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## Spectrum of some regular graphs with widely spaced modifications

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

Tipo: Tese de Doutorado
Formato: 72 p.; 3795180 bytes; 3794939 bytes; application/pdf; application/pdf

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This thesis has two parts. The first part studies the spectrum of a family of growing trees, we show that the eigenvalues of the adjacency matrix and Laplacian matrix have high multiplicities. As the trees grow, the graphs of those eigenvalues approach a piecewise-constant "Cantor function", which is different from the corresponding properties of the infinite tree. The second part studies the effect of "widely spaced" modifications on the spectrum of some type of structured matrices. We show that by applying those modifications, new eigenvectors that are localized near the components that correspond to the modified rows appear. By knowing the approximate form of those eigenvectors, we also determine a very close (and simple) approximation to the eigenvalues, and then we show that this approximation is indeed the limit as the matrix grows.; by Xiangwei Liu.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (p. 71-72).

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## The involution principle and h-positive symmetric functions

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

Tipo: Tese de Doutorado
Formato: 65 p.; 3844151 bytes; 3843908 bytes; application/pdf; application/pdf

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The criterion of h-positivity corresponds to the criterion that a polynomial representation of the general linear group of V is a sum of tensor products of symmetric powers of V. Expanding the iterated exponential function as a power series yields coefficients whose positivity implies the h-positivity of the characteristic of the symmetric group character whose value on the permutation w is the number of labeled forests with c(w) vertices, where c(w) is the number of cycles of w. Another example of an h-positive symmetric function is the characteristic of the top homology of the even-ranked subposet of the partition lattice. In this case, the positive coefficients of the characteristic refine the tangent number E₂nâ₁ into sums of powers of two.; by Benjamin S. Joseph.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (p. 65).

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## Modeling and prediction of sunspot cycles

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

Tipo: Tese de Doutorado
Formato: 165 p.; 8880896 bytes; 8880652 bytes; application/pdf; application/pdf

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Solar activity, as indexed by sunspots, has a cycle of length varying from about 9 to 13 years. Statisticians have fitted several models to predict sunspot numbers one year ahead. We instead focus on predicting the magnitudes of the next cycle maximum, the next cycle minimum, the time from the initial minimum of a cycle to its maximum, called the rise time, and the time from a cycle maximum to the next cycle minimum, called the fall time. The predictions are based on sunspot numbers just far enough into a new cycle to establish that a cycle has started. We propose parsimonious regression models for the maximum and rise time. For the fall time and minimum we propose crude models, based on the sample means and variances for all past cycles. We compare our models to simulation results for many models we found in the literature including autoregressive (AR) models, subset AR (SAR) models, and related nonlinear models including a threshold AR model, transformed by squaring (TTAR model), a bilinear model, a so-called ASTAR model, and a neural network (CNAR) model. We also consider a model proposed by solar physicists, based on a functional form for cycles.; (cont.) Numerical results show that among all the models considered for sunspot numbers...

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## Eta-invariants and Molien series for unimodular group

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

Tipo: Tese de Doutorado
Formato: 58 p.; 3714695 bytes; 3714453 bytes; application/pdf; application/pdf

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We look at the singularity Cn/[Gamma], for [Gamma] finite subgroup of SU(n), from two perspectives. From a geometrical point of view, Cn/[Gamma] is an orbifold with boundary S2n-1/[Gamma]. We define and compute the corresponding orbifold [eta]-invariant. From an algebraic point of view, we look at the algebraic variety Cn/[Gamma] and we analyze the associated Molien series. The main result is formula which relates the two notions: [eta]-invariant and Molien series. Along the way computations of the spectrum of the Dirac operator on the sphere are performed.; by Anda Degeratu.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (p. 57-58).

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## Rigidity and invariance properties of certain geometric frameworks

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

Tipo: Tese de Doutorado
Formato: 60 leaves; 3229421 bytes; 3229179 bytes; application/pdf; application/pdf

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Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called "k-cables"); the volume of the k-faces in the 2nd group can not decrease ("k-struts"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn.; by Lizhao Zhang.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (leaves 59-60).

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## Geometry of cone-beam reconstruction

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

Tipo: Tese de Doutorado
Formato: 91 p.; 5942218 bytes; 5941980 bytes; application/pdf; application/pdf

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Geometry is the synthetic tool we use to unify all existing analytical cone-beam reconstruction methods. These reconstructions are based on formulae derived by Tuy [Tuy, 1983], Smith [Smith, 1985] and Grangeat [Grangeat, 1991] which explicitly link the cone-beam data to some intermediate functions in the Radon transform domain. However, the essential step towards final reconstruction, that is, differential-backprojection, has not yet achieved desired efficiency. A new inversion formula is obtained directly from the 3D Radon inverse [Radon, 1917, Helgason, 1999]. It incorporates the cone-beam scanning geometry and allows the theoretical work mentioned above to be reduced to exact and frugal implementations. Extensions can be easily carried out to 2D fan-beam reconstruction as well as other scanning modalities such as parallel scans by allowing more abstract geometric description on the embedding subspace of the Radon manifold. The new approach provides a canonical inverse procedure for computerized tomography in general, with applications ranging from diagnostic medical imaging to industrial testing, such as X-ray CT, Emission CT, Ultrasound CT, etc. It also suggests a principled frame for approaching other 3D reconstruction problems related to the Radon transform. The idea is simple: as was spelled out by Helgason on the opening page of his book...

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## A complexity theoretic approach to learning

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

Tipo: Tese de Doutorado
Formato: 138 leaves; 9348960 bytes; 9348716 bytes; application/pdf; application/pdf

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This thesis details a new vantage point for attacking longstanding problems in machine learning. We use tools from computational complexity theory to make progress on problems from computational learning theory. Our methods yield the fastest and most expressive algorithms to date for learning several fundamental concept classes: * We show that any s-term DNF over n variables can be computed by a polynomial threshold function of order O(n1/3 log s). As an immediate consequence we obtain the fastest known DNF learning algorithm which runs in time 2O(n1/3). * We give the first polynomial time algorithm to learn an intersection of a constant number of halfspaces under the uniform distribution to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any function of a constant number of halfspaces with polynomial bounded weights under any distribution. * We give an algorithm to learn constant-depth polynomial-size circuits augmented with majority gates under the uniform distribution using random examples only. For circuits which contain a polylogarithmic number of majority gates the algorithm runs in quasipolynomial time. Under a suitable cryptographic assumption we show that these are the most expressive circuits which will admit a non-trivial learning algorithm. Our approach relies heavily on giving novel representations of well known concept classes via complexity theoretic reductions. We exploit the fact that many results in computational learning theory have a complexity theoretic analogue or implication. As such...

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## Rational families of vector bundles on curves

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

Tipo: Tese de Doutorado
Formato: 163 p.; 10427492 bytes; 10427251 bytes; application/pdf; application/pdf

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We find and describe the irreducible components of the space of rational curves on moduli spaces M of rank 2 stable vector bundles with odd determinant on curves C of genus g [greater than or equal to] 2. We prove that the maximally rationally connected quotient of such a component is either the Jacobian J(C) or a direct sum of two copies of the Jacobian. We show that moduli spaces of rational curves on M are in one-to-one correspondence with moduli of rank 2 vector bundles on the surface P[set]1 x C.; by Ana-Maria Castravet.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 163).

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## Ricci flow on 3-manifolds with symmetry

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

Tipo: Tese de Doutorado
Formato: 53 p.; 2968454 bytes; 2968214 bytes; application/pdf; application/pdf

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In this thesis, we proved that for a 3-manifold S2 x S1 with warped product metric, the isoperimetric ratio on the base manifold S2 has a low positive bound away from zero, if the scalar curvature on the 3-manifold is positive. We also obtained a monotonicity result under the condition that the length of the optimal curve for isoperimetric ratio shrinks to zero under Ricci flow. This result excludes the product of a cigar soliton [Sigma]² with R¹ as the dilation limit of the Ricci flow equation. We also obtained an inequality of the curvature ratio Rmin/Rmax on the dilation limit for compact 3-manifold with positive scalar curvature.; by Xiaodong Cao.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 51-53).

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## Running in circles : packet routing on ring networks

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

Tipo: Tese de Doutorado
Formato: 156 p.; 11749248 bytes; 11749004 bytes; application/pdf; application/pdf

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I analyze packet routing on unidirectional ring networks, with an eye towards establishing bounds on the expected length of the queues. Suppose we route packets by a greedy "hot potato" protocol. If packets are inserted by a Bernoulli process and have uniform destinations around the ring, and if the nominal load is kept fixed, then I can construct an upper bound on the expected queue length per node that is independent of the size of the ring. If the packets only travel one or two steps, I can calculate the exact expected queue length for rings of any size. I also show some stability results under more general circumstances. If the packets are inserted by any ergodic hidden Markov process with nominal loads less than one, and routed by any greedy protocol, I prove that the ring is ergodic.; by William F. Bradley.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 151-154) and index.

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## The homotopy type of the matroid Grassmannian

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

Tipo: Tese de Doutorado
Formato: 38 p.; 2576462 bytes; 2576223 bytes; application/pdf; application/pdf

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In this thesis, I establish a homotopy equivalence between the matroid Grassmannian [parallel] MacP(k, n) [parallel] and the real Grassmannian G(k, n) of k-planes in [Real set]n. This is accomplished by finding a Schubert stratification of the former space and analyzing its relationship to the ordinary Schubert cell decomposition of the Grassmannian. Since the classifying spaces for rank k matroid bundles and rank k vector bundles are, respectively, obtained by taking colimits of the above spaces as n grows, this result provides a natural equivalence between the functors of matroid bundles and vector bundles. This, in turn, has implications for the interplay between combinatorics and topology, particularly concerning the Gelfand-MacPherson combinatorial formula for rational Pontrjagin classes.; by Daniel Kálmán Biss.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 37-38).

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## [rho]-compact groups as framed manifolds

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

Tipo: Tese de Doutorado
Formato: 59 p.; 2652696 bytes; 2652459 bytes; application/pdf; application/pdf

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We describe a natural way to associate to any [rho]-compact group an element of the [rho]-local stable stems, which, applied to the [rho]-completion of a compact Lie group G, coincides with the element represented by the manifold G with its left-invariant framing. To this end, we construct a d-dimensional sphere SG with a stable G-action for every d-dimensional [rho]-compact group G, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with SG.; by Tilman Bauer.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; In title on t.p. "[rho]" appears as the lower-case Greek letter.; Includes bibliographical references (p. 57-59).

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## Localization genus of classifying spaces

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

Tipo: Tese de Doutorado
Formato: 37 p.; 2110385 bytes; 2110143 bytes; application/pdf; application/pdf

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We show that for a large class of torsionfree classifying spaces, K-theory filtered ring is an invariant of the genus. We apply this result in two ways. First, we use it to show that the powerseries ring on n indeterminates over the integers admits uncountably many mutually non-isomorphic [lambda]-ring structures. Second, we use it to study the genus of infinite quaternionic projective space. In particular, we describe spaces in the genus of infinite quaternionic projective space which occur as targets of essential maps from infinite complex projective space, and we compute explicitly the homotopy classes of maps in these cases.; by Donald Y. Yau.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 35-37).

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## Support Vector Machine algorithms : analysis and applications; SVM algorithms : analysis and applications

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

Tipo: Tese de Doutorado
Formato: 97 p.; 6704574 bytes; 6704331 bytes; application/pdf; application/pdf

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Support Vector Machines (SVMs) have attracted recent attention as a learning technique to attack classification problems. The goal of my thesis work is to improve computational algorithms as well as the mathematical understanding of SVMs, so that they can be easily applied to real problems. SVMs solve classification problems by learning from training examples. From the geometry, it is easy to formulate the finding of SVM classifiers as a linearly constrained Quadratic Programming (QP) problem. However, in practice its dual problem is actually computed. An important property of the dual QP problem is that its solution is sparse. The training examples that determine the SVM classifier are known as support vectors (SVs). Motivated by the geometric derivation of the primal QP problem, we investigate how the dual problem is related to the geometry of SVs. This investigation leads to a geometric interpretation of the scaling property of SVMs and an algorithm to further compress the SVs. A random model for the training examples connects the Hessian matrix of the dual QP problem to Wishart matrices. After deriving the distributions of the elements of the inverse Wishart matrix Wn-1(n, nI), we give a conjecture about the summation of the elements of Wn-1(n...

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## Affine quantum algebras, Weyl groups and constructible functions

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

Tipo: Tese de Doutorado
Formato: 51 p.; 3924339 bytes; 3924098 bytes; application/pdf; application/pdf

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This thesis consists of two parts. In the first part we study the affine quantum group of type A, giving a geometric description of its natural inner product, and studying the theory of cells attached to the canonical basis. In the second part we study a realization of the group algebra of the Weyl group in a convolution algebra of constructible functions on the Steinberg variety, and examine how this may be used to see Springer representations.; by Kevin McGerty.; Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.; Includes bibliographical references (p. 49-51).

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## Equivariant cohomology, homogeneous spaces and graphs

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

Tipo: Tese de Doutorado
Formato: 100 p.; 6762305 bytes; 6762062 bytes; application/pdf; application/pdf

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The focus of this thesis is manifolds with group actions, in particular symplectic manifolds with Hamiltonian torus actions. We investigate the relationship between the equivariant cohomology of the manifold M and the fixed point data of the torus action. We are interested in understanding the topology of the space of T-orbits in M. In particular, we explore aspects of this topology which are determined by data from the image of a moment map [Phi] : M [right arrow] t* associated to the Hamiltonian action. To better understand the orbit space, we apply the algebraic techniques of equivariant cohomology to the study these systems further. Equivariant cohomology associates to a manifold with a G-action a ring H*G(M). Much of the topology of the orbit space is encoded in the equivariant cohomology ring H*G(M). In 1998, Goresky, Kottwitz and MacPherson provided a new method for computing this ring. Their method associates to this orbit space a graph [Gamma] whose vertices are the zero-dimensional orbits and edges the connected components of the set of one-dimensional orbits. The ring H*G(M) can then be computed combinatorially in terms of the data incorporated in [Gamma]. The strength of this construction is that it makes the computation of equivariant cohomology into a combinatorial computation...

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## Decidable prime models

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

Tipo: Tese de Doutorado
Formato: 31 leaves; 2928275 bytes; 2928034 bytes; application/pdf; application/pdf

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by Jessica Millar Young.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (leaf 31).

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## Character sheaves on symmetric spaces

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

Tipo: Tese de Doutorado
Formato: 77, [3] p.; 6112746 bytes; 6112507 bytes; application/pdf; application/pdf

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Inspired by the work of Lusztig, we apply the theory of character sheaves on a symmetric space (h la Ginzburg and Grojnowski) to the problem of determining the spherical functions (averages of the irreducible characters) of the associated symmetric space over a finite field. A crucial result about the filtration of character sheaves by cells enables us to compute the spherical functions explicitly in the cases GLn((Fq2)/GLn(]Fq) and GL2n(1Fq)/Sp2n((Fq), as well as other closely related examples. We include partial results about the symmetric space GLn/(GLm x GLn-m), to illustrate some of the obstacles to further generalization.; by Anthony Henderson.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (p. [79]-[80]).

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## Noncommutative symmetric functions of type B; BSym

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

Tipo: Tese de Doutorado
Formato: 107 p.; 5724108 bytes; 5723868 bytes; application/pdf; application/pdf

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The noncommutative symmetric functions Sym of Gelfand et al. give not only a lifting of the well-developed commutative theory of symmetric functions to the non-commutative level, but also relate the descent algebras of Solomon and the quasi-symmetric functions, where the latter are dual to the noncommutative symmetric functions equipped with the internal product, which are anti-isomorphic to the descent algebras. Using this anti-isomorphism, properties of both noncommutative symmetric functions and of descent algebras can be studied. Generalizations of the above theory are made in the present work. The starting point is the quasi-symmetric functions of type B, BQSym, which are shown to have an algebra, a comodule, and a coalgebra structures. The noncommutative symmetric functions BSym are then introduced as a module over Sym dual to the comodule structure of BQSym. It is then made into a coalgebra dual to the algebra structure of BQSym, and into an algebra dual to the coalgebra structure of BQSym. The latter duality defines the internal product *B on BSym, which makes (BSym, *B) anti-isomorphic to the descent algebra [Sigma]Bn of the hyperoctahedral groups Bn, studied by Bergeron and Bergeron.; (cont.) Lie idempotents of both BSym and [Sigma]Bn are then studied via the anti-isomorphism. In particular...

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## Mathematical methods for protein structural motif recognition

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

Tipo: Tese de Doutorado
Formato: 137 leaves; 11667830 bytes; 11667590 bytes; application/pdf; application/pdf

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In this thesis, two algorithms for protein structural motif recognition are presented. A program is described which successfully recognizes the occurrence of the right-handed parallel 8-helix fold from protein sequence data alone. When run on sequences of unknown structure the program identifies the fold in a significant number of microbial outer membrane and cell-surface proteins implicated in infectious disease. A search algorithm is introduced for unsupervised discovery of protein sequence patterns that are significantly correlated with structural motifs. The algorithm identifies known, biologically relevant patterns as well as several potentially novel motifs. It is hoped that both algorithms may contribute to a better understanding of the connection between protein sequence and structure.; by Philip H. Bradley.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.; Includes bibliographical references (leaves 129-137).

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