Elliptic multiple polylogarithms in open string theory

In dieser Dissertation wird eine Methode zur Berechnung der genus-eins Korrekturen von offenen Strings zu Feldtheorie-Amplituden konstruiert. Hierzu werden Vektoren von Integralen definiert, die ein elliptisches Knizhnik-Zamolodchikov-Bernard (KZB) System auf dem punktierten Torus erfüllen, und die entsprechenden Matrizen aus dem KZB System berechnet. Der elliptische KZB Assoziator erzeugt eine Relation zwischen zwei regulierten Randwerten dieser Vektoren. Die Randwerte enthalten die genus-null und genus-eins Korrekturen. Das führt zu einer Rekursion im Genus und der Anzahl externer Zustände, die einzig algebraische Operationen der bekannten Matrizen aus dem KZB System umfasst. Geometrisch werden zwei externe Zustände der genus-null Weltfläche der offenen Strings zu einer genus-eins Weltfläche zusammengeklebt. Die Herleitung dieser genus-eins Rekursion und die Berechnung der relevanten Matrizen wird durch eine graphische Methode erleichtert, mit der die Kombinatorik strukturiert werden kann. Sie wurde durch eine erneute Untersuchung der auf Genus null bekannten Rekursion entwickelt, bei welcher der Drinfeld Assoziator Korrekturen offener Strings auf Genus null auf solche mit einem zusätzlichen externen Zustand abbildet. Diese genus-null Rekursion umfasst ebenfalls ausschliesslich Matrixoperationen und basiert auf einem Vektor von Integralen, der eine Knizhnik-Zamolodchikov (KZ) Gleichung erfüllt. Die in der Rekursion gebrauchten Matrizen aus der KZ Gleichung werden als Darstellungen einer Zopfgruppe identifiziert und rekursiv berechnet. Der elliptische KZB Assoziator ist die Erzeugendenreihe der elliptischen Multiplen Zeta-Werte. Die Konstruktion der genus-eins Rekursion benötigt verschiedene Eigenschaften dieser Werte und ihren definierenden Funktionen, den elliptischen Multiplen Polylogarithmen. So werden Relationen verschiedener Klassen von elliptischen Polylogarithmen und Funktionalrelationen erzeugt durch elliptische Funktionen hergeleitet.In this thesis, a method to calculate the genus-one, open-string corrections to the field-theory amplitudes is constructed. For this purpose, vectors of integrals satisfying an elliptic Knizhnik-Zamolodchikov-Bernard (KZB) system on the punctured torus are defined and the matrices from the KZB system are calculated. The elliptic KZB associator is used to relate two regularised boundary values of these vectors. The boundary values are shown to contain the open-string corrections at genus zero and genus one. This yields a recursion in the genus and the number of external states, solely involving algebraic operations on the known matrices from the KZB system. Geometrically, two external states of the genus-zero, open-string worldsheet are glued together to form a genus-one, open-string worldsheet. The derivation of this genus-one recursion and the calculation of the relevant matrices is facilitated by a graphical method to structure the combinatorics involved. It is motivated by the reinvestigation of the recursion in the number of external states known at genus zero, where the Drinfeld associator maps the genus-zero, open-string corrections to the corrections with one more external state. This genus-zero recursion also involves matrix operations only and is based on a vector of integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. The matrices in the KZ equation and used in the recursion are shown to be braid matrices and a recursive method for their calculation is provided. The elliptic KZB associator is the generating series of elliptic multiple zeta values. The construction of the genus-one recursion requires various properties of these values and their defining functions, the elliptic multiple polylogarithms. Thus, the third part of this thesis consists of an analysis of elliptic multiple polylogarithms, which in particular leads to relations among different classes of elliptic polylogarithms and functional relations generated by elliptic functions

Your God is Too Small: The Impact of the College Experience on God Image

Today’s American college student enters higher education with the expectation college will play a role in spiritual development. Spiritual development is an ambiguous term to many student affairs professionals. To qualify students’ experiences of spiritual development in college, this study examined the aspects of the college experience affecting a student’s God image. The God image, a construct encompassing myriad ways of viewing and understanding God, serves as a working model or metaphor of the person of God. The researcher chose God image as a construct for this study because it changes as a result of experiences. Using a qualitative approach and phenomenological design, this study explored the essence of the aspects of college having the greatest effect on a student’s God image. Major themes from the data included relationships, settings, and specific conditions within which students noted their understanding of God changed. The implications of this research provide direction to student affairs professionals as they seek pathways for faith development among college students

Rough Path Perspectives on the Itô-Stratonovich Dilemma

This thesis is comprised of six distinct research projects which share the theme of rough and stochastic integration theory. Chapter 1 deals with the problem of approximating an SDE X in R^d with one Y defined on a specified submanifold, so as to minimise quantities such as E[|Y_t − X_t|^2] for small t: this is seen to be best performed when using Itô instead of Stratonovich calculus. Chapter 2 develops the theory of not necessarily geometric 3 > p-rough paths on manifolds. Drawing on [FH14, É89, É90] we define controlled rough integration and RDEs both in the local and extrinsic framework, with the latter generalising [CDL15]. Finally, we lay out the theory of parallel transport and Cartan development, for which non-geometricity results in second-order conditions and corrections to the classical formulae. In Chapter 3 we treat the theory of geometric rough paths of arbitrary roughness in the framework of controlled paths of [Gub04], from an algebraic and combinatorial point of view, and avoiding the smooth approximation arguments used in [FV10b]. As an application, we show how our emphasis on functoriality allows for a simple transposition of the theory to the manifold setting. The goal of Chapter 4 is to treat the theory of branched rough paths on manifolds. Drawing on [HK15, Kel12], we show how to lift a controlled path to a rough path. The “transfer principle”, intended in the sense of Malliavin and Emery, refers to the expression of a connection-dependent “intrinsic differential” d_∇X that defines integration in a coordinate-invariant manner, which we derive by combining Kelly’s bracket corrections with certain higher-order Christoffel symbols. In reviewing branched rough paths, special attention is given to those that can be defined on Hoffman’s quasi-shuffle algebra [Hof00], for which some of the relations simplify. The final two chapters do not involve any differential geometry. Chapter 5 is a report on work in progress, the aim of which is to compute the Wiener chaos decomposition (and in particular the expectation) of the signature of certain multidimensional Gaussian processes such as 1/3 < H-fractional Brownian motion (fBm). This generalises the results of [BC07], arrived at through a piecewise-linear approximation argument which fails when 1/4 < H ≤ 1/2. Furthermore, our calculation restricts to that of [Bau04] in the case of Brownian motion, and can be applied to other semimartingales, such as the Brownian bridge. Our novel approach makes use of Malliavin calculus and the recent rough-Skorokhod conversion formula of [CL19]. Finally, in Chapter 6 we combine the topics of the previous two to define a branched rough path above multidimensional 1/4 < H-fBm, and compute its terms and correction terms. Our rough path is defined intrinsically and canonically in terms of the stochastic process, restricts to the Itô rough path when H = 1/2, has the property that its integrals of one-forms vanish in mean, and is not quasi-geometric when H ∈ (1/4, 1/3].Open Acces

Computable Diagonalizations and Turing's Cardinality Paradox

A. N. Turing's 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing's Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing's objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing's effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing's Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing's discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing's two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers

Symplectic cohomology and duality for the wrapped Fukaya category

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.Comment: v1: 166 pages, 26 figures. Feedback and comments are welcome! This paper will (eventually) be split into two papers. arXiv admin note: text overlap with arXiv:1001.4593 by other author

Communicative emergence and cultural evolution of word meanings

The question of how language evolved has received an increasing amount of attention in recent years. Compared to seemingly more complex phenomena such as syntax, word meanings are usually seen as relatively easy to explain. Mainstream accounts in psycholinguistics and evolutionary linguistics assume that word meanings correspond to stable concepts which are prior to language and derive straightforwardly from human perception of structure in the world. Taking a cognitive linguistic approach based on psycholinguistic evidence, I argue instead that word meanings are conventions, grounded, learned and used in the context of communication. The meaning of a word is the sum of its contexts of use, with particular features of these contexts made more or less salient by mechanisms of attentional learning and communicative inference. Evolutionarily, word meanings arise as an emergent product of humans’ adapted tendency to infer each other’s intentions using contextual cues. They are then shaped over cultural evolution by the need to be learnable and useful for communication. This thesis presents a series of experiments that test the effect of these pressures on the origins and development of word meanings. Experiment 1 investigates the origins of strong tendencies for words to specify features on particular dimensions (such as the shape bias). The results show that these tendencies arise via attentional learning effects amplified by iterated learning. Dimensions which are less salient in contexts of learning and use drop out of word meanings as they are passed down a chain of learners. Experiments 2, 3 and 4 investigate the structure of word meanings produced during either paired communication games or individual labelling of images by similarity. While communication alone leads to word meanings that are unstructured and poorly aligned within pairs, communication plus iterated learning leads to word meanings that increase in structure and alignment over generations. Finally, Experiment 5 investigates the interaction of event structure and developing conventions in shaping word meanings. The structure of events in an artificial world is shown to influence lexicalisation patterns in the languages conventionalised by communicating pairs. Event features that are less predictable across communicative contexts tend to be more strongly associated with the conventions in the language. Overall, the experiments show that rather than straightforwardly reflecting pre-linguistic conceptualisation, word meanings are also dynamically shaped by learning and communication. In addition, these processes are constrained by the conventions that already exist within a language. This illuminates the mixture of convergence and diversity we see in word meanings in natural languages, and gives insight into their evolutionary origins

Symplectic cohomology and duality for the wrapped Fukaya Category

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 313-315).Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.by Sheel Ganatra.Ph.D