The stochastic exit problem for dynamical systems

The problem of "exit against a flow" for dynamical systems subject to small Gaussian white noise excitation is studied. Here the word "flow" refers to the behavior in phase space of the unperturbed system's state variables. "Exit against a flow" occurs if a perturbation causes the phase point to leave a phase space region within which it would normally be confined. In particular, there are two components of the problem of exit against a flow:

i) the mean exit time

ii) the phase-space distribution of exit locations.

When the noise perturbing the dynamical systems is small, the solution of each component of the problem of exit against a flow is, in general, the solution of a singularly perturbed, degenerate elliptic-parabolic boundary value problem.

Singular perturbation techniques are used to express the asymptotic solution in terms of an unknown parameter. The unknown parameter is determined using the solution of the adjoint boundary value problem.

The problem of exit against a flow for several dynamical systems of physical interest is considered, and the mean exit times and distributions of exit positions are calculated. The systems are then simulated numerically, using Monte Carlo techniques, in order to determine the validity of the asymptotic solutions.

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).

Optimal scaling in ductile fracture

This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. We also put forth a physical argument that identifies the intrinsic length and suggests a linear growth of the nonlocal energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, i.e., it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material. Next, we present an experimental assessment of the optimal scaling laws. We show that when the specific fracture energy is renormalized in a manner suggested by the optimal scaling laws, the data falls within the bounds predicted by the analysis and, moreover, they ostensibly collapse---with allowances made for experimental scatter---on a master curve dependent on the hardening exponent, but otherwise material independent.

A study of the proposed Diamond Creek Reservoir

Politically the Colorado river is an interstate as well as an international stream. Physically the basin divides itself distinctly into three sections. The upper section from head waters to the mouth of San Juan comprises about 40 percent of the total of the basin and affords about 87 percent of the total runoff, or an average of about 15 000 000 acre feet per annum. High mountains and cold weather are found in this section. The middle section from the mouth of San Juan to the mouth of the Williams comprises about 35 percent of the total area of the basin and supplies about 7 percent of the annual runoff. Narrow canyons and mild weather prevail in this section. The lower third of the basin is composed of mainly hot arid plains of low altitude. It comprises some 25 percent of the total area of the basin and furnishes about 6 percent of the average annual runoff.

The proposed Diamond Creek reservoir is located in the middle section and is wholly within the boundary of Arizona. The site is at the mouth of Diamond Creek and is only 16 m. from Beach Spring, a station on the Santa Fe railroad. It is solely a power project with a limited storage capacity. The dam which creats the reservoir is of the gravity type to be constructed across the river. The walls and foundation are of granite. For a dam of 290 feet in height, the back water will be about 25 m. up the river.

The power house will be placed right below the dam perpendicular to the axis of the river. It is entirely a concrete structure. The power installation would consist of eighteen 37 500 H.P. vertical, variable head turbines, directly connected to 28 000 kwa. 110 000 v. 3 phase, 60 cycle generators with necessary switching and auxiliary apparatus. Each unit is to be fed by a separate penstock wholly embedded into the masonry.

Concerning the power market, the main electric transmission lines would extend to Prescott, Phoenix, Mesa, Florence etc. The mining regions of the mountains of Arizona would be the most adequate market. The demand of power in the above named places might not be large at present. It will, from the observation of the writer, rapidly increase with the wonderful advancement of all kinds of industrial development.

All these things being comparatively feasible, there is one difficult problem: that is the silt. At the Diamond Creek dam site the average annual silt discharge is about 82 650 acre feet. The geographical conditions, however, will not permit silt deposites right in the reservoir. So this design will be made under the assumption given in Section 4.

The silt condition and the change of lower course of the Colorado are much like those of the Yellow River in China. But one thing is different. On the Colorado most of the canyon walls are of granite, while those on the Yellow are of alluvial loess: so it is very hard, if not impossible, to get a favorable dam site on the lower part. As a visitor to this country, I should like to see the full development of the Colorado: but how about THE YELLOW!

A computer-aided investigation of motion detection units in the fly

This work deals with two related areas: processing of visual information in the central nervous system, and the application of computer systems to research in neurophysiology.

Certain classes of interneurons in the brain and optic lobes of the blowfly Calliphora phaenicia were previously shown to be sensitive to the direction of motion of visual stimuli. These units were identified by visual field, preferred direction of motion, and anatomical location from which recorded. The present work is addressed to the questions: (1) is there interaction between pairs of these units, and (2) if such relationships can be found, what is their nature. To answer these questions, it is essential to record from two or more units simultaneously, and to use more than a single recording electrode if recording points are to be chosen independently. Accordingly, such techniques were developed and are described.

One must also have practical, convenient means for analyzing the large volumes of data so obtained. It is shown that use of an appropriately designed computer system is a profitable approach to this problem. Both hardware and software requirements for a suitable system are discussed and an approach to computer-aided data analysis developed. A description is given of members of a collection of application programs developed for analysis of neuro-physiological data and operated in the environment of and with support from an appropriate computer system. In particular, techniques developed for classification of multiple units recorded on the same electrode are illustrated as are methods for convenient graphical manipulation of data via a computer-driven display.

By means of multiple electrode techniques and the computer-aided data acquisition and analysis system, the path followed by one of the motion detection units was traced from open optic lobe through the brain and into the opposite lobe. It is further shown that this unit and its mirror image in the opposite lobe have a mutually inhibitory relationship. This relationship is investigated. The existence of interaction between other pairs of units is also shown. For pairs of units responding to motion in the same direction, the relationship is of an excitatory nature; for those responding to motion in opposed directions, it is inhibitory.

Experience gained from use of the computer system is discussed and a critical review of the current system is given. The most useful features of the system were found to be the fast response, the ability to go from one analysis technique to another rapidly and conveniently, and the interactive nature of the display system. The shortcomings of the system were problems in real-time use and the programming barrier—the fact that building new analysis techniques requires a high degree of programming knowledge and skill. It is concluded that computer system of the kind discussed will play an increasingly important role in studies of the central nervous system.

Algebraic Techniques in Coding Theory: Entropy Vectors, Frames, and Constrained Coding

The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.

The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.

The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.

The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes.

Study of the problem of the Los Angeles sewage disposal

No abstract.

Special Sewage Problem : Report on the Sewage Problem of the City of Los Angeles with Conclusions and Recommendations

Not available.

Extension theorems for functions of vanishing mean oscillation

A locally integrable function is said to be of vanishing mean oscillation (VMO) if its mean oscillation over cubes in R^d converges to zero with the volume of the cubes. We establish necessary and sufficient conditions for a locally integrable function defined on a bounded measurable set of positive measure to be the restriction to that set of a VMO function.

We consider the similar extension problem pertaining to BMO(ρ) functions; that is, those VMO functions whose mean oscillation over any cube is O(ρ(l(Q))) where l(Q) is the length of Q and ρ is a positive, non-decreasing function with ρ(0⁺) = 0.

We apply these results to obtain sufficient conditions for a Blaschke sequence to be the zeros of an analytic BMO(ρ) function on the unit disc.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.

A multiple scattering problem

The present work deals with the problem of the interaction of the electromagnetic radiation with a statistical distribution of nonmagnetic dielectric particles immersed in an infinite homogeneous isotropic, non-magnetic medium. The wavelength of the incident radiation can be less, equal or greater than the linear dimension of a particle. The distance between any two particles is several wavelengths. A single particle in the absence of the others is assumed to scatter like a Rayleigh-Gans particle, i.e. interaction between the volume elements (self-interaction) is neglected. The interaction of the particles is taken into account (multiple scattering) and conditions are set up for the case of a lossless medium which guarantee that the multiple scattering contribution is more important than the self-interaction one. These conditions relate the wavelength λ and the linear dimensions of a particle a and of the region occupied by the particles D. It is found that for constant λ/a, D is proportional to λ and that |Δχ|, where Δχ is the difference in the dielectric susceptibilities between particle and medium, has to lie within a certain range.

The total scattering field is obtained as a series the several terms of which represent the corresponding multiple scattering orders. The first term is a single scattering term. The ensemble average of the total scattering intensity is then obtained as a series which does not involve terms due to products between terms of different orders. Thus the waves corresponding to different orders are independent and their Stokes parameters add.

The second and third order intensity terms are explicitly computed. The method used suggests a general approach for computing any order. It is found that in general the first order scattering intensity pattern (or phase function) peaks in the forward direction Θ = 0. The second order tends to smooth out the pattern giving a maximum in the Θ = π/2 direction and minima in the Θ = 0 , Θ = π directions. This ceases to be true if ka (where k = 2π/λ) becomes large (> 20). For large ka the forward direction is further enhanced. Similar features are expected from the higher orders even though the critical value of ka may increase with the order.

The first order polarization of the scattered wave is determined. The ensemble average of the Stokes parameters of the scattered wave is explicitly computed for the second order. A similar method can be applied for any order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. If the latter is elliptically polarized then the first order scattered wave is elliptically polarized, but in the Θ = π/2 direction is linearly polarized. If the incident wave is circularly polarized the first order scattered wave is elliptically polarized except for the directions Θ = π/2 (linearly polarized) and Θ = 0, π (circularly polarized). The handedness of the Θ = 0 wave is the same as that of the incident whereas the handedness of the Θ = π wave is opposite. If the incident wave is linearly polarized the first order scattered wave is also linearly polarized. The second order makes the total scattered wave to be elliptically polarized for any Θ no matter what the incident wave is. However, the handedness of the total scattered wave is not altered by the second order. Higher orders have similar effects as the second order.

If the medium is lossy the general approach employed for the lossless case is still valid. Only the algebra increases in complexity. It is found that the results of the lossless case are insensitive in the first order of k_imD where k_im = imaginary part of the wave vector k and D a linear characteristic dimension of the region occupied by the particles. Thus moderately extended regions and small losses make (k_imD)² ≪ 1 and the lossy character of the medium does not alter the results of the lossless case. In general the presence of the losses tends to reduce the forward scattering.

Towards in situ Single Cell Systems Biology

Systems-level studies of biological systems rely on observations taken at a resolution lower than the essential unit of biology, the cell. Recent technical advances in DNA sequencing have enabled measurements of the transcriptomes in single cells excised from their environment, but it remains a daunting technical problem to reconstruct in situ gene expression patterns from sequencing data. In this thesis I develop methods for the routine, quantitative in situ measurement of gene expression using fluorescence microscopy.

The number of molecular species that can be measured simultaneously by fluorescence microscopy is limited by the pallet of spectrally distinct fluorophores. Thus, fluorescence microscopy is traditionally limited to the simultaneous measurement of only five labeled biomolecules at a time. The two methods described in this thesis, super-resolution barcoding and temporal barcoding, represent strategies for overcoming this limitation to monitor expression of many genes in a single cell. Super-resolution barcoding employs optical super-resolution microscopy (SRM) and combinatorial labeling via-smFISH (single molecule fluorescence in situ hybridization) to uniquely label individual mRNA species with distinct barcodes resolvable at nanometer resolution. This method dramatically increases the optical space in a cell, allowing a large numbers of barcodes to be visualized simultaneously. As a proof of principle this technology was used to study the S. cerevisiae calcium stress response. The second method, sequential barcoding, reads out a temporal barcode through multiple rounds of oligonucleotide hybridization to the same mRNA. The multiplexing capacity of sequential barcoding increases exponentially with the number of rounds of hybridization, allowing over a hundred genes to be profiled in only a few rounds of hybridization.

The utility of sequential barcoding was further demonstrated by adapting this method to study gene expression in mammalian tissues. Mammalian tissues suffer both from a large amount of auto-fluorescence and light scattering, making detection of smFISH probes on mRNA difficult. An amplified single molecule detection technology, smHCR (single molecule hairpin chain reaction), was developed to allow for the quantification of mRNA in tissue. This technology is demonstrated in combination with light sheet microscopy and background reducing tissue clearing technology, enabling whole-organ sequential barcoding to monitor in situ gene expression directly in intact mammalian tissue.

The methods presented in this thesis, specifically sequential barcoding and smHCR, enable multiplexed transcriptional observations in any tissue of interest. These technologies will serve as a general platform for future transcriptomic studies of complex tissues.