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## Multi-period pricing for perishable products : uncertainty and competition

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

Tipo: Tese de Doutorado
Formato: 109 p.

Português

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The pricing problem in a multi-period setting is a challenging problem and has attracted much attention in recent years. In this thesis, we consider a monopoly and an oligopoly pricing problem. In the latter, several sellers simultaneously seek an optimal pricing policy for their products. The products are assumed to be differentiated and substitutable. Each seller has the option to set prices for her products at each time period, and her goal is to find a pricing policy that will yield the maximum overall profit. Each seller has a fixed initial inventory of each product to be allocated over the entire time horizon and does not have the option to produce additional inventory between periods. There are no holding costs or back-order costs. In addition, the products are perishable and have no salvage costs. This means that at the end of the entire time horizon, any remaining products will be worthless. The demand function each seller faces for each product is uncertain and is affected by both the prices at the current period and past pricing history for her and her competitors. In this thesis, we address both the uncertain and the competitive aspect of the problem. First, we study the uncertain aspect of the problem in a simplified setting...

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## Reduced basis method for 2nd order wave equation : application to one-dimensional seismic problem; Reduced basis method for second order wave equation : application to 1D seismic problem

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

Tipo: Tese de Doutorado
Formato: 95 p.

Português

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In this thesis, we solve the 2nd order wave equation, which is hyperbolic and linear in nature, to determine the pressure distribution for a one-dimensional seismic problem with smooth initial pressure and rate of pressure change with time. With Dirichlet and Neumann boundary conditions, the pressure distribution is solved for a total of 500 time steps, which is slighter more than a periodic cycle. Our focus is on the dependence of the output, the average surface pressure as it varies with time, on the system parameters ,u, which consist of the earthquake source x8 and the occurring time T. The reduced basis method, the offline-online computational procedures and the associated a posteriori error estimation are developed. We have shown that the reduced basis pressure distribution is an accurate approximation to the finite element pressure distribution. The greedy algorithm, the procedure of selecting the basis vectors which span the reduced basis space, works reasonably well although a period of slow convergence is experienced: this is because the finite element pressure distribution along the edges of the earthquake source-time space are fairly "unique" and cannot be accurately represented as a linear combination of the existing basis vectors;; (cont.) hence...

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## Low rank decompositions for sum of squares optimization

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

Tipo: Tese de Doutorado
Formato: 79 leaves

Português

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In this thesis, we investigate theoretical and numerical advantages of a novel representation for Sum of Squares (SOS) decomposition of univariate and multivariate polynomials. This representation formulates a SOS problem by interpolating a polynomial at a finite set of sampling points. As compared to the conventional coefficient method of SOS, the formulation has a low rank property in its constraints. The low rank property is desirable as it improves computation speed for calculations of barrier gradient and Hessian assembling in many semidefinite programming (SDP) solvers. Currently, SDPT3 solver has a function to store low rank constraints to explore its numerical advantages. Some SOS examples are constructed and tested on SDPT3 to a great extent. The experimental results demonstrate that the computation time decreases significantly. Moreover, the solutions of the interpolation method are verified to be numerically more stable and accurate than the solutions yielded from the coefficient method.; by Jia Li Sun.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.; Includes bibliographical references (leaves 77-79).

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## Domain partitioning to bound moments of differential equations using semidefinite optimization

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

Tipo: Tese de Doutorado
Formato: 95 leaves

Português

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In this thesis, we present a modification of an existing methodology to obtain a hierarchy of lower and upper bounds on moments of solutions of linear differential equations. The motivation for change is to obtain tighter bounds by solving smaller semidefinite problems. The modification we propose involves partitioning the domain and normalizing each partition to ensure numerical stability. Using the adjoint operator, linear constraints involving the boundary conditions and moments of the solution are developed for each partition. Semidefinite constraints are imposed on the moments, and an optimization problem is solved to obtain the bounds. We have demonstrated the algorithm by calculating bounds on moments of various one-dimensional case differential equations including the Bessel ODE, and Legendre polynomials. In the two-dimensional case we have demonstrated the algorithm by calculating bounds on various PDEs including the Helmholtz equation, and heat equation. In both cases, the results were encouraging with tighter bounds on moments being obtained by solving smaller problems with domain partitioning.; by Sandeep Sethuraman.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.; Includes bibliographical references (leaf 95).

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## Efficient reduced-basis approximation of scalar nonlinear time-dependent convection-diffusion problems, and extension to compressible flow problems

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

Tipo: Tese de Doutorado
Formato: 65 p.

Português

Relevância na Pesquisa

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In this thesis, the reduced-basis method is applied to nonlinear time-dependent convection-diffusion parameterized partial differential equations (PDEs). A proper orthogonal decomposition (POD) procedure is used for the construction of reduced-basis approximation for the field variables. In the presence of highly nonlinear terms, conventional reduced-basis would be inefficient and no longer superior to classical numerical approaches using advanced iterative techniques. To recover the computational advantage of the reduced-basis approach, an empirical interpolation approximation method is employed to define the coefficient-function approximation of the nonlinear terms. Next, the coefficient-function approximation is incorporated into the reduced-basis method to obtain a reduced-order model of nonlinear time-dependent parameterized convection-diffusion PDEs. Two formulations for the reduced-order models are proposed, which construct the reduced-basis space for the nonlinear functions and residual vector respectively. Finally, an offline-online procedure for rapid and inexpensive evaluation of the reduced-order model solutions and outputs, as well as associated asymptotic a posterior error estimators are developed.; (cont.) The operation count for the online stage depends only on the dimension of our reduced-basis approximation space and the dimension of our coefficient-function approximation space. The extension of the reduced-order model to a system of equations is also explored.; by Han Men.; Thesis (S.M.)--Massachusetts Institute of Technology...

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## Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation; RKDG method for the Boltzmann equation

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

Tipo: Tese de Doutorado
Formato: 87 p.

Português

Relevância na Pesquisa

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In this thesis we investigate the ability of the Runge-Kutta Discontinuous Galerkin (RKDG) method to provide accurate and efficient solutions of the Boltzmann equation. Solutions of the Boltzmann equation are desirable in connection to small scale science and technology because when characteristic flow length scales become of the order of, or smaller than, the molecular mean free path, the Navier-Stokes description fails. The prevalent Boltzmann solution method is a stochastic particle simulation scheme known as Direct Simulation Monte Carlo (DSMC). Unfortunately, DSMC is not very effective in low speed flows (typical of small scale devices of interest) because of the high statistical uncertainty associated with the statistical sampling of macroscopic quantities employed by this method. This work complements the recent development of an efficient low noise method for calculating the collision integral of the Boltzmann equation, by providing a high-order discretization method for the advection operator balancing the collision integral in the Boltzmann equation. One of the most attractive features of the RKDG method is its ability to combine high-order accuracy, both in physical space and time, with the ability to capture discontinuous solutions.; (cont.) The validity of this claim is thoroughly investigated in this thesis. It is shown that...

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## Semidefinite relaxation based branch-and-bound method for nonconvex quadratic programming

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

Tipo: Tese de Doutorado
Formato: 75 leaves

Português

Relevância na Pesquisa

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In this thesis, we use a semidefinite relaxation based branch-and-bound method to solve nonconvex quadratic programming problems. Firstly, we show an interval branch-and-bound method to calculate the bounds for the minimum of bounded polynomials. Then we demonstrate four SDP relaxation methods to solve nonconvex Box constrained Quadratic Programming (BoxQP) problems and the comparison of the four methods. For some lower dimensional problems, SDP relaxation methods can achieve tight bounds for the BoxQP problem; whereas for higher dimensional cases (more than 20 dimensions), the bounds achieved by the four Semidefinite programming (SDP) relaxation methods are always loose. To achieve tight bounds for higher dimensional BoxQP problems, we combine the branch-and-bound method and SDP relaxation method to develop an SDP relaxation based branch-and-bound (SDPBB) method. We introduce a sensitivity analysis method for the branching process of SDPBB. This sensitivity analysis method can improve the convergence speed significantly.; (cont.) Compared to the interval branch-and-bound method and the global optimization software BARON, SDPBB can achieve better bounds and is also much more efficient. Additionally, we have developed a multisection algorithm for SDPBB and the multisection algorithm has been parallelized using Message Passing Interface (MPI). By parallelizing the program...

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## A fast 3D full-wave solver for nanophotonics; fast three-dimensional full-wave solver for nanophotonics

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

Tipo: Tese de Doutorado
Formato: 61 p.

Português

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Conventional fast integral equation solvers seem to be ideal approaches for simulating 3-D nanophotonic devices, as these devices are considered to be open structures, generating fields in both an interior channel and in the infinite exterior domain. However, many devices of interest, such as optical ring resonator filters or waveguides, have channels that can not be terminated without generating numerical reflections. Therefore, designing absorbers for these channels is a new problem for integral equation methods, as integral equation methods were initially developed for problems with finite surfaces. In this thesis we present a technique to eliminate reflections, making the channel volume conductive outside the domain of interest. The surface integral equation (SIE) method is employed to take advantage of the piecewise homogeneous medium. The Poggio-Miller-Chang-Harrington-Wu (PM-CHW) formulation is formed and the boundary element method is employed to construct and solve a linear system. Moreover, the block Toeplitz matrix property and using FFT helps reduce memory requirement, and accelerate the circulant matrix vector product. Numerical experiments are presented to demonstrate that this method can effectively reduce reflections to 1%...

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## Surrogate modeling for large-scale black-box systems

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

Tipo: Tese de Doutorado
Formato: 110 p.

Português

Relevância na Pesquisa

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This research introduces a systematic method to reduce the complexity of large-scale blackbox systems for which the governing equations are unavailable. For such systems, surrogate models are critical for many applications, such as Monte Carlo simulations; however, existing surrogate modeling methods often are not applicable, particularly when the dimension of the input space is very high. In this research, we develop a systematic approach to represent the high-dimensional input space of a large-scale system by a smaller set of inputs. This collection of representatives is called a multi-agent collective, forming a surrogate model with which an inexpensive computation replaces the original complex task. The mathematical criteria used to derive the collective aim to avoid overlapping of characteristics between representatives, in order to achieve an effective surrogate model and avoid redundancies. The surrogate modeling method is demonstrated on a light inventory that contains light data corresponding to 82 aircraft types. Ten aircraft types are selected by the method to represent the full light inventory for the computation of fuel burn estimates, yielding an error between outputs from the surrogate and full models of just 2.08%. The ten representative aircraft types are selected by first aggregating similar aircraft types together into agents...

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## A shifting method for dynamic system Model Order Reduction

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

Tipo: Tese de Doutorado
Formato: 86 p.

Português

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Model Order Reduction (MOR) is becoming increasingly important in computational applications. At the same time, the need for more comprehensive models of systems is generating problems with increasing numbers of outputs and inputs. Classical methods, which were developed for Single-Input Single-Output (SISO) systems, generate reduced models that are too computationally inefficient for large Multiple-Input Multiple-Output (MIMO) systems. Although many approaches exclusively designed for MIMO systems have emerged during the past decade, they cannot satisfy the overall needs for maintaining the characteristics of systems. This research investigates the reasons for the poor performances of the proposed approaches, using specific examples. Inspired by these existing methods, this research develops a novel way to extract information from MIMO systems, by means of system transfer functions. The approach, called Shifting method, iteratively extracts time-constant shifts from the system and splits the transfer function into several simple systems referred to as contour terms that outline the system structure, and a reducible system referred to as remainder system that complement the Contour Terms. This algorithm produces a remainder system that existing approaches can reduce more effectively. This approach works particularly well for systems with either tightly clustered or well separated modes...

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## Robust transportation network design under user equilibrium

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

Tipo: Tese de Doutorado
Formato: 63 p.

Português

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195.74%

We address the problem of designing a transportation network in the presence of demand uncertainty, multiple origin-destination pairs and a budget constraint for the overall construction cost, under the behavioral assumption that travelers optimize their own travel costs (i.e., the "user-equilibrium" condition). Under deterministic demand, we propose an exact integer optimization approach that leads to a quadratic objective, linear constraints optimization problem. As a result, the problem is efficiently solvable via commercial software, when the costs are linear functions of traffic flows. We then use an iterative algorithm to address the case of nonlinear cost functions. While the problem is intractable under probabilistic assumptions on demand uncertainty, we extend the previous model and propose an iterative algorithm using a robust optimization approach that models demand uncertainty. We finally report extensive numerical results to illustrate that our approach leads to tractable solutions for large scale networks.; by Yun Lu.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.; Includes bibliographical references (p. 59-63).

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## Computational issues and related mathematics of an exponential annealing homotropy for conic optimization

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

Tipo: Tese de Doutorado
Formato: 106 p.

Português

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We present a further study and analysis of an exponential annealing based algorithm for convex optimization. We begin by developing a general framework for applying exponential annealing to conic optimization. We analyze the hit-and-run random walk from the perspective of convergence and develop (partially) an intuitive picture that views it as the limit of a sequence of finite state Markov chains. We then establish useful results that guide our sampling. Modifications are proposed that seek to raise the computational practicality of exponential annealing for convex optimization. In particular, inspired by interior-point methods, we propose modifying the hit-and-run random walk to bias iterates away from the boundary of the feasible region and show that this approach yields a substantial reduction in computational cost. We perform computational experiments for linear and semidefinite optimization problems. For linear optimization problems, we verify the correlation of phase count with the Renegar condition measure (described in [13]); for semidefinite optimization, we verify the correlation of phase count with a geometry measure (presented in [4]).; by Jeremy Chen.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program...

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## A generalized precorrected-FFT method for electromagnetic analysis

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

Tipo: Tese de Doutorado
Formato: 119 p.

Português

Relevância na Pesquisa

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Boundary Element Methods (BEM) can be ideal approaches for simulating the behavior of physical systems in which the volumes have homogeneous properties. These, especially the so-called "fast" or "accelerated" BEM approaches often have significant computational advantages over other well-known methods which solve partial differential equations on a volume domain. However, the implementation of techniques used to accelerate BEM approaches often comes at a loss of some generality, reducing their applicability to many problems and preventing engineers and researchers from easily building on a common, popular base of code. In this thesis we create a BEM solver which uses the Pre-Corrected FFT technique for accelerating computation, and uses a novel approach which allows users to provide arbitrary basis functions. We demonstrate its utility for both electrostatic and full-wave electromagnetic problems in volumes with homogeneous isotropic permittivity, bounded by arbitrarily complex surface geometries. The code is shown to have performance characteristics similar to the best known approaches for these problems. It also provides an increased level of generality, and is designed in such a way that should allow it to easily be extended by other researchers.; by Stephen Gerald Leibman.; Thesis (S.M.)--Massachusetts Institute of Technology...

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## Model reduction for dynamic sensor steering : a Bayesian approach to inverse problems

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

Tipo: Tese de Doutorado
Formato: 101 p.

Português

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In many settings, distributed sensors provide dynamic measurements over a specified time horizon that can be used to reconstruct information such as parameters, states or initial conditions. This estimation task can be posed formally as an inverse problem: given a model and a set of measurements, estimate the parameters of interest. We consider the specific problem of computing in real-time the prediction of a contamination event, based on measurements obtained by mobile sensors. The spread of the contamination is modeled by the convection diffusion equation. A Bayesian approach to the inverse problem yields an estimate of the probability density function of the initial contaminant concentration, which can then be propagated through the forward model to determine the predicted contaminant field at some future time and its associated uncertainty distribution. Sensor steering is effected by formulating and solving an optimization problem that seeks the sensor locations that minimize the uncertainty in this prediction. An important aspect of this Dynamic Sensor Steering Algorithm is the ability to execute in real-time. We achieve this through reduced-order modeling, which (for our two-dimensional examples) yields models that can be solved two orders of magnitude faster than the original system...

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## LP-based subgradient algorithm for joint pricing and inventory control problems; Linear programming-based subgradient algorithm for joint pricing and inventory control problems

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

Tipo: Tese de Doutorado
Formato: 94 p.

Português

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It is important for companies to manage their revenues and -reduce their costs efficiently. These goals can be achieved through effective pricing and inventory control strategies. This thesis studies a joint multi-period pricing and inventory control problem for a make-to-stock manufacturing system. Multiple products are produced under shared production capacity over a finite time horizon. The demand for each product is a function of the prices and no back orders are allowed. Inventory and production costs are linear functions of the levels of inventory and production, respectively. In this thesis, we introduce an iterative gradient-based algorithm. A key idea is that given a demand realization, the cost minimization part of the problem becomes a linear transportation problem. Given this idea, if we knew the optimal demand, we could solve the production problem efficiently. At each iteration of the algorithm, given a demand vector we solve a linear transportation problem and use its dual variables in order to solve a quadratic optimization problem that optimizes the revenue part and generates a new pricing policy. We illustrate computationally that this algorithm obtains the optimal production and pricing policy over the finite time horizon efficiently. The computational experiments in this thesis use a wide range of simulated data. The results show that the algorithm we study in this thesis indeed computes the optimal solution for the joint pricing and inventory control problem and is efficient as compared to solving a reformulation of the problem directly using commercial software. The algorithm proposed in this thesis solves large scale problems and can handle a wide range of nonlinear demand functions.; by Tingting Rao.; Thesis (S.M.)--Massachusetts Institute of Technology...

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## A simulation-based resource optimization and time reduction model using design structure matrix

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

Tipo: Tese de Doutorado
Formato: 95 p.

Português

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Project scheduling is an important research and application area in engineering management. Recent research in this area addresses resource constraints as well as stochastic durations. This thesis presents a simulation-based optimization model for solving resource-constrained product development project scheduling problems. The model uses design structure matrix (DSM) to represent the information exchange among various tasks of a project. Instead of a simple binary precedence relationship, DSM is able to quantify the extent of interactions as well. In particular, these interactions are characterized by rework probabilities, rework impacts and learning. As a result, modeling based on DSM allows iterations to take place. This stochastic characteristic is not well addressed in earlier literatures of project scheduling problems. Adding resource factors to DSM simulation is a relatively new topic. We not only model the constraints posed by resource requirements, but also explore the effect of allocating different amount of resources on iterations. Genetic algorithm (GA) is chosen to optimize the model over a weighted sum of a set of heuristics. GA is known for its robustness in solving many types of problems. While the normal branch-and-bound method depends on problem specific information to generate tight bounds...

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## Empirical comparison of robust, data driven and stochastic optimization

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

Tipo: Tese de Doutorado
Formato: 49 leaves

Português

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In this thesis, we compare computationally four methods for solving optimization problems under uncertainty: * Robust Optimization (RO) * Adaptive Robust Optimization (ARO) * Data Driven Optimization (DDO) * stochastic Programming (SP) We have implemented several computation experiments to demonstrate the different performance of these methods. We conclude that ARO outperform RO, which has a comparable performance with DDO. SP has a comparable performance with RO when the assumed distribution is the same as the true underlying distribution, but under performs RO when the assumed distribution is different from the true distribution.; by Wang, Yanbo.; Includes bibliographical references (leaf 49).; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.

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## A comparison of discrete and flow-based models for air traffic flow management

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

Tipo: Tese de Doutorado
Formato: 74 leaves

Português

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The steady increase of congestion in air traffic networks has resulted in significant economic losses and potential safety issues in the air transportation. A potential way to reduce congestion is to adopt efficient air traffic management policies, such as, optimally scheduling and routing air traffic throughout the network. In recent years, several models have been proposed to predict and manage air traffic. This thesis focuses on the comparison of two such approaches to air traffic flow management: (i) a discrete Mixed Integer Program model, and (ii) a continuous flow-based model. The continuous model is applied in a multi-commodity setting to take into account the origins and destinations of the aircraft. Sequential quadratic programming is used to optimize the continuous model. A comparison of the performance of the two models based on a set of large scale test cases is provided. Preliminary results suggest that the linear programming relaxation of the discrete model provides results similar to the continuous flow-based model for high volumes of air traffic.; by Thi Vu Phu.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.; Includes bibliographical references (leaves 73-74).

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## Blast mitigation strategies for vehicles using shape optimization methods

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

Tipo: Tese de Doutorado
Formato: 72 p.

Português

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by Ganesh Gurumurthy.; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.; Includes bibliographical references (p. 69-72).

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## A nonsmooth exclusion test for finding all solutions of nonlinear equations

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

Tipo: Tese de Doutorado
Formato: 94 p.

Português

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A new approach is proposed for finding all solutions of systems of nonlinear equations with bound constraints. The zero finding problem is converted to a global optimization problem whose global minima with zero objective value, if any, correspond to all solutions of the initial problem. A branch-and-bound algorithm is used with McCormick's nonsmooth convex relaxations to generate lower bounds. An inclusion relation between the solution set of the relaxed problem and that of the original non-convex problem is established which motivates a method to generate automatically reasonably close starting points for a local Newton-type method. A damped-Newton method with natural level functions employing the restrictive monotonicity test is employed to find solutions robustly and rapidly. Due to the special structure of the objective function, the solution of the convex lower bounding problem yields a nonsmooth root exclusion test which is found to perform better than earlier interval based exclusion tests. The Krawczyk operator based root inclusion and exclusion tests are also embedded in the proposed algorithm to refine the variable bounds for efficient fathoming of the search space. The performance of the algorithm on a variety of test problems from the literature is presented and for most of them the first solution is found at the first iteration of the algorithm due to the good starting point generation.; by Vinay Kumar.; Thesis (S.M.)--Massachusetts Institute of Technology...

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