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Texas A&M University
1.
Huang, Yanhe.
Injectivity Properties of Generalized Wronski Map.
Degree: MS, Mathematics, 2016, Texas A&M University
URL: http://hdl.handle.net/1969.1/157103 
► Given a complex vector space V, consider the quotient map of the image of the Plucker embedding of the Grassmannian of m-planes of V by…
(more)
▼ Given a complex vector space V, consider the quotient map of the image of the Plucker embedding of the Grassmannian of
m-planes of V by a certain subspace of P ∧
m V . Such maps generalize the classical Wronski maps on Grassmannians of spaces of polynomials, the Wronski maps on Grassmanians of spaces of solutions of linear homogeneous differential equations, pole-placement maps of input-output linear systems and their realizations as linear control systems. We are interested in finding the degree of such maps, i.e. in determining the number of points in the preimage of the generic point of the image. We distinguish a special subclass of these maps, called self-adjoint, for which the degree of the corresponding Wronski map is at least two. In the case of Wronski maps on Grassmanians of spaces of solutions of linear homogeneous differential equations our self-adjoint generalized Wronski maps correspond to the classical self-adjoint linear differential operators, up to a natural equivalence. In the case of linear control systems, they correspond to control system with symmetric transfer function, up to a state-feedback equivalence. The main question is whether there are non-selfadjoint generalized Wronski maps with the degree greater than 1. We give a negative answer to this question in the case
m = 2 and
m = 3 under some natural assumptions.
Advisors/Committee Members: Zelenko, Igor (advisor), Sottile, Frank (committee member), Vadali, Srinivas (committee member).
Subjects/Keywords: Strongly Noninjective; Wronski Map
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APA (6th Edition):
Huang, Y. (2016). Injectivity Properties of Generalized Wronski Map. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157103
Chicago Manual of Style (16th Edition):
Huang, Yanhe. “Injectivity Properties of Generalized Wronski Map.” 2016. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/157103.
MLA Handbook (7th Edition):
Huang, Yanhe. “Injectivity Properties of Generalized Wronski Map.” 2016. Web. 10 Apr 2021.
Vancouver:
Huang Y. Injectivity Properties of Generalized Wronski Map. [Internet] [Masters thesis]. Texas A&M University; 2016. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/157103.
Council of Science Editors:
Huang Y. Injectivity Properties of Generalized Wronski Map. [Masters Thesis]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157103
Texas A&M University
2.
Wendel, Eric.
Receding Horizon Covariance Control.
Degree: MS, Aerospace Engineering, 2012, Texas A&M University
URL: http://hdl.handle.net/1969.1/ETD-TAMU-2012-08-11691
► Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to…
(more)
▼ Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time.
The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?.
These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.
Advisors/Committee Members: Bhattacharya, Raktim (advisor), Robles, Colleen (committee member), Zelenko, Igor (committee member).
Subjects/Keywords: receding horizon control; Lie group actions; optimal control theory; Hamiltonian systems; covariance control theory
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APA (6th Edition):
Wendel, E. (2012). Receding Horizon Covariance Control. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/ETD-TAMU-2012-08-11691
Chicago Manual of Style (16th Edition):
Wendel, Eric. “Receding Horizon Covariance Control.” 2012. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/ETD-TAMU-2012-08-11691.
MLA Handbook (7th Edition):
Wendel, Eric. “Receding Horizon Covariance Control.” 2012. Web. 10 Apr 2021.
Vancouver:
Wendel E. Receding Horizon Covariance Control. [Internet] [Masters thesis]. Texas A&M University; 2012. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/ETD-TAMU-2012-08-11691.
Council of Science Editors:
Wendel E. Receding Horizon Covariance Control. [Masters Thesis]. Texas A&M University; 2012. Available from: http://hdl.handle.net/1969.1/ETD-TAMU-2012-08-11691
Texas A&M University
3.
Desai, Uri.
A Comparative Study of Estimation Models for Satellite Relative Motion.
Degree: MS, Aerospace Engineering, 2013, Texas A&M University
URL: http://hdl.handle.net/1969.1/149297
► The problem of relative spacecraft motion estimation is considered with application to various reference and relative orbits. Mean circular and elliptic orbits are analyzed, with…
(more)
▼ The problem of relative spacecraft motion estimation is considered with application to various reference and relative orbits. Mean circular and elliptic orbits are analyzed, with relative orbits ranging in size from 1 km to 10 km. Estimators are built for three propagation models: (i) Gim-Alfriend State Transition Matrix, (ii) the J2-Linearized Equations of Motion for Circular Orbits, and (iii) the Clohessy-Wiltshire Equations of Motion. Two alternative models were developed in an attempt to ac- count for unmodeled nonlinearities: (i) Biased Clohessy-Whiltshire Equations, and (ii) J2 -Linearized State Transition Matrix. Two estimation techniques are presented in an attempt to explore and determine which propagation model minimizes the error residual: the linear Kalman filter is presented under the assumption of vector based, GPS-type measurements; the extended Kalman filter is analyzed assuming angle-range, optical-type measurements. Sampling time is varied to look at the effect of measurement frequency. It is assumed that the orbit of one of the satellites, the chief, is known reasonably well.
This work showed that the error residuals from the state estimates were minimized when the propagation technique utilized was the Gim-Alfriend State Transition Matrix. This supports conclusions that are obtained outside of the estimation problem. Additionally, the error residuals obtained when the propagation technique was the Clohessy-Wiltshire Equations is comparable to the more complicated models. Unmodeled nonlinearities affect the magnitude of the error residuals. As expected, the Gim-Alfriend STM comes closest to the truth; for smaller eccentricities (0.005), the Clohessy-Wiltshire EOM show minor deviations from the truth. As the eccentricity increases, the linear models begin to diverge greatly from the true response. The additional two models (the biased CW equations, and the linear STM) show decent performance under specific conditions. The former accounts for some of the unaccounted for nonlinearities. The latter exhibits comparable performance to the Gim-Alfrien STM for circular reference orbits. However, in each case, as the nonlinearity of the problem increases, the accuracy of the model decreases.
Advisors/Committee Members: Vadali, Srinivas R (advisor), Junkins, John (committee member), Zelenko, Igor (committee member).
Subjects/Keywords: relative navigation; Clohessy-Wiltshire Equations; Gim-Alfriend State Transition Matrix (GA-STM); estimation
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APA (6th Edition):
Desai, U. (2013). A Comparative Study of Estimation Models for Satellite Relative Motion. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/149297
Chicago Manual of Style (16th Edition):
Desai, Uri. “A Comparative Study of Estimation Models for Satellite Relative Motion.” 2013. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/149297.
MLA Handbook (7th Edition):
Desai, Uri. “A Comparative Study of Estimation Models for Satellite Relative Motion.” 2013. Web. 10 Apr 2021.
Vancouver:
Desai U. A Comparative Study of Estimation Models for Satellite Relative Motion. [Internet] [Masters thesis]. Texas A&M University; 2013. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/149297.
Council of Science Editors:
Desai U. A Comparative Study of Estimation Models for Satellite Relative Motion. [Masters Thesis]. Texas A&M University; 2013. Available from: http://hdl.handle.net/1969.1/149297
Texas A&M University
4.
Mahajan, Bharat.
Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics.
Degree: PhD, Aerospace Engineering, 2018, Texas A&M University
URL: http://hdl.handle.net/1969.1/173295
► In 1959, Dirk Brouwer pioneered the use of the Hamiltonian perturbation methods for constructing artificial satellite theories with effects due to nonspherical gravitational perturbations included.…
(more)
▼ In 1959, Dirk Brouwer pioneered the use of the Hamiltonian perturbation methods for constructing
artificial satellite theories with effects due to nonspherical gravitational perturbations included.
His solution specifically accounted for the effects of the first few zonal spherical harmonics.
However, the development of a closed-form (in the eccentricity) satellite theory that accounts
for any arbitrary spherical harmonic perturbation remains a challenge to this day. In the present
work, the author has obtained novel solutions for the absolute and relative motion of artificial satellites
(absolute motion in this work refers to the motion relative to the central gravitational body)
for an arbitrary zonal or tesseral spherical harmonic by using Hamiltonian perturbation methods,
without resorting to expansions in either the eccentricity or the small ratio of the satellite’s mean
motion and the angular velocity of the central body. First, generalized closed-form expressions
for the secular, long-period, and short-period variations of the equinoctial orbital elements due
to an arbitrary zonal harmonic are derived, along with the explicit expressions for the first six
zonal harmonics. Next, similar closed-form expressions are obtained for the sectorial and tesseral
(collectively referred to as tesserals henceforth) harmonics by using a new approach for the exact
Delaunay normalization of the perturbed Keplerian Hamiltonian. This approach reduces the solution
for the tesseral periodic perturbations to quadratures. It is shown that the existing approximate
approaches for the normalization of the tesseral problem, such as the method of relegation, can be
derived from the proposed exact solution. Moreover, the exact solution for the periodic variations
due to the tesseral harmonics produces a unified artificial satellite theory for the sub-synchronous
and super-synchronous orbit regimes without any singularities for the resonant orbits. The closedform
theories developed for the absolute motion are then used to develop analytic solutions in the
form of state transition matrices for the satellite relative motion near a perturbed elliptic reference
orbit. The expressions for differential equinoctial orbital elements for establishing a general circular
orbit type satellite formation are also derived to avoid singularities for the equatorial and
circular reference orbits. In order to negate the along-track drifts in satellite formations, an ana-
lytic expression for the differential semimajor axis is derived by taking into account the secular
effects due to all the zonal harmonics. The potential applications of the proposed satellite theories
range from fuel-efficient guidance and control algorithms, formation design, faster trade and
parametric studies to catalog maintenance, conjunction analysis, and covariance propagation for
space situational awareness. Two specific applications, one for solving a perturbed multiple revolution
Lambert’s problem and the other for rapid nonlinear propagation of orbit uncertainties using
point…
Advisors/Committee Members: Vadali, Srinivas R. (advisor), Alfriend, Kyle T. (advisor), Hurtado, John E. (committee member), Zelenko, Igor (committee member).
Subjects/Keywords: Artificial Satellite Theories; Zonal and Tesseral Spherical Harmonics; Canonical Perturbation Methods; Satellite Relative Motion; Formation Flying; Relative Motion STM; Perturbed Lambert Solver; Formation Drift Mitigation
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APA (6th Edition):
Mahajan, B. (2018). Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/173295
Chicago Manual of Style (16th Edition):
Mahajan, Bharat. “Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics.” 2018. Doctoral Dissertation, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/173295.
MLA Handbook (7th Edition):
Mahajan, Bharat. “Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics.” 2018. Web. 10 Apr 2021.
Vancouver:
Mahajan B. Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics. [Internet] [Doctoral dissertation]. Texas A&M University; 2018. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/173295.
Council of Science Editors:
Mahajan B. Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics. [Doctoral Dissertation]. Texas A&M University; 2018. Available from: http://hdl.handle.net/1969.1/173295
Texas A&M University
5.
Johnson, Kirk Wayne.
Approaches for Modeling Satellite Relative Motion.
Degree: PhD, Aerospace Engineering, 2016, Texas A&M University
URL: http://hdl.handle.net/1969.1/158945
► This dissertation explores new approaches for modeling perturbed and unperturbed satellite relative motion. It extends Hoots orbit theory, an analytical averaging-method perturbation solution to the…
(more)
▼ This dissertation explores new approaches for modeling perturbed and unperturbed satellite relative motion. It extends Hoots orbit theory, an analytical averaging-method perturbation solution to the Zonal Problem, to second order. In addition, this study develops a new hybrid numerical/analytical algorithm for converting initial conditions from osculating elements to mean elements, so that a single set of osculating initial conditions may be taken as simulation inputs. Also, this study develops a new version of the Gim-Alfriend State Transition Matrix (GA STM) for linearized perturbed relative motion, in terms of the variables from Hoots theory. These variables, the Hoots elements, are advantageous (although not unique) in that they have no singularities for orbit eccentricity or inclination and they require only one solution of Kepler's Equation at each time step, even when using the GA STM. The new models are compared by simulation with orbit theories and GA STMs using the so-called nonsingular elements (which are in fact singular for zero inclination) and the equinoctial elements. This study predicts and verifies the order of magnitude of modeling error due to various sources.
This study also considers two special applications in satellite relative motion modeling. First, Projected Circular Orbit (PCO) formations, originally defined for unperturbed motion about a circular reference orbit, have important applications and are widely studied. This dissertation removes the singularity for zero inclination by implementing the PCO initial conditions in equinoctial elements, allowing PCO formations to be initialized about equatorial orbits. Furthermore, this study reveals how the choice of variables for writing the PCO initial conditions changes the geometric interpretation of the PCO phase angle parameter α. Second, this study develops an alternative to the standard methods for mitigating along-track drift in perturbed satellite formations. The new method eliminates all along-track secular motion to first order by sacrificing one degree of freedom in the formation design.
Advisors/Committee Members: Alfriend, Kyle T. (advisor), Vadali, Srinivas R. (advisor), Hurtado, John E. (committee member), Zelenko, Igor (committee member).
Subjects/Keywords: Satellite Relative Motion; Satellite Formation Flying; Zonal Problem; Astrodynamics
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APA (6th Edition):
Johnson, K. W. (2016). Approaches for Modeling Satellite Relative Motion. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/158945
Chicago Manual of Style (16th Edition):
Johnson, Kirk Wayne. “Approaches for Modeling Satellite Relative Motion.” 2016. Doctoral Dissertation, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/158945.
MLA Handbook (7th Edition):
Johnson, Kirk Wayne. “Approaches for Modeling Satellite Relative Motion.” 2016. Web. 10 Apr 2021.
Vancouver:
Johnson KW. Approaches for Modeling Satellite Relative Motion. [Internet] [Doctoral dissertation]. Texas A&M University; 2016. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/158945.
Council of Science Editors:
Johnson KW. Approaches for Modeling Satellite Relative Motion. [Doctoral Dissertation]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/158945
Texas A&M University
6.
Nadubettu Yadukumar, Shishir 1986-.
Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics.
Degree: MS, Electrical Engineering, 2012, Texas A&M University
URL: http://hdl.handle.net/1969.1/148284
► The thesis shows how to achieve bipedal robotic walking on flat-ground, up-slope and rough terrain by using Human-Inspired control. We begin by considering human walking…
(more)
▼ The thesis shows how to achieve bipedal robotic walking on flat-ground, up-slope and rough terrain by using Human-Inspired control. We begin by considering human walking data and find outputs (or virtual constraints) that, when calculated from the human data, are described by simple functions of time (termed canonical walking functions). Formally, we construct a torque controller, through model inversion, that drives the outputs of the robot to the outputs of the human as represented by the canonical walking function; while these functions fit the human data well, they do not apriori guarantee robotic walking (due to do the physical differences between humans and robots). An optimization problem is presented that determines the best fit of the canonical walking function to the human data, while guaranteeing walking for a specific bipedal robot; in addition, constraints can be added that guarantee physically realizable walking. We consider a physical bipedal robot, AMBER, and considering the special property of the motors used in the robot, i.e., low leakage inductance, we approximate the motor model and use the formal controllers that satisfy the constraints and translate into an efficient voltage-based controller that can be directly implemented on AMBER. The end result is walking on flat-ground and up-slope which is not just human-like, but also amazingly robust. Having obtained walking on specific well defined terrains separately, rough terrain walking is achieved by dynamically changing the extended canonical walking functions (ECWF) that the robot outputs should track at every step. The state of the robot, after every non-stance foot strike, is actively sensed and the new CWF is constructed to ensure Hybrid Zero Dynamics is respected in the next step. Finally, the technique developed is tried on different terrains in simulation and in AMBER showing how the walking gait morphs depending on the terrain.
Advisors/Committee Members: Ames, Aaron D (advisor), Bhattacharyya, Shankar P (committee member), Ehsani, Mehrdad (committee member), Zelenko, Igor (committee member).
Subjects/Keywords: Legged Walking; Bipedal Walking; Humanoid
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APA (6th Edition):
Nadubettu Yadukumar, S. 1. (2012). Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/148284
Chicago Manual of Style (16th Edition):
Nadubettu Yadukumar, Shishir 1986-. “Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics.” 2012. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/148284.
MLA Handbook (7th Edition):
Nadubettu Yadukumar, Shishir 1986-. “Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics.” 2012. Web. 10 Apr 2021.
Vancouver:
Nadubettu Yadukumar S1. Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics. [Internet] [Masters thesis]. Texas A&M University; 2012. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/148284.
Council of Science Editors:
Nadubettu Yadukumar S1. Bipedal Robotic Walking on Flat-Ground, Up-Slope and Rough Terrain with Human-Inspired Hybrid Zero Dynamics. [Masters Thesis]. Texas A&M University; 2012. Available from: http://hdl.handle.net/1969.1/148284
Texas A&M University
7.
Porter, Curtis Wade.
The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type.
Degree: PhD, Mathematics, 2016, Texas A&M University
URL: http://hdl.handle.net/1969.1/157877
► We apply E. Cartan’s method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds M up to local CR equivalence in the case that the cubic…
(more)
▼ We apply E. Cartan’s method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds
M up to local CR equivalence in the case that the cubic form of
M satisfies a certain symmetry property with respect to the Levi form of
M. The solution to the equivalence problem is given by a parallelism on a principal bundle over
M which takes values in su(2, 2) or su(3, 1), depending on the signature of the nondegenerate part of the Levi form. Differentiating this parallelism provides a complete set of local invariants of
M. We exhibit an explicit example of a real hypersurface in C
4 whose invariants are nontrivial.
Advisors/Committee Members: Landsberg, Joseph M (advisor), Zelenko, Igor (advisor), Robles, Colleen (committee member), Becker, Melanie (committee member), Boas, Harold (committee member), Lima-Filho, Paulo (committee member).
Subjects/Keywords: CR geometry; method of equivalence; exterior differential systems
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APA (6th Edition):
Porter, C. W. (2016). The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157877
Chicago Manual of Style (16th Edition):
Porter, Curtis Wade. “The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type.” 2016. Doctoral Dissertation, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/157877.
MLA Handbook (7th Edition):
Porter, Curtis Wade. “The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type.” 2016. Web. 10 Apr 2021.
Vancouver:
Porter CW. The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type. [Internet] [Doctoral dissertation]. Texas A&M University; 2016. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/157877.
Council of Science Editors:
Porter CW. The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type. [Doctoral Dissertation]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157877
8.
Gomez Constante, Juan Pablo.
FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS.
Degree: PhD, Mechanical Engineering, 2018, Texas A&M University
URL: http://hdl.handle.net/1969.1/173653
► Inspired in the work of Poiseuille, and taking advantage of the much more realistic model for describing the mechanical properties of blood in large vessels…
(more)
▼ Inspired in the work of Poiseuille, and taking advantage of the much more realistic model
for describing the mechanical properties of blood in large vessels derived by K. R. Rajagopal,
this dissertation explains the behavior of blood on non circular cross sections, given an Implicit
Constitutive Relation.
First, by assuming that the axial direction is dominant in this phenomenom, an axial flow
problem is solved. Given the non linear and implicit nature of the problem, a non linear system
of coupled partial differential equations will be solved. Using the Minty-Browder Theorem and
the theory for linear elliptic operators, some elementary results about existence and uniqueness of
solutions can be stated.
Next, assuming a lower order velocity field in the other two directions, a secondary flow problem
in the cross sectional area is stated. Taking into account just the terms of the same order and
the solution of the previous step, the problem becomes a linear coupled system of partial differential
equations. Using the theorem of existence of a streamfunction in a two dimensional problem,
we can actually prove that there are no secondary flows for this model.
Finally, a numerical approximation directly based on the Lions-Mercier Splitting Algorithm is
given. Some generalizations of the problem are proposed as future work.
Advisors/Committee Members: Rajagopal, Kumbakonam R (advisor), Zelenko, Igor (committee member), Freed, Alan (committee member), Lazarov, Raytcho (committee member).
Subjects/Keywords: Fluid Mechanics; Implicit Power-Law FLuid
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APA (6th Edition):
Gomez Constante, J. P. (2018). FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/173653
Chicago Manual of Style (16th Edition):
Gomez Constante, Juan Pablo. “FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS.” 2018. Doctoral Dissertation, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/173653.
MLA Handbook (7th Edition):
Gomez Constante, Juan Pablo. “FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS.” 2018. Web. 10 Apr 2021.
Vancouver:
Gomez Constante JP. FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS. [Internet] [Doctoral dissertation]. Texas A&M University; 2018. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/173653.
Council of Science Editors:
Gomez Constante JP. FLOW OF STRESS POWER LAW FLUIDS IN CYLINDERS OF NON CIRCULAR CROSS SECTIONS. [Doctoral Dissertation]. Texas A&M University; 2018. Available from: http://hdl.handle.net/1969.1/173653
9.
Wendel, Eric Duong Ba.
On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube.
Degree: MS, Mathematics, 2015, Texas A&M University
URL: http://hdl.handle.net/1969.1/155576
► This thesis is devoted to aspects of the local differential geometry of regular, bracket-generating distributions. The classical discrete in variants of such distributions at a…
(more)
▼ This thesis is devoted to aspects of the local differential geometry of regular, bracket-generating distributions. The classical discrete in variants of such distributions at a point are its degree of nonholonomy and small growth vector, both of which encode how quickly the iterated Lie brackets of the distribution saturate the tangent space of the manifold at a point. Recently, Boris Doubrov and
Igor Zelenko introduced a new discrete in variant of these distributions at a point called the Jacobi symbol, and constructed canonical frames for all distributions with given Jacobisymbol. Their constructions, however, require an additional generic assumption called the maximal class condition. This condition can be formulated in purely control-theoretic language as a property of the end point map along special curves called abnormal trajectories. There is a strong belief that the condition of maximal class is essentially redundant, which is to say that all bracket-generating distributions are of maximal class or become maximal class after a natural reduction process. The aim of this thesis is to develop general tools for proving this conjecture and to apply these tools for the verification of the conjecture for a number of cases. We begin by proving that the maximal class condition is essentially determined by its Tanaka symbol, which is the graded space associated with the filtration of the distribution. We are most interested in proving that rank-2 distributions of dimension n ≥ 6 and small growth vector (2, 3, 5, ...) are of maximal class, and make our first steps towards this goal by proving that all (2, n)-distributions with Tanaka symbol isomorphic to the free truncated graded Lie algebra with 2 generators and degree of nilpotency 4 or 5 are of maximal class. We then provide calculations proving that all (2, n)-distributions associated with Monge ODEs areof maximal class.
Advisors/Committee Members: Zelenko, Igor (advisor), Sottile, Frank (committee member), Bhattacharya, Raktim (committee member).
Subjects/Keywords: nonholonomic distributions; abnormal extremals; differential invariants
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APA (6th Edition):
Wendel, E. D. B. (2015). On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/155576
Chicago Manual of Style (16th Edition):
Wendel, Eric Duong Ba. “On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube.” 2015. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/155576.
MLA Handbook (7th Edition):
Wendel, Eric Duong Ba. “On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube.” 2015. Web. 10 Apr 2021.
Vancouver:
Wendel EDB. On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube. [Internet] [Masters thesis]. Texas A&M University; 2015. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/155576.
Council of Science Editors:
Wendel EDB. On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube. [Masters Thesis]. Texas A&M University; 2015. Available from: http://hdl.handle.net/1969.1/155576
10.
Liu, Bingjin.
Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication.
Degree: MS, Mathematics, 2016, Texas A&M University
URL: http://hdl.handle.net/1969.1/157048
► Matrix multiplication is commonly used in scientific computation. Given matrices A = (aij ) of size l × m and B = (bij ) of…
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▼ Matrix multiplication is commonly used in scientific computation. Given matrices A = (aij ) of size l ×
m and B = (bij ) of size
m × n, the standard way to compute the product C := AB is computing cij = Σ
m k=1 aikbkj . In this case, lmn multiplications and ln(
m − 1) additions are used. In 1969, V. Strassen found a surprising algorithm to multiply 2 × 2 matrices using 7 multiplications instead of 8 in the standard algorithm. In this way, n × n matrix multiplication can be computed using O(n
log7 2 ) scalar multiplication operations. If n is large, the Strassen algorithm is much more efficient than the standard algorithm. After Strassen’s algorithm, numerous efforts were made to reduce the complexity for n × n matrix multiplication. By 1986, the bound was reduced to O(n
2.
38) by Coppersmith and Winograd. However this is an asymptotic result rather than an implementable algorithm. The complexity has not been significantly improved for 30 years.
Matrix multiplication is a tensor and one way to measure the complexity is using its tensor rank. Any tensor can be written as finite sum of rank one tensors and the rank for a tensor is the least number of rank-one tensors needed in the sum. A theorem due to Strassen shows the tensor rank is a good measurement for the complexity. One Bini’s theorem demonstrates that the border rank of the matrix multiplication tensor
M <n> is a complexity measurement for matrix multiplication. Even though the problem may sound simple, the border ranks of small matrix multiplication tensors are still unknown. Suppose one wants to compute the border rank of the tensor for the matrix multiplication of size
m × 2 and 2 × 2 denoted by R(
M<
m,2,2>). R(
M<
m,2,2>) is closely related to the border rank of reduced matrix multiplication tensor TBCLRS,
m, where one entry is set equal to zero. For small
m like 2 and 3, there are good geometric configurations in the border rank algorithms for the tensor TBCLRS,
m. My project is to understand the geometry of the good existing algorithms in the cases
m = 2, 3. In the configuration of case
m = 2, the limit 5-plane in the Grassmannian plane in the algorithm intersects with the Segre variety in three special lines. For the case
m = 3, the intersection of the limiting 8-plane and the Segre variety consists of the union of a family of lines passing through a plane conic and a special sub-Segre variety. I also try to find analogous algorithms to the
m = 2 case or disprove the existence of such algorithms.
Advisors/Committee Members: Landsberg, Joseph M (advisor), Zelenko, Igor (committee member), Welch, Jennifer Lundelius (committee member).
Subjects/Keywords: complexity for matrix multiplication; geometric configurations for algorithms
Record Details
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Record Details
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Liu, B. (2016). Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication. (Masters Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157048
Chicago Manual of Style (16th Edition):
Liu, Bingjin. “Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication.” 2016. Masters Thesis, Texas A&M University. Accessed April 10, 2021.
http://hdl.handle.net/1969.1/157048.
MLA Handbook (7th Edition):
Liu, Bingjin. “Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication.” 2016. Web. 10 Apr 2021.
Vancouver:
Liu B. Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication. [Internet] [Masters thesis]. Texas A&M University; 2016. [cited 2021 Apr 10].
Available from: http://hdl.handle.net/1969.1/157048.
Council of Science Editors:
Liu B. Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication. [Masters Thesis]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157048
. 
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