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A place for no place is a practice based research project that investigates the notion of in-between space. The focus of this project is to consider the nature of binary oppositions and the thresholds between, through the designing of memorial space. The key issue of this project is aimed at investigation of the nature of thresholds of dualism, contrasting dualities and binary oppositions.With help of this project, I attempt to discuss a design methodology by balancing the binarization in natural environment in terms of applying their principles to creating the space, meanwhile, making the space from outside Visually the practical part of the exegesis will grow from process of spatial analysis and detailing by consulting with the key spatial binary oppositions of the site specific projects – Flat Alleyway and Wen-chuan Earthquake Memorial Park. Spatial arrangement and function, along with their codified details will be domesticated with concerns of the special character of the site and the consideration of scale, function, body movement, interior and exterior decoration aspects of the space. The result will be a cohesive range that represents new spaces by the meaning of In-Between-Ness. Heuristic and practical projects are my main methodological approaches. My work does not seek to fix a solution by the notion of in-between, but open up area of ongoing discovery. The practical spatial design projects are regarded as a process of meditation for a self-development, which allows me to reconsider, renegotiate, reflect and renew my work throughout the practical process. In this way, hidden spatial codes can be brought out to the surface.

<p>The main achievement of this work is the development of a duality theory for optimal control problems with stochastic differential equations. Incipient with the Hamilton-Jacobi-Bellman equation we established a dual problem to a given stochastic control problem and were also able to generalise the assembled theory.</p>

We study the target space duality of classical two dimensional sigma models. The models with dilaton and tachyon field are analyzed. As a motivating example the historical electric-magnetic duality is presented. We review the construction of the duality transformation and the integrability conditions for the nonlinear sigma models with target spaces described by general metrics and antisymmetric two-forms. We generalize the formalism for the models whose actions contain the dilaton and tachyon field. For the dilaton field case it is required that the duality is a property solely of the target manifolds, independent of the world-sheet geometry. For both cases the duality transformation is established and the integrability conditions are calculated. The set of restrictions on geometrical data describing the models is obtained, the previously calculated condition on connections on target spaces is maintained in both cases.

The main achievement of this work is the development of a duality theory for optimal control problems with stochastic differential equations. Incipient with the Hamilton-Jacobi-Bellman equation we established a dual problem to a given stochastic control problem and were also able to generalise the assembled theory.

Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theory on R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S². We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state.

Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theory on R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S². We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1975. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 42-44.).

Data types are a widely-used feature of functional programming languages that allow programmers to create abstractions and control branching computations. Instances of data types are introduced by applying one of a disjoint set of constructors and are eliminated by pattern matching on the constructor used. Dually, codata types are defined by their destructors, are introduced by copattern matching on their context, and eliminated by applying destructors. We extend motivation for codata types to include adding types that satisfy the extensional laws and adding an abstraction for constraining clients of code. We also improve on work implementing codata by developing an untyped compilation technique for codata that works for both call-by-name and call-by-value evaluation strategies and scales to simple and indexed type systems. We demonstrate the practicality of our technique by implementing a prototype compiler and a Haskell language extension.

Doctor of Philosophy / Department of Mathematics / Pietro Poggi-Corradini / This thesis contains six chapters. In the first chapter, the continuous and the discrete cases of p-modulus is introduced. We present properties of p-modulus and its connection to classical quantities. We also introduce use Arne Beurling's criterion for extremality to build insight and intuition regarding the modulus. After building an intuitive understanding of the p-modulus, we then proceed to switch perspectives to that of convex analysis. Using the theory of convex analysis, the uniqueness and existence of extremal densities is shown. We end this chapter with the introduction of the probabilistic interpretation of Modulus. In the second chapter, we introduce the Fulkerson duality. After defining the Fulkerson dual, we will investigate the blocking duality for different families of objects that the NODE research group has been studying and has been established. An important result that connects the Fulkerson dual and modulus is given at the end of this chapter. This important theorem will be used in proving one of the main results that [delta]p (introduced in Chapter 4) is a metric on graphs. The third chapter will discuss about metrics and ultrametrics on networks. Among these metrics, effective resistance is given special attention because the proof of [delta]p metric also serves as a new proof that effective resistance is a metric on graphs. We define effective resistance and give two different proves that show it is a metric, namely flows and the Laplacian. Two new families of metrics on graphs that arises through modulus are introduced in the fourth chapter. We also show how the two families are related as the d_p metric is viewed as a snowflaked version of the [delta]p metric. We end this chapter with some numerical examples that proves this connection and also serves as a set of plentiful examples of modulus calculations. Clutters and blockers is also another topic that is very much related to families of objects. While it has different rules and conditions, the study of clutters and blockers can give more insights to both modulus and clutters. We explore these relations in chapter 5. We provide some examples of clutters and blockers and finally reveal the relationship between the blocker and Fulkerson dual. Finally, in chapter 6, we end the thesis by presenting some of the open questions that we would like to explore and find answers in the future. In the second chapter, we introduce the Fulkerson duality. After defining the Fulkerson dual, we will investigate the blocking duality for different families of objects that the NODE research group has been studying and has been established. An important result that connects the Fulkerson dual and modulus is given at the end of this chapter. This important theorem will be used in proving one of the main results that delta_p (introduced in Chapter 4) is a metric on graphs. The third chapter will discuss about metrics and ultrametrics on networks. Among these metrics, effective resistance is given special attention because the proof of delta_p metric also serves as a new proof that effective resistance is a metric on graphs. We define effective resistance and give two different proves that show it is a metric, namely flows and the Laplacian. Two new families of metrics on graphs that arises through modulus are introduced in the fourth chapter. We also show how the two families are related as the d_p metric is viewed as a snowflaked version of the delta_p metric. We end this chapter with some numerical examples that proves this connection and also serves as a set of plentiful examples of modulus calculations. Clutters and blockers is also another topic that is very much related to families of objects. While it has different rules and conditions, the study of clutters and blockers can give more insights to both modulus and clutters. We explore these relations in chapter 5. We provide some examples of clutters and blockers and finally reveal the relationship between the blocker and Fulkerson dual. Finally, in chapter 6, we end the thesis by presenting some of the open questions that we would like to explore and find answers in the future.