collector accessing a random subset of the nodes. This optimization problem is challenging in general because of its combinatorial nature, despite its simple formulation. We study several variations of the problem, assuming different allocation models and access models, and determine the optimal allocation and the optimal symmetric allocation (in which all nonempty nodes store the same amount of data) for a variety of cases. Although the optimal allocation can have nonintuitive structure and can be difficult to find in general, our results suggest that, as a simple heuristic, reliable storage can be achieved by spreading the budget maximally over all nodes when the budget is large, and spreading it minimally over a few nodes when it is small. Coding would therefore be beneficial in the former case, while uncoded replication would suffice in the latter case.

Second, we study how distributed storage allocations affect the recovery delay in a mobile setting. Specifically, two recovery delay optimization problems are considered for a network of mobile storage nodes: the maximization of the probability of successful recovery by a given deadline, and the minimization of the expected recovery delay. We show that the first problem is closely related to the earlier allocation problem, and solve the second problem completely for the case of symmetric allocations. It turns out that the optimal allocations for the two problems can be quite different. In a simulation study, we evaluated the performance of a simple data dissemination and storage protocol for mobile delay-tolerant networks, and observed that the choice of allocation can have a significant impact on the recovery delay under a variety of scenarios.

Third, we consider a real-time streaming system where messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link; the receiver must subsequently decode each message within a given delay from its creation time. For erasure models containing a limited number of erasures per coding window, per sliding window, and containing erasure bursts whose maximum length is sufficiently short or long, we show that a time-invariant intrasession code asymptotically achieves the maximum message size among all codes that allow decoding under all admissible erasure patterns. For the bursty erasure model, we also show that diagonally interleaved codes derived from specific systematic block codes are asymptotically optimal over all codes in certain cases. We also study an i.i.d. erasure model in which each transmitted packet is erased independently with the same probability; the objective is to maximize the decoding probability for a given message size. We derive an upper bound on the decoding probability for any time-invariant code, and show that the gap between this bound and the performance of a family of time-invariant intrasession codes is small when the message size and packet erasure probability are small. In a simulation study, these codes performed well against a family of random time-invariant convolutional codes under a number of scenarios.

Finally, we consider the joint problems of routing and caching for named data networking. We propose a backpressure-based policy that employs virtual interest packets to make routing and caching decisions. In a packet-level simulation, the proposed policy outperformed a basic protocol that combines shortest-path routing with least-recently-used (LRU) cache replacement.

Expanding the repertoire of DNA-mediated signaling in DNA repair

DNA damage is extremely detrimental to the cell and must be repaired to protect the genome. DNA is capable of conducting charge through the overlapping π-orbitals of stacked bases; this phenomenon is extremely sensitive to the integrity of the π-stack, as perturbations attenuate DNA charge transport (CT). Based on the E. coli base excision repair (BER) proteins EndoIII and MutY, it has recently been proposed that redox-active proteins containing metal clusters can utilize DNA CT to signal one another to locate sites of DNA damage.

To expand our repertoire of proteins that utilize DNA-mediated signaling, we measured the DNA-bound redox potential of the nucleotide excision repair (NER) helicase XPD from Sulfolobus acidocaldarius. A midpoint potential of 82 mV versus NHE was observed, resembling that of the previously reported BER proteins. The redox signal increases in intensity with ATP hydrolysis in only the WT protein and mutants that maintain ATPase activity and not for ATPase-deficient mutants. The signal increase correlates directly with ATP activity, suggesting that DNA-mediated signaling may play a general role in protein signaling. Several mutations in human XPD that lead to XP-related diseases have been identified; using SaXPD, we explored how these mutations, which are conserved in the thermophile, affect protein electrochemistry.

To further understand the electrochemical signaling of XPD, we studied the yeast S. cerevisiae Rad3 protein. ScRad3 mutants were incubated on a DNA-modified electrode and exhibited a similar redox potential to SaXPD. We developed a haploid strain of S. cerevisiae that allowed for easy manipulation of Rad3. In a survival assay, the ATPase- and helicase-deficient mutants show little survival, while the two disease-related mutants exhibit survival similar to WT. When both a WT and G47R (ATPase/helicase deficient) strain were challenged with different DNA damaging agents, both exhibited comparable survival in the presence of hydroxyurea, while with methyl methanesulfonate and camptothecin, the G47R strain exhibits a significant change in growth, suggesting that Rad3 is involved in repairing damage beyond traditional NER substrates. Together, these data expand our understanding of redox-active proteins at the interface of DNA repair.

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.

A Cerenkov-ΔΕ-Cerenkov detector for high energy cosmic ray isotopes and an accelerator study of ^(40)Ar & ^(56)Fe fragmentation

This thesis has two major parts. The first part of the thesis will describe a high energy cosmic ray detector -- the High Energy Isotope Spectrometer Telescope (HEIST). HEIST is a large area (0.25 m²sr) balloon-borne isotope spectrometer designed to make high-resolution measurements of isotopes in the element range from neon to nickel (10 ≤ Z ≤ 28) at energies of about 2 GeV/nucleon. The instrument consists of a stack of 12 NaI(Tl) scintilla tors, two Cerenkov counters, and two plastic scintillators. Each of the 2-cm thick NaI disks is viewed by six 1.5-inch photomultipliers whose combined outputs measure the energy deposition in that layer. In addition, the six outputs from each disk are compared to determine the position at which incident nuclei traverse each layer to an accuracy of ~2 mm. The Cerenkov counters, which measure particle velocity, are each viewed by twelve 5-inch photomultipliers using light integration boxes.

HEIST-2 determines the mass of individual nuclei by measuring both the change in the Lorentz factor (Δγ) that results from traversing the NaI stack, and the energy loss (ΔΕ) in the stack. Since the total energy of an isotope is given by Ε = γM, the mass M can be determined by M = ΔΕ/Δγ. The instrument is designed to achieve a typical mass resolution of 0.2 amu.

The second part of this thesis presents an experimental measurement of the isotopic composition of the fragments from the breakup of high energy ⁴⁰Ar and ⁵⁶Fe nuclei. Cosmic ray composition studies rely heavily on semi-empirical estimates of the cross-sections for the nuclear fragmentation reactions which alter the composition during propagation through the interstellar medium. Experimentally measured yields of isotopes from the fragmentation of ⁴⁰Ar and ⁵⁶Fe are compared with calculated yields based on semi-empirical cross-section formulae. There are two sets of measurements. The first set of measurements, made at the Lawrence Berkeley Laboratory Bevalac using a beam of 287 MeV/nucleon ⁴⁰Ar incident on a CH₂ target, achieves excellent mass resolution (σ_m ≤ 0.2 amu) for isotopes of Mg through K using a Si(Li) detector telescope. The second set of measurements, also made at the Lawrence Berkeley Laboratory Bevalac, using a beam of 583 MeV/nucleon ⁵⁶FeFe incident on a CH₂ target, resolved Cr, Mn, and Fe fragments with a typical mass resolution of ~ 0.25 amu, through the use of the Heavy Isotope Spectrometer Telescope (HIST) which was later carried into space on ISEE-3 in 1978. The general agreement between calculation and experiment is good, but some significant differences are reported here.

Theory of the positive column of a gas discharge

An attempt is made to provide a theoretical explanation of the effect of the positive column on the voltage-current characteristic of a glow or an arc discharge. Such theories have been developed before, and all are based on balancing the production and loss of charged particles and accounting for the energy supplied to the plasma by the applied electric field. Differences among the theories arise from the approximations and omissions made in selecting processes that affect the particle and energy balances. This work is primarily concerned with the deviation from the ambipolar description of the positive column caused by space charge, electron-ion volume recombination, and temperature inhomogeneities.

The presentation is divided into three parts, the first of which involved the derivation of the final macroscopic equations from kinetic theory. The final equations are obtained by taking the first three moments of the Boltzmann equation for each of the three species in the plasma. Although the method used and the equations obtained are not novel, the derivation is carried out in detail in order to appraise the validity of numerous approximations and to justify the use of data from other sources. The equations are applied to a molecular hydrogen discharge contained between parallel walls. The applied electric field is parallel to the walls, and the dependent variables—electron and ion flux to the walls, electron and ion densities, transverse electric field, and gas temperature—vary only in the direction perpendicular to the walls. The mathematical description is given by a sixth-order nonlinear two-point boundary value problem which contains the applied field as a parameter. The amount of neutral gas and its temperature at the walls are held fixed, and the relation between the applied field and the electron density at the center of the discharge is obtained in the process of solving the problem. This relation corresponds to that between current and voltage and is used to interpret the effect of space charge, recombination, and temperature inhomogeneities on the voltage-current characteristic of the discharge.

The complete solution of the equations is impractical both numerically and analytically, and in Part II the gas temperature is assumed uniform so as to focus on the combined effects of space charge and recombination. The terms representing these effects are treated as perturbations to equations that would otherwise describe the ambipolar situation. However, the term representing space charge is not negligible in a thin boundary layer or sheath near the walls, and consequently the perturbation problem is singular. Separate solutions must be obtained in the sheath and in the main region of the discharge, and the relation between the electron density and the applied field is not determined until these solutions are matched.

In Part III the electron and ion densities are assumed equal, and the complicated space-charge calculation is thereby replaced by the ambipolar description. Recombination and temperature inhomogeneities are both important at high values of the electron density. However, the formulation of the problem permits a comparison of the relative effects, and temperature inhomogeneities are shown to be important at lower values of the electron density than recombination. The equations are solved by a direct numerical integration and by treating the term representing temperature inhomogeneities as a perturbation.

The conclusions reached in the study are primarily concerned with the association of the relation between electron density and axial field with the voltage-current characteristic. It is known that the effect of space charge can account for the subnormal glow discharge and that the normal glow corresponds to a close approach to an ambipolar situation. The effect of temperature inhomogeneities helps explain the decreasing characteristic of the arc, and the effect of recombination is not expected to appear except at very high electron densities.

Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn's inequality under various side conditions is considered, with emphasis on the Korn's constant. In the "second case" of Korn's inequality, a variational approach leads to an eigenvalue problem; it is shown that, for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to finding the values of Poisson's ratio for which the displacement boundary-value problem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue for the latter problem is examined and the results applied to the spectrum of the Korn eigenvalue problem. In this way, further information on the Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introduced and the associated eigenvalue problem is a gain related to the displacement boundary-value problem of linear elastostatics in two dimensions.

Investigations in the theory of electromagnetic scattering from expanding dielectric obstacles

An equation for the reflection which results when an expanding dielectric slab scatters normally incident plane electromagnetic waves is derived using the invariant imbedding concept. The equation is solved approximately and the character of the solution is investigated. Also, an equation for the radiation transmitted through such a slab is similarly obtained. An alternative formulation of the slab problem is presented which is applicable to the analogous problem in spherical geometry. The form of an equation for the modal reflections from a nonrelativistically expanding sphere is obtained and some salient features of the solution are described. In all cases the material is assumed to be a nondispersive, nonmagnetic dielectric whose rest frame properties are slowly varying.

On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.

The ionic basis of frequency selectivity in hair cells of the bullfrog's sacculus

Hair cells from the bull frog's sacculus, a vestibular organ responding to substrate-borne vibration, possess electrically resonant membrane properties which maximize the sensitivity of each cell to a particular frequency of mechanical input. The electrical resonance of these cells and its underlying ionic basis were studied by applying gigohm-seal recording techniques to solitary hair cells enzymatically dissociated from the sacculus. The contribution of electrical resonance to frequency selectivity was assessed from microelectrode recordings from hair cells in an excised preparation of the sacculus.

Electrical resonance in the hair cell is demonstrated by damped membrane-potential oscillations in response to extrinsic current pulses applied through the recording pipette. This response is analyzed as that of a damped harmonic oscillator. Oscillation frequency rises with membrane depolarization, from 80-160 Hz at resting potential to asymptotic values of 200-250 Hz. The sharpness of electrical tuning, denoted by the electrical quality factor, Q_e, is a bell-shaped function of membrane voltage, reaching a maximum value around eight at a membrane potential slightly positive to the resting potential.

In whole cells, three time-variant ionic currents are activated at voltages more positive than -60 to -50 mV; these are identified as a voltage-dependent, non-inactivating Ca current (I_ca), a voltage-dependent, transient K current (I_a), and a Ca-dependent K current (I_c). The C channel is identified in excised, inside-out membrane patches on the basis of its large conductance (130-200 pS), its selective permeability to Kover Na or Cl, and its activation by internal Ca ions and membrane depolarization. Analysis of open- and closed-lifetime distributions suggests that the C channel can assume at least two open and three closed kinetic states.

Exposing hair cells to external solutions that inhibit the Ca or C conductances degrades the electrical resonance properties measured under current-clamp conditions, while blocking the A conductance has no significant effect, providing evidence that only the Ca and C conductances participate in the resonance mechanism. To test the sufficiency of these two conductances to account for electrical resonance, a mathematical model is developed that describes I_ca, I_c, and intracellular Ca concentration during voltage-clamp steps. I_ca activation is approximated by a third-order Hodgkin-Huxley kinetic scheme. Ca entering the cell is assumed to be confined to a small submembrane compartment which contains an excess of Ca buffer; Ca leaves this space with first-order kinetics. The Ca- and voltage-dependent activation of C channels is described by a five-state kinetic scheme suggested by the results of single-channel observations. Parameter values in the model are adjusted to fit the waveforms of I_ca and I_c evoked by a series of voltage-clamp steps in a single cell. Having been thus constrained, the model correctly predicts the character of voltage oscillations produced by current-clamp steps, including the dependencies of oscillation frequency and Q_e on membrane voltage. The model shows quantitatively how the Ca and C conductances interact, via changes in intracellular Ca concentration, to produce electrical resonance in a vertebrate hair cell.