Spatial Aggregation: Theory and Applications

Visual thinking plays an important role in scientific reasoning. Based on the research in automating diverse reasoning tasks about dynamical systems, nonlinear controllers, kinematic mechanisms, and fluid motion, we have identified a style of visual thinking, imagistic reasoning. Imagistic reasoning organizes computations around image-like, analogue representations so that perceptual and symbolic operations can be brought to bear to infer structure and behavior. Programs incorporating imagistic reasoning have been shown to perform at an expert level in domains that defy current analytic or numerical methods. We have developed a computational paradigm, spatial aggregation, to unify the description of a class of imagistic problem solvers. A program written in this paradigm has the following properties. It takes a continuous field and optional objective functions as input, and produces high-level descriptions of structure, behavior, or control actions. It computes a multi-layer of intermediate representations, called spatial aggregates, by forming equivalence classes and adjacency relations. It employs a small set of generic operators such as aggregation, classification, and localization to perform bidirectional mapping between the information-rich field and successively more abstract spatial aggregates. It uses a data structure, the neighborhood graph, as a common interface to modularize computations. To illustrate our theory, we describe the computational structure of three implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the spatial aggregation generic operators by mixing and matching a library of commonly used routines.Comment: See http://www.jair.org/ for any accompanying file

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or Π11-CA—do not have least transitive models while weaker theories—from GBC to GBC + ETROrd —do have least transitive models

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of $T$-realizations of a fixed countable model of $\mathsf{ZFC}$, where $T$ is a reasonable second-order set theory such as $\mathsf{GBC}$ or $\mathsf{KM}$, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from $\mathsf{KM}$ to weaker theories. They showed that every model of $\mathsf{KM}$ plus the Class Collection schema "unrolls" to a model of $\mathsf{ZFC}^-$ with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as $\mathsf{GBC} + \mathsf{ETR}$. I also show that being $T$-realizable goes down to submodels for a broad selection of second-order set theories $T$. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from $\mathsf{GBC}$ to $\mathsf{KM}$. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories---such as $\mathsf{KM}$ or $\Pi^1_1\text{-}\mathsf{CA}$---do not have least transitive models while weaker theories---from $\mathsf{GBC}$ to $\mathsf{GBC} + \mathsf{ETR}_\mathrm{Ord}$---do have least transitive models.Comment: This is my PhD dissertatio

Generalized Domination.

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113539/1/danhath_1.pd

Shaking loose mushy magma: the effect of seismic waves on magma mush bodies and the potential for triggering an eruption

A central goal within volcanology is understanding how eruptions are triggered. Statistical analyses of earthquake and eruption data indicates that volcanoes show heightened activity after major earthquakes within ~750 km of the source, caused by transient (dynamic) and permanent (static) stress, particularly on gases within the reservoir, such as the accumulation of bubbles, and crustal extension and relaxation. Refinement of the volcanic plumbing structure via geophysical imaging reveals reservoirs are largely comprised of crystal mush, however the effect of earthquakes on crystal movement within this mush is unknown. This thesis explores whether seismic shaking encourages compaction and melt expulsion within mush, and whether energy produced by seismic waves is sufficient to form melt ‘caps’ at the top of mush columns, like in crystal-poor rhyolitic melts. Building on previous studies using saturated particle “packs” as synthesised mush, particle movement under oscillation is analysed using Stokes’ Law, combined with the acceleration of waves via Γ=A⁄g, where Γ is the effective wave acceleration, A is the shaking parameters (amplitude and frequency) and g=9.81 m/s^2. Within a few hundred kilometres, accelerations (PGA) produced by seismic waves are sufficient to encourage compaction (i.e. Γ>1), as applied to six case studies from locations such as Chile and Indonesia. However, not all volcanic bodies within these case studies fall within this effective distance, as waves decay over distance via an inverse square law. Γ at the volcanoes is <1, but above Γ=0.2, meaning minor compaction and expulsion from the mush may occur, but is not of significant volume. Hence, shaking alone may not be responsible for triggering volcanic activity, and melt segregation and dynamic stress work with other triggering mechanisms. Reservoirs must already be at a critical state of instability (within 99% of the maximum overpressure) if any process, including seismic forcing, is to affect the activity

Gaussian Process-based Optimization using Mutual Information for Computer Experiments. Application to Storm Surge extremes

The computational burden of running a complex computer model can make optimization impractical. Gaussian Processes (GPs) are statistical surrogates (also known as emulators) that alleviate this issue since they cheaply replace the computer model. As a result, the exploration vs. exploitation trade-off strategy can be accelerated by building a GP surrogate. In the current study, we propose a new surrogate-based optimization scheme that minimizes the number of evaluations of the computationally expensive function. Taking advantage of parallelism of the evaluation of the unknown function, the uncertain regions are explored simultaneously, and a batch of input points is chosen using Mutual Information for Computer Experiments (MICE), a sequential design algorithm which maximizes the in- formation theoretic Mutual Information over the input space. The computational efficiency of interweaving the optimization scheme with MICE (optim-MICE) is examined and demonstrated on test functions. Optim-MICE is compared with state- of-the-art heuristics. We demonstrate that optim-MICE outperforms the alternative schemes on a large range of computational experiments. The proposed algorithm is also employed to study the extrema of coastal storm waves, such as the ones that ob- served during Typhoon Haiyan (2013, Philippines). A stretch of coral reef near the coast, which was expected to protect the coastal communities, actually amplified the waves. The propagation and breaking process of such large nearshore waves can be successfully captured by a phase-resolving wave model. However, the computational complexity of the simulator makes optimization tasks impractical. The optim-MICE algorithm is therefore used to find the maximum breaking wave (bore) height and the maximum run-up. In two idealised settings, we efficiently identify the conditions that create the largest storm waves at the coast using a minimal number of simulations. This is the first surrogate-based optimization of storm waves and it opens the door to previously inconceivable coastal risk assessments

Part 1 of Martin's Conjecture for order-preserving and measure-preserving functions

Martin's Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first classifies functions which are not above the identity and the second of classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which preserve Turing reducibility). We prove the first part of the conjecture for all order-preserving functions. We do this by introducing a class of functions on the Turing degrees which we call "measure-preserving" and proving that part 1 of Martin's Conjecture holds for all measure-preserving functions and also that all non-trivial order-preserving functions are measure-preserving. Our result on measure-preserving functions has several other consequences for Martin's Conjecture, including an equivalence between part 1 of the conjecture and a statement about the structure of the Rudin-Keisler order on ultrafilters on the Turing degrees.Comment: 44 pages; updated to correct some attributions and fix some typo