Some central limit theorems for doubly restricted partitions

Let P_{K, L}(N) be the number of unordered partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form

Ʃ/N≤x P_K,L(N)

is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.

The main result is the asymptotic behavior of P_K,K(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

Stationary absolute distributions for chains of infinite order

Let {Ƶ_n}^∞_{n = -∞} be a stochastic process with state space S₁ = {0, 1, …, D – 1}. Such a process is called a chain of infinite order. The transitions of the chain are described by the functions

Q_i(i⁽⁰⁾) = Ƥ(Ƶ_n = i | Ƶ_{n - 1} = i ⁽⁰⁾₁, Ƶ_{n - 2} = i ⁽⁰⁾₂, …) (i ɛ S₁), where i⁽⁰⁾ = (i⁽⁰⁾₁, i⁽⁰⁾₂, …) ranges over infinite sequences from S₁. If i⁽ⁿ⁾ = (i⁽ⁿ⁾₁, i⁽ⁿ⁾₂, …) for n = 1, 2,…, then i⁽ⁿ⁾ → i⁽⁰⁾ means that for each k, i⁽ⁿ⁾_k = i⁽⁰⁾_k for all n sufficiently large.

Given functions Q_i(i⁽⁰⁾) such that

(i) 0 ≤ Q_i(i⁽⁰) ≤ ξ ˂ 1

(ii)D – 1/Ʃ/i = 0 Q_i(i⁽⁰⁾) Ξ 1

(iii) Q_i(i⁽ⁿ⁾) → Q_i(i⁽⁰⁾) whenever i⁽ⁿ⁾ → i⁽⁰⁾,

we prove the existence of a stationary chain of infinite order {Ƶ_n} whose transitions are given by

Ƥ (Ƶ_n = i | Ƶ_{n - 1}, Ƶ_{n - 2}, …) = Q_i(Ƶ_{n - 1}, Ƶ_{n - 2}, …)

With probability 1. The method also yields stationary chains {Ƶ_n} for which (iii) does not hold but whose transition probabilities are, in a sense, “locally Markovian.” These and similar results extend a paper by T.E. Harris [Pac. J. Math., 5 (1955), 707-724].

Included is a new proof of the existence and uniqueness of a stationary absolute distribution for an Nth order Markov chain in which all transitions are possible. This proof allows us to achieve our main results without the use of limit theorem techniques.

An arithmetical theorem for partially ordered sets

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d. This is the meet operation, defined as greatest lower bound. Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.

The central nervous control of walking in the locust, Schistocerca Americana

Rhythmic motor behaviors in all animals appear to be under the control of "central pattern generator" circuits, neural circuits which can produce output patterns appropriate for behavior even when isolated from their normal peripheral inputs. Insects have been a useful model system in which to study the control of legged terrestrial locomotion. Much is known about walking in insects at the behavioral level, but to date there has been no clear demonstration that a central pattern generator for walking exists. The focus of this thesis is to explore the central neural basis for locomotion in the locust, Schistocerca americana.

Rhythmic motor patterns could be evoked in leg motor neurons of isolated thoracic ganglia of locusts by the muscarinic agonist pilocarpine. These motor patterns would be appropriate for the movement of single legs during walking. Rhythmic patterns could be evoked in all three thoracic ganglia, but the segmental rhythms differed in their sensitivities to pilocarpine, their frequencies, and the phase relationships of motor neuron antagonists. These different patterns could be generated by a simple adaptable model circuit, which was both simulated and implemented in VLSI hardware. The intersegmental coordination of leg motor rhythms was then examined in preparations of isolated chains of thoracic ganglia. Correlations between motor patterns in different thoracic ganglia indicated that central coupling between segmental pattern generators is likely to contribute to the coordination of the legs during walking.

The work described here clearly demonstrates that segmental pattern generators for walking exist in insects. The pattern generators produce motor outputs which are likely to contribute to the coordination of the joints of a limb, as well as the coordination of different limbs. These studies lay the groundwork for further studies to determine the relative contributions of central and sensory neural mechanisms to terrestrial walking.

Conformal geometry processing

This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. Conformal transformations are valuable in applications because they naturally preserve the integrity of geometric data. To date, however, there has been no clearly stated and consistent theory of conformal transformations that can be used to develop general-purpose geometry processing algorithms: previous methods for computing conformal maps have been restricted to the flat two-dimensional plane, or other spaces of constant curvature. In contrast, our formulation can be used to produce---for the first time---general surface deformations that are perfectly conformal in the limit of refinement. It is for this reason that we commandeer the title Conformal Geometry Processing.

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in our theory. Given an immersed surface, we wish to construct new immersions that (i) induce a conformally equivalent metric and (ii) exhibit a prescribed change in extrinsic curvature. Curvature determines the potential in the Dirac equation; the solution of this equation determines the geometry of the new surface. We derive the precise conditions under which curvature is allowed to evolve, and develop efficient numerical algorithms for solving the Dirac equation on triangulated surfaces.

From a practical perspective, this theory has a variety of benefits: conformal maps are desirable in geometry processing because they do not exhibit shear, and therefore preserve textures as well as the quality of the mesh itself. Our discretization yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications. We also present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces.

Aseismic deformation in subduction megathrusts : central Andes and north-east Japan

We aim to characterize fault slip behavior during all stages of the seismic cycle in subduction megathrust environments with the eventual goal of understanding temporal and spatial variations of fault zone rheology, and to infer possible causal relationships between inter-, co- and post-seismic slip, as well as implications for earthquake and tsunami hazard. In particular we focus on analyzing aseismic deformation occurring during inter-seismic and post-seismic periods of the seismic cycle. We approach the problem using both Bayesian and optimization techniques. The Bayesian approach allows us to completely characterize the model parameter space by searching a posteriori estimates of the range of allowable models, to easily implement any kind of physically plausible a priori information and to perform the inversion without regularization other than that imposed by the parameterization of the model. However, the Bayesian approach computational expensive and not currently viable for quick response scenarios. Therefore, we also pursue improvements in the optimization inference scheme. We present a novel, robust and yet simple regularization technique that allows us to infer robust and somewhat more detailed models of slip on faults. We apply such methodologies, using simple quasi-static elastic models, to perform studies of inter- seismic deformation in the Central Andes subduction zone, and post-seismic deformation induced by the occurrence of the 2011 Mw 9.0 Tohoku-Oki earthquake in Japan. For the Central Andes, we present estimates of apparent coupling probability of the subduction interface and analyze its relationship to past earthquakes in the region. For Japan, we infer high spatial variability in material properties of the megathrust offshore Tohoku. We discuss the potential for a large earthquake just south of the Tohoku-Oki earthquake where our inferences suggest dominantly aseismic behavior.

Geological evolution of two crustal scale shear zones. Part I. The Rand thrust complex, northwestern Mojave Desert, California. Part II. The Magdalena metamorphic core complex, north central Sonora, Mexico

The geology and structure of two crustal scale shear zones were studied to understand the partitioning of strain within intracontinental orogenic belts. Movement histories and regional tectonic implications are deduced from observational data. The two widely separated study areas bear the imprint of intense Late Mesozoic through Middle Cenozoic tectonic activity. A regional transition from Late Cretaceous-Early Tertiary plutonism, metamorphism, and shortening strain to Middle Tertiary extension and magmatism is preserved in each area, with contrasting environments and mechanisms. Compressional phases of this tectonic history are better displayed in the Rand Mountains, whereas younger extensional structures dominate rock fabrics in the Magdalena area.

In the northwestern Mojave desert, the Rand Thrust Complex reveals a stack of four distinctive tectonic plates offset along the Garlock Fault. The lowermost plate, Rand Schist, is composed of greenschist facies metagraywacke, metachert, and metabasalt. Rand Schist is structurally overlain by Johannesburg Gneiss (= garnet-amphibolite grade orthogneisses, marbles and quartzites), which in turn is overlain by a Late Cretaceous hornblende-biotite granodiorite. Biotite granite forms the fourth and highest plate. Initial assembly of the tectonic stack involved a Late Cretaceous? south or southwest vergent overthrusting event in which Johannesburg Gneiss was imbricated and attenuated between Rand Schist and hornblende-biotite granodiorite. Thrusting postdated metamorphism and deformation of the lower two plates in separate environments. A post-kinematic stock, the Late Cretaceous Randsburg Granodiorite, intrudes deep levels of the complex and contains xenoliths of both Rand Schist and mylonitized Johannesburg? gneiss. Minimum shortening implied by the map patterns is 20 kilometers.

Some low angle faults of the Rand Thrust Complex formed or were reactivated between Late Cretaceous and Early Miocene time. South-southwest directed mylonites derived from Johannesburg Gneiss are commonly overprinted by less penetrative north-northeast vergent structures. Available kinematic information at shallower structural levels indicates that late disturbance(s) culminated in northward transport of the uppermost plate. Persistence of brittle fabrics along certain structural horizons suggests a possible association of late movement(s) with regionally known detachment faults. The four plates were juxtaposed and significant intraplate movements had ceased prior to Early Miocene emplacement of rhyolite porphyry dikes.

In the Magdalena region of north central Sonora, components of a pre-Middle Cretaceous stratigraphy are used as strain markers in tracking the evolution of a long lived orogenic belt. Important elements of the tectonic history include: (1) Compression during the Late Cretaceous and Early Tertiary, accompanied by plutonism, metamorphism, and ductile strain at depth, and thrust driven? syntectonic sedimentation at the surface. (2) Middle Tertiary transition to crustal extension, initially recorded by intrusion of leucogranites, inflation of the previously shortened middle and upper crustal section, and surface volcanism. (3) Gravity induced development of a normal sense ductile shear zone at mid crustal levels, with eventual detachment and southwestward displacement of the upper crustal stratigraphy by Early Miocene time.

Elucidation of the metamorphic core complex evolution just described was facilitated by fortuitous preservation of a unique assemblage of rocks and structures. The "type" stratigraphy utilized for regional correlation and strain analysis includes a Jurassic volcanic arc assemblage overlain by an Upper Jurassic-Lower Cretaceous quartz pebble conglomerate, in turn overlain by marine strata with fossiliferous Aptian-Albian limestones. The Jurassic strata, comprised of (a) rhyolite porphyries interstratified with quartz arenites, (b) rhyolite cobble conglomerate, and (c) intrusive granite porphyries, are known to rest on Precambrian basement north and east of the study area. The quartz pebble conglomerate is correlated with the Glance Conglomerate of southeastern Arizona and northeastern Sonora. The marine sequence represents part of an isolated arm? of the Bisbee Basin.

Crosscutting structural relationships between the pre-Middle Cretaceous supracrustal section, younger plutons, and deformational fabrics allow the tectonic sequence to be determined. Earliest phases of a Late Cretaceous-Early Tertiary orogeny are marked by emplacement of the 78 ± 3 Ma Guacomea Granodiorite (U/Pb zircon, Anderson et al., 1980) as a sill into deep levels of the layered Jurassic series. Subsequent regional metamorphism and ductile strain is recorded by a penetrative schistosity and lineation, and east-west trending folds. These fabrics are intruded by post-kinematic Early Tertiary? two mica granites. At shallower crustal levels, the orogeny is represented by north directed thrust faulting, formation of a large intermontane basin, and development of a pronounced unconformity. A second important phase of ductile strain followed Middle Tertiary? emplacement of leucogranites as sills and northwest trending dikes into intermediate levels of the deformed section (surficial volcanism was also active during this transitional period to regional extension). Gravitational instabilities resulting from crustal swelling via intrusion and thermal expansion led to development of a ductile shear zone within the stratigraphic horizon occupied by a laterally extensive leucogranite sill. With continued extension, upper crustal brittle normal faults (detachment faults) enhanced the uplift and tectonic denudation of this mylonite zone, ultimately resulting in southwestward displacement of the upper crustal stratigraphy.

Strains associated with the two ductile deformation events have been successfully partitioned through a multifaceted analysis. R_f/Ø measurements on various markers from the "type" stratigraphy allow a gradient representing cumulative strain since Middle Cretaceous time to be determined. From this gradient, noncoaxial strains accrued since emplacement of the leucogranites may be removed. Irrotational components of the postleucogranite strain are measured from quartz grain shapes in deformed granites; rotational components (shear strains) are determined from S-C fabrics and from restoration of rotated dike and vein networks. Structural observations and strain data are compatable with a deformation path of: (1) coaxial strain (pure shear?), followed by (2) injection of leucogranites as dikes (perpendicular to the minimum principle stress) and sills (parallel to the minimum principle stress), then (3) southwest directed simple shear. Modeling the late strain gradient as a simple shear zone permits a minimum displacement of 10 kilometers on the Magdalena mylonite zone/detachment fault system. Removal of the Middle Tertiary noncoaxial strains yields a residual (or pre-existing) strain gradient representative of the Late Cretaceous-Early Tertiary deformation. Several partially destrained cross sections, restored to the time of leucogranite emplacement, illustrate the idea that the upper plate of the core complex bas been detached from a region of significant topographic relief. 50% to 100% bulk extension across a 50 kilometer wide corridor is demonstrated.

Late Cenozoic tectonics of the Magdalena region are dominated by Basin and Range style faulting. Northeast and north-northwest trending high angle normal faults have interacted to extend the crust in an east-west direction. Net extension for this period is minor (10% to 15%) in comparison to the Middle Tertiary detachment related extensional episode.

Seismic structure along transitions from flat to normal subduction: central Mexico, southern Peru, and southwest Japan

The fine-scale seismic structure of the central Mexico, southern Peru, and southwest Japan subduction zones is studied using intraslab earthquakes recorded by temporary and permanent regional seismic arrays. The morphology of the transition from flat to normal subduction is explored in central Mexico and southern Peru, while in southwest Japan the spatial coincidence of a thin ultra-slow velocity layer (USL) atop the flat slab with locations of slow slip events (SSEs) is explored. This USL is also observed in central Mexico and southern Peru, where its lateral extent is used as one constraint on the nature of the flat-to-normal transitions.

In western central Mexico, I find an edge to this USL which is coincident with the western boundary of the projected Orozco Fracture Zone (OFZ) region. Forward modeling of the 2D structure of the subducted Cocos plate using a finite-difference algorithm provides constraints on the velocity and geometry of the slab’s seismic structure in this region and confirms the location of the USL edge. I propose that the Cocos slab is currently fragmenting into a North Cocos plate and a South Cocos plate along the projection of the OFZ, by a process analogous to that which occurred when the Rivera plate separated from the proto-Cocos plate 10 Ma.

In eastern central Mexico, observations of a sharp transition in slab dip near the abrupt end of the Trans Mexican Volcanic Belt (TMVB) suggest a possible slab tear located within the subducted South Cocos plate. The eastern lateral extent of the USL is found to be coincident with these features and with the western boundary of a zone of decreased seismicity, indicating a change in structure which I interpret as evidence of a possible tear. Analysis of intraslab seismicity patterns and focal mechanism orientations and faulting types provides further support for a possible tear in the South Cocos slab. This potential tear, together with the tear along the projection of the OFZ to the northwest, indicates a slab rollback mechanism in which separate slab segments move independently, allowing for mantle flow between the segments.

In southern Peru, observations of a gradual increase in slab dip coupled with a lack of any gaps or vertical offsets in the intraslab seismicity suggest a smooth contortion of the slab. Concentrations of focal mechanisms at orientations which are indicative of slab bending are also observed along the change in slab geometry. The lateral extent of the USL atop the horizontal Nazca slab is found to be coincident with the margin of the projected linear continuation of the subducting Nazca Ridge, implying a causal relationship, but not a slab tear. Waveform modeling of the 2D structure in southern Peru provides constraints on the velocity and geometry of the slab’s seismic structure and confirms the absence of any tears in the slab.

In southwest Japan, I estimate the location of a possible USL along the Philippine Sea slab surface and find this region of low velocity to be coincident with locations of SSEs that have occurred in this region. I interpret the source of the possible USL in this region as fluids dehydrated from the subducting plate, forming a high pore-fluid pressure layer, which would be expected to decrease the coupling on the plate interface and promote SSEs.

Source characteristics of recent and historic earthquakes in Central and Southern California: results from forward modeling

The long- and short-period body waves of a number of moderate earthquakes occurring in central and southern California recorded at regional (200-1400 km) and teleseismic (> 30°) distances are modeled to obtain the source parameters-focal mechanism, depth, seismic moment, and source time history. The modeling is done in the time domain using a forward modeling technique based on ray summation. A simple layer over a half space velocity model is used with additional layers being added if necessary-for example, in a basin with a low velocity lid.

The earthquakes studied fall into two geographic regions: 1) the western Transverse Ranges, and 2) the western Imperial Valley. Earthquakes in the western Transverse Ranges include the 1987 Whittier Narrows earthquake, several offshore earthquakes that occurred between 1969 and 1981, and aftershocks to the 1983 Coalinga earthquake (these actually occurred north of the Transverse Ranges but share many characteristics with those that occurred there). These earthquakes are predominantly thrust faulting events with the average strike being east-west, but with many variations. Of the six earthquakes which had sufficient short-period data to accurately determine the source time history, five were complex events. That is, they could not be modeled as a simple point source, but consisted of two or more subevents. The subevents of the Whittier Narrows earthquake had different focal mechanisms. In the other cases, the subevents appear to be the same, but small variations could not be ruled out.

The recent Imperial Valley earthquakes modeled include the two 1987 Superstition Hills earthquakes and the 1969 Coyote Mountain earthquake. All are strike-slip events, and the second 1987 earthquake is a complex event With non-identical subevents.

In all the earthquakes studied, and particularly the thrust events, constraining the source parameters required modeling several phases and distance ranges. Teleseismic P waves could provide only approximate solutions. P_(nl) waves were probably the most useful phase in determining the focal mechanism, with additional constraints supplied by the SH waves when available. Contamination of the SH waves by shear-coupled PL waves was a frequent problem. Short-period data were needed to obtain the source time function.

In addition to the earthquakes mentioned above, several historic earthquakes were also studied. Earthquakes that occurred before the existence of dense local and worldwide networks are difficult to model due to the sparse data set. It has been noticed that earthquakes that occur near each other often produce similar waveforms implying similar source parameters. By comparing recent well studied earthquakes to historic earthquakes in the same region, better constraints can be placed on the source parameters of the historic events.

The Lompoc earthquake (M=7) of 1927 is the largest offshore earthquake to occur in California this century. By direct comparison of waveforms and amplitudes with the Coalinga and Santa Lucia Banks earthquakes, the focal mechanism (thrust faulting on a northwest striking fault) and long-period seismic moment (10^(26) dyne cm) can be obtained. The S-P travel times are consistent with an offshore location, rather than one in the Hosgri fault zone.

Historic earthquakes in the western Imperial Valley were also studied. These events include the 1942 and 1954 earthquakes. The earthquakes were relocated by comparing S-P and R-S times to recent earthquakes. It was found that only minor changes in the epicenters were required but that the Coyote Mountain earthquake may have been more severely mislocated. The waveforms as expected indicated that all the events were strike-slip. Moment estimates were obtained by comparing the amplitudes of recent and historic events at stations which recorded both. The 1942 event was smaller than the 1968 Borrego Mountain earthquake although some previous studies suggested the reverse. The 1954 and 1937 earthquakes had moments close to the expected value. An aftershock of the 1942 earthquake appears to be larger than previously thought.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.

On concentrated loads and Green's functions in elastostatics

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.

Computational Microscopy: Breaking the Limit of Conventional Optics

Computational imaging is flourishing thanks to the recent advancement in array photodetectors and image processing algorithms. This thesis presents Fourier ptychography, which is a computational imaging technique implemented in microscopy to break the limit of conventional optics. With the implementation of Fourier ptychography, the resolution of the imaging system can surpass the diffraction limit of the objective lens's numerical aperture; the quantitative phase information of a sample can be reconstructed from intensity-only measurements; and the aberration of a microscope system can be characterized and computationally corrected. This computational microscopy technique enhances the performance of conventional optical systems and expands the scope of their applications.