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## Orbifold genera, product formulas and power operations

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 56 p.; 2082437 bytes; 2082243 bytes; application/pdf; application/pdf

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There is a formula by the string theorists Dijkgraaf, Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous p-typical statement follows as an easy corollary from the fact that the map of spectra corresponding to the genus preserves power operations. We define higher chromatic versions of the notion of orbifold genus, involving h-tuples rather than pairs of commuting elements. Using homotopy theoretic methods we are able to prove an integrality result and show that our definition is independent of the representation of the orbifold. Our setup is so simple, that it allows us to prove DMVV-type product formulas for these higher chromatic orbifold genera in the same way that the product formula for the topological Todd genus is proved. More precisely, we show that any genus induced by an H[omega]-map into one of the Morava-Lubin-Tate cohomology theories Eh has such a product formula and that the formula depends only on h and not on the genus. Since the complex H[omega]-genera into Eh have been classified in [And95], a large family of genera to which our results apply is completely understood. Loosely speaking...

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## Chain and antichain enumeration in posets, and b-ary partitions

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 72 leaves; 2213961 bytes; 2213767 bytes; application/pdf; application/pdf

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The Greene-Kleitman theorem says that the lengths of chains and antichains in any poset are intimately related via an integer partition, but very little is known about the partition [lambda](P) for most posets P. Our first goal is to develop a method for calculating values of [lambda]k(P) for certain posets. We find the size of the largest union of two or three chains in the lattice of partitions of n under dominance order, and in the Tamari lattice. Similar techniques are then applied to the k-equal partition lattice. We also present some partial results and conjectures on chains and antichains in these lattices. We give an elementary proof of the rank-unimodality of L(2, n, m), and find a symmetric chain decomposition of L(2, 2, m). We also present some partial results and conjectures about related posets, including a theorem on the size of the largest union of k chains in these posets and a bijective proof of the symmetry of the H-vector for 2 x n. We answer a question of Knuth about the existence of a Gray path for binary partitions, and generalize to b-ary partitions when b is even. We also discuss structural properties of the posets Rb(n), and compute some chain and antichain lengths in the subposet of join-irreducibles.; by Edward Fielding Early.; Thesis (Ph. D.)--Massachusetts Institute of Technology...

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## On the 2-typical de Rham-Witt complex; On the two-typical de Rham-Witt complex

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 55 p.; 1537194 bytes; 1535736 bytes; application/pdf; application/pdf

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In this thesis we introduce the 2-typical de Rham-Witt complex for arbitrary commutative, unital rings and log-rings. We describe this complex for the rings Z and Z(2), for the log-ring (Z(2), M) with the canonical log-structure, and we describe its behaviour under polynomial extensions. In an appendix we also describe the p-typical de Rham-Witt complex of (Z(p), M) for p odd.; by Viorel Costeanu.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; Includes bibliographical references (p. 55).

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## Geometry of Ricci-flat Kähler manifolds and some counterexamples

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 64 leaves; 2632174 bytes; 2630560 bytes; application/pdf; application/pdf

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In this work, we study geometry of Ricci-flat Kähler manifolds, and also provide some counterexample constructions. We study asymptotic behavior of complete Ricci-flat metrics at infinity and consider a construction of approximate Ricci-flat metrics on quasiprojective manifolds with a divisor with normal crossings removed, by means of reducing torsion of a non-Kähler metric with the right volume form. Next, we study special Lagrangian fibrations using methods of geometric function theory. In particular, we generalize the method of extremal length and prove a generaliziation of the Teichmiiller theorem. We relate extremal problems to the existence of special Lagrangian fibrations in the large complex structure limit of Calabi-Yau manifolds. We proceed to some problems in the theory of minimal surfaces, disproving the Schoen-Yau conjecture and providing a first example of a proper harmonic map from the unit disk to a complex plane. In the end, we prove that the union closed set conjecture is equivalent to a strengthened version, giving a construction which might lead to a counterexample.; by Vladimir BoÅ¾in.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; Includes bibliographical references (leaves 61-64).

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## Frobenius transfers and p-local finite groups

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 44 p.; 1679211 bytes; 1677870 bytes; application/pdf; application/pdf

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In this thesis we explore the possibility of defining the p-local finite groups of Broto, Levi and Oliver in terms of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be recovered by studying maps f: BS --> X from the classifying space of a finite p-group S to a p-complete space X equipped with a stable retract t satisfying a form of Frobenius reciprocity. In the case where S is elementary abelian, we answer this question in the affirmative, by showing that under some finiteness conditions such a triple (f, t, X) does indeed induce a p-local finite group over S. We also discuss the converse in some detail for general S.; by KaÅi Ragnarsson.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; Includes bibliographical references (p. 43-44).

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## Constraining the QSO luminosity function using gravitational lensing statistics; Constraining the Quasi-Stellar Object luminosity function using gravitational lensing statistics

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 87 leaves; 4673795 bytes; 4677885 bytes; application/pdf; application/pdf

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In this thesis we use gravitational lensing statistics to constrain the QSO luminosity function at a variety of redshifts. We present a theoretical discussion of gravitational lensing statistics and illustrate how high resolution QSO imagery can be used to constrain the QSO luminosity function. We then discuss the selection and observation of the 1073 QSO exposures in our sample. The sample covers a redshift range of 0.7

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## Total-to-peak ratios of high purity germanium gamma ray detector

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 22 leaves; 878708 bytes; 876736 bytes; application/pdf; application/pdf

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This study is concerned with the percentage of [gamma]-rays of a certain energy having their energy correctly measured by a high purity Germanium [gamma]-ray detector. The ratio between the total counts and the counts within the energy peak (total-to-peak ratio) is determined for seven energies ranging from 89 keV to 1275 keV. A Monte Carlo based on the physical parameters of the detector was used to extrapolate between these points and after an energy independent scaling factor fit the data with a reduced [chi]² slightly below 1. The same experiment was repeated with a lead brick and then a β detector near the Ge detector and these objects were found to not have an effect on the total-to-peak ratios within the precision of the experiment.; by Justin Matthew Nelson.; Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics; and, (S.B.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; MIT Institute Archives copy: leaves 2-22 bound in reverse order.; Includes bibliographical references (leaf 22).

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## Approximation algorithms for distributed and selfish agents

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 165 p.; 9483194 bytes; 9492590 bytes; application/pdf; application/pdf

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Many real-world systems involve distributed and selfish agents who optimize their own objective function. In these systems, we need to design efficient mechanisms so that system-wide objective is optimized despite agents acting in their own self interest. In this thesis, we develop approximation algorithms and decentralized mechanisms for various combinatorial optimization problems in such systems. First, we investigate the distributed caching and a general set of assignment problems. We develop an almost tight LP-based ... approximation algorithm and a local search ... approximation algorithm for these problems. We also design efficient decentralized mechanisms for these problems and study the convergence of the corresponding games. In the following chapters, we study the speed of convergence to high quality solutions on (random) best-response paths of players. First, we study the average social value on best response paths in basic-utility, market sharing, and cut games. Then, we introduce the sink equilibrium as a new equilibrium concept. We argue that, unlike Nash equilibria, the selfish behavior of players converges to sink equilibria and all strategic games have a sink equilibrium. To illustrate the use of this new concept, we study the social value of sink equilibria in weighted selfish routing (or weighted congestion) games and valid-utility (or submodular-utility) games. In these games...

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## The stochastic operator approach to random matrix theory

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 156 p.; 6534500 bytes; 6543035 bytes; application/pdf; application/pdf

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Classical random matrix models are formed from dense matrices with Gaussian entries. Their eigenvalues have features that have been observed in combinatorics, statistical mechanics, quantum mechanics, and even the zeros of the Riemann zeta function. However, their eigenvectors are Haar-distributed-completely random. Therefore, these classical random matrices are rarely considered as operators. The stochastic operator approach to random matrix theory, introduced here, shows that it is actually quite natural and quite useful to view random matrices as random operators. The first step is to perform a change of basis, replacing the traditional Gaussian random matrix models by carefully chosen distributions on structured, e.g., tridiagonal, matrices. These structured random matrix models were introduced by Dumitriu and Edelman, and of course have the same eigenvalue distributions as the classical models, since they are equivalent up to similarity transformation. This dissertation shows that these structured random matrix models, appropriately rescaled, are finite difference approximations to stochastic differential operators. Specifically, as the size of one of these matrices approaches infinity, it looks more and more like an operator constructed from either the Airy operator...

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## Twisted stable homotopy theory

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 137 p.; 7023296 bytes; 7030385 bytes; application/pdf; application/pdf

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There are two natural interpretations of a twist of stable homotopy theory. The first interpretation of a twist is as a nontrivial bundle whose fibre is the stable homotopy category. This kind of radical global twist forms the basis for twisted parametrized stable homotopy theory, which is introduced and explored in Part I of this thesis. The second interpretation of a twist is as a nontrivial bundle whose fibre is a particular element in the stable homotopy category. This milder notion of twisting leads to twisted generalized homology and cohomology and is central to the well established field of parametrized stable homotopy theory. Part II of this thesis concerns a computational problem in parametrized stable homotopy, namely the determination of the twisted K-homology of the simple Lie groups. In more detail, the contents of the two parts of the thesis are as follows. Part I: I describe a general framework for twisted forms of parametrized stable homotopy theory. An ordinary parametrized spectrum over a space X is a map from X into the category Spec of spectra; in other words, it is a section of the trivial Spec- bundle over X. A twisted parametrized spectrum over X is a section of an arbitrary bundle whose fibre is the category of spectra. I present various ways of characterizing and classifying these twisted parametrized spectra in terms of invertible sheaves and local systems of categories of spectra. I then define homotopy-theoretic invariants of twisted parametrized spectra and describe a spectral sequence for computing these invariants.; (cont.) In a more geometric vein...

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## The p-adic local langlands conjecture

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 47 leaves; 2125656 bytes; 2127516 bytes; application/pdf; application/pdf

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Let k be a p-adic field. Split reductive groups over k can be described up to k- isomorphism by a based root datum alone, but other groups, called rational forms of the split group, involve an action of the Galois group of k. The Galois action on the based root datum is shared by members of an inner class of k-groups, in which one k--isomorphism class is quasi-split. Other forms of the inner class can be called pure or impure, depending on the Galois action. Every form of an adjoint group is pure, but only the quasi-split forms of simply connected groups are pure. A p-adic Local Langlands correspondence would assign an L-packet, consisting of finitely many admissible representations of a p-adic group, to each Langlands parameter. To identify particular representations, data extending a Langlands parameter is needed to make "completed Langlands parameters." Data extending a Langlands parameter has been utilized by Lusztig and others to complete portions of a Langlands classification for pure forms of reductive p- adic groups, and in applications such as endoscopy and the trace formula, where an entire L-packet of representations contributes at once.; (cont.) We consider a candidate for completed Langlands parameters to classify representations of arbitrary rational forms...

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## Traveling salesman path problems

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 155 p.; 9104947 bytes; 9111456 bytes; application/pdf; application/pdf

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In the Traveling Salesman Path Problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. The problem is a generalization of the Traveling Salesman Problem with many important applications. In this thesis, we study polyhedral and combinatorial properties of a variant we call the Traveling Salesman Walk Problem, in which the minimum cost walk from the source to destination visits all cities at least once. Using the approach of linear programming, we study properties of the polyhedron corresponding to a linear programming relaxation of the traveling salesman walk problem. Our results relate the structure of the underlying graph of the problem instance with polyhedral properties of the corresponding fractional walk polyhedron. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively.; (cont.) We extend these results to relate the underlying graph structure to the integrality gap of the corresponding fractional walk polyhedron. We present several graph operations which preserve integrality gap; these operations allow us to find the integrality gap of graphs built from smaller bricks...

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## Asymptotic analysis of extreme electrochemical transport

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 244 p.; 11555675 bytes; 11566560 bytes; application/pdf; application/pdf

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In the study of electrochemical transport processes, experimental exploration currently outpaces theoretical understanding of new phenomena. Classical electrochemical transport theory is not equipped to explain the behavior of electrochemical systems in the extreme operating conditions required by modern devices. In this thesis, we extend the classical theory to examine the response of two electrochemical systems that form the basis for novel electrochemical devices. We first examine the DC response of an electrochemical thin film, such as the separator in a micro-battery, driven by current applied through reactive electrodes. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer which mediates Faradaic reactions with Butler-Volmer kinetics. Our analysis differs from previous studies in two significant ways. First, we impose the full nonlinear, reactive boundary conditions appropriate for electrolytic/galvanic cells.; (cont.) Since surface effects become important for physically small systems, the use of reactive boundary conditions is critical in order to gain insight into the behavior of actual electrochemical thin films that are sandwiched between reactive electrodes...

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## Quantifier rank spectrum of L-infinity-omega

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 321 p.; 1495538 bytes; 1490131 bytes; application/pdf; application/pdf

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In Part A we will study the quantifier rank spectrum of sentences of L!1,!. We will show that there are scattered sentences with models of arbitrarily high but bounded quantifier rank. We will also consider the case of weakly scattered and almost scattered sentences, and we will make some conjectures. In Part B we will look at a new method of induction in the case of sheaves. We will then use this method to generalize the classical proof of the Suslin-Kleene Separation Theorem to the context of sheaves on a partial Grothendieck topology.; by Nathaniel Leedom Ackerman.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.; Includes bibliographical references (p. 321) and index.

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## An application of the Malliavin calculus to infinite dimensional diffusions

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 28 leaves; 692038 bytes; 693152 bytes; application/pdf; application/pdf

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by Laura E. Clemens.; Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1984.; MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.; Bibliography: leaf 28.

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## Branching from K to M for split classical groups

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 91 p.; 3771837 bytes; 3775584 bytes; application/pdf; application/pdf

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We provide two algorithms to solve branching from K to M for the real split reductive group of type A, one inductive and one related to semistandard Young tableaux. The results extend to branching from Ke to M Ke for the real split reductive groups of type Bn and Dn.; by Nicholas McCarthy.; Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.; Includes bibliographical references (p. 91).

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## Computational methods for higher real K-theory with applications to tmf

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 69 p.; 3600879 bytes; 3603677 bytes; application/pdf; application/pdf

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We begin by present a new Hopf algebra which can be used to compute the tmf homology of a space or spectrum at the prime 3. Generalizing work of Mahowald and Davis, we use this Hopf algebra to compute the tmf homology of the classifying space of the symmetric group on three elements. We also discuss the E3 Tate spectrum of tmf at the prime 3. We then build on work of Hopkins and his collaborators, first computing the Adams-Novikov zero line of the homotopy of the spectrum eo4 at 5 and then generalizing the Hopf algebra for tmf to a family of Hopf algebras, one for each spectrum eop_l at p. Using these, and using a K(p - 1)-local version, we further generalize the Davis-Mahowald result, computing the eop_1 homology of the cofiber of the transfer map [...]. We conclude by computing the initial computations needed to understand the homotopy groups of the Hopkins-Miller real K-theory spectra for heights large than p- 1 at p. The basic computations are supplemented with conjectures as to the collapse of the spectral sequences used herein to compute the homotopy.; by Michael Anthony Hill.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; Includes bibliographical references (p. 67-69).

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## Super symmetric vertex algebras and supercurves

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 151 p.; 7925705 bytes; 7932023 bytes; application/pdf; application/pdf

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We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Given a strongly conformal SUSY vertex algebra V and a supercurve X, we construct a vector bundle [ ... ] on X, the fiber of which, is isomorphic to V. Moreover, the state-field correspondence of V canonically gives rise to (local) sections of these vector bundles. We also define chiral algebras on any supercurve X, and show that the vector bundle [ ... ] corresponding to a SUSY vertex algebra, carries the structure of a chiral algebra.; by Reimundo Heluani.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; Includes bibliographical references (p. 149-151).

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## Noncommutative ring spectra

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 91 p.; 5236865 bytes; 5241226 bytes; application/pdf; application/pdf

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Let A be an Ax ring spectrum. We give an explicit construction of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. Using this construction we can then study how THH(A) varies over the moduli space of AO structures on A, a problem which seems largely intractable using strictly associative replacements of A. We study how topological Hochschild cohomology of any 2-periodic Morava K-theory varies over the moduli space of AO structures and show that in the generic case, when a certain matrix describing the multiplication is invertible, the result is the corresponding Morava E-theory. If this matrix is not invertible, the result is some extension of Morava E-theory, and exactly which extension we get depends on the AO structure. To make sense of our constructions, we first set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-E operad P in C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects.; (cont.) For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.; by Vigleik Angeltveit.; Thesis (Ph. D.)--Massachusetts Institute of Technology...

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## Thread-wire surfaces

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 190 p.; 13231503 bytes; 13230927 bytes; application/pdf; application/pdf

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This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are C1 at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows.; by Benjamin K. Stephens.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; Includes bibliographical references (p. 183-190) and Index.

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