You searched for +publisher:"Vanderbilt University" +contributor:("Akram Aldroubi").
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Vanderbilt University
1.
Jiang, Jiayi.
Quantization in Signal Processing with Frame Theory.
Degree: PhD, Mathematics, 2016, Vanderbilt University
URL: http://hdl.handle.net/1803/11347 
► Quantization is an important part of signal processing. Several issues influence the performance of a quantization algorithm. One is the “basis” we choose to represent…
(more)
▼ Quantization is an important part of signal processing. Several issues influence the performance
of a quantization algorithm. One is the “basis” we choose to represent the signal, another is how
we quantize the “basis” coefficients. We desire to explore these two things in this thesis. In the first chapter, we review frame theory and fusion frame theory. In the second chapter, we introduce a
popular quantization algorithm known as Sigma-Delta quantization and show how to apply it
to finite frames. Then we give the definition and properties of Sobolev duals which are optimized
duals associated to Sigma-Delta quantization. The contraction Sobolev duals depends on the frame, and in chapter 3, we prove that for any finite unit-norm frame, the best error bound that can be achieved
from the reconstruction with Sobolev duals in rth Sigma-Delta quantization is equal to order O(N^-r), where the error bound can be related to both operator norm and Frobenius norm. In the final chapter, we develop Sigma-Delta quantization for fusion frames. We construct stable first-order and high-order Sigma-Delta algorithms for quantizing fusion frame projections of f onto W_n, where W_n is an M_n dimension subspace of R^d. Our stable 1st-order quantizer uses only log2(Mn+1) bits per subspace. Besides, we give an algorithm to calculate the Kashin representations for fusion frames to improve the performance of the high-order Sigma-Delta quantization algorithm. Then by defining the left inverse and the canonical left inverse for fusion frames, we prove the property that the canonical left inverse has the minimal operator norm and Frobenius norm. Based on this property, we give the idea of Sobolev left inverses for fusion
frames and prove it leads to minimal squared error.
Advisors/Committee Members: Doug Hardin (committee member), Akram Aldroubi (committee member), Brett Byram (committee member), Alexander Powell (Committee Chair).
Subjects/Keywords: Frame; Fusion Frame; Sigma-Delta Quantization; Sobolev Dual
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APA (6th Edition):
Jiang, J. (2016). Quantization in Signal Processing with Frame Theory. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11347
Chicago Manual of Style (16th Edition):
Jiang, Jiayi. “Quantization in Signal Processing with Frame Theory.” 2016. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11347.
MLA Handbook (7th Edition):
Jiang, Jiayi. “Quantization in Signal Processing with Frame Theory.” 2016. Web. 23 Jan 2021.
Vancouver:
Jiang J. Quantization in Signal Processing with Frame Theory. [Internet] [Doctoral dissertation]. Vanderbilt University; 2016. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11347.
Council of Science Editors:
Jiang J. Quantization in Signal Processing with Frame Theory. [Doctoral Dissertation]. Vanderbilt University; 2016. Available from: http://hdl.handle.net/1803/11347
Vanderbilt University
2.
Tuo, Shengquan.
Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector.
Degree: PhD, Physics, 2015, Vanderbilt University
URL: http://hdl.handle.net/1803/10900
► The azimuthal correlations of charged particles produced in lead-lead collisions at a nucleon-nucleon center-of-mass energy of 2.76 TeV and proton-lead collisions at 5.02 TeV have…
(more)
▼ The azimuthal correlations of charged particles produced in lead-lead collisions at a nucleon-nucleon center-of-mass energy of 2.76 TeV and proton-lead collisions at 5.02 TeV have been measured using multiparticle correlations. In PbPb collisions, the second, fourth and sixth Fourier coefficients of the charged-particle azimuthal distributions are studied as a function of the particle transverse momentum, pseudorapidity, and collision centrality. The results from several experimental methods that have different sensitivity to the fluctuations in the initial-state of ultra-relativistic heavy ion collisions are compared. The combined results provide access to the properties of the quark-gluon plasma through comparisons with the predictions of viscous hydrodynamic calculations. A near-perfect fluid behavior is observed in PbPb collisions at sqrt{s
_{NN}} = 2.76 TeV at the LHC, similar to previous findings at lower center-of-mass energies from RHIC. The system-size dependence of the collective flow is studied in peripheral PbPb collisions and in high-multiplicity pPb collisions. Remarkable similarities are found when the two systems are compared for collisions with the same final-state particle multiplicity. Four-particle correlations emerge in pPb collisions with more than 40 particles detected in the final state. The elliptic anisotropies derived from four-particle or all-particle correlations are found to agree within 10%, which indicates a high degree of collectivity in high-multiplicity pPb collisions and might be an evidence for the smallest liquid droplet ever produced in the laboratory.
Advisors/Committee Members: Senta Greene (committee member), Will Johns (committee member), Robert Scherrer (committee member), Akram Aldroubi (committee member), Julia Velkovska (Committee Chair).
Subjects/Keywords: Heavy ion collision; Multiparticle Correlation; LHC; Flow; CMS
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APA (6th Edition):
Tuo, S. (2015). Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/10900
Chicago Manual of Style (16th Edition):
Tuo, Shengquan. “Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector.” 2015. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/10900.
MLA Handbook (7th Edition):
Tuo, Shengquan. “Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector.” 2015. Web. 23 Jan 2021.
Vancouver:
Tuo S. Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector. [Internet] [Doctoral dissertation]. Vanderbilt University; 2015. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/10900.
Council of Science Editors:
Tuo S. Multiparticle Azimuthal Correlation Measurements in Lead-Lead and Proton-Lead Collisions at LHC with the CMS Detector. [Doctoral Dissertation]. Vanderbilt University; 2015. Available from: http://hdl.handle.net/1803/10900
Vanderbilt University
3.
Sekmen, Ali Safak.
Subspace Segmentation and High-Dimensional Data Analysis.
Degree: PhD, Mathematics, 2012, Vanderbilt University
URL: http://hdl.handle.net/1803/11935
► This thesis developed theory and associated algorithms to solve subspace segmentation problem. Given a set of data W={w1,...,wN} in RD that comes from a union…
(more)
▼ This thesis developed theory and associated algorithms to solve subspace segmentation problem. Given a set of data W={w
1,...,w
N} in R
D that comes from a union of subspaces, we focused on determining a nonlinear model of the form U={S
i}
i in I, where S
i is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our first approach is based on the binary reduced row echelon form of data matrix. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace S
i. We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise. Our second approach is based on nearness to local subspaces approach and it can handle noise effectively, but it works only in special cases of the general subspace segmentation problem (i.e., subspaces of equal and known dimensions). Our approach is based on the computation of a binary similarity matrix for the data points. A local subspace is first estimated for each data point. Then, a distance matrix is generated by computing the distances between the local subspaces and points. The distance matrix is converted to the similarity matrix by applying a data-driven threshold. The problem is then transformed to segmentation of subspaces of dimension 1 instead of subspaces of dimension d. The algorithm was applied to the Hopkins 155 Dataset and generated the best results to date.
Advisors/Committee Members: Douglas Hardin (committee member), Alexander Powell (committee member), Larry Schumaker (committee member), Mitchell Wilkes (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: motion segmentation; subspace segmentation; union of subspaces
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APA (6th Edition):
Sekmen, A. S. (2012). Subspace Segmentation and High-Dimensional Data Analysis. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11935
Chicago Manual of Style (16th Edition):
Sekmen, Ali Safak. “Subspace Segmentation and High-Dimensional Data Analysis.” 2012. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11935.
MLA Handbook (7th Edition):
Sekmen, Ali Safak. “Subspace Segmentation and High-Dimensional Data Analysis.” 2012. Web. 23 Jan 2021.
Vancouver:
Sekmen AS. Subspace Segmentation and High-Dimensional Data Analysis. [Internet] [Doctoral dissertation]. Vanderbilt University; 2012. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11935.
Council of Science Editors:
Sekmen AS. Subspace Segmentation and High-Dimensional Data Analysis. [Doctoral Dissertation]. Vanderbilt University; 2012. Available from: http://hdl.handle.net/1803/11935
Vanderbilt University
4.
Chen, Xuemei.
Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm.
Degree: PhD, Mathematics, 2012, Vanderbilt University
URL: http://hdl.handle.net/1803/12443
► This dissertation consists of two topics: compressed sensing and the Kaczmarz algorithm. Compressed sensing addresses the problem of recovering an unknown signal z0in ℝd from…
(more)
▼ This dissertation consists of two topics: compressed sensing and the Kaczmarz algorithm.
Compressed sensing addresses the problem of recovering an unknown signal z
0in ℝ
d from a small number of linear measurements based on an underlying structure of sparsity or compressibility. This dissertation will focus on the ell
q minimization approach. We show that the null space property is a necessary and sufficient condition on the measurement matrix for stable recovery. Moreover, we consider the compressed sensing problem when signals are sparse in a dictionary. Some basic conditions are given for this problem to be meaningful. It is known that under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ell
q minimization can provide an effective signal recovery tool even when the dictionary is coherent. We propose that a modified null space property for the dictionary is also sufficient to stably recover the signal. Perturbations on the measurement matrices and the dictionary are also considered.
The second part of this dissertation is concerned with the almost sure convergence rate of the Kaczmarz algorithm. The Kaczmarz algorithm is an iterative method for reconstructing a signal ξnℝ
d from an overcomplete collection of linear measurements y
n = langle x, varphi
n
angle, n geq 1. This algorithm is widely used in image processing and computer tomography.
We prove quantitative bounds on the rate of almost sure exponential convergence in the Kaczmarz algorithm for suitable classes of random measurement vectors {varphi
n}
n=1infty subset ℝ
d.
Refined convergence results are given for the special case when each varphi
n has i.i.d. Gaussian entries
and, more generally, when each varphi
n/|varphi
n| is uniformly distributed on mathbb{S}
d-1.
Advisors/Committee Members: Douglas Hardin (committee member), Edward Saff (committee member), Xenofon Koutsoukos (committee member), Alexander Powell (Committee Chair), Akram Aldroubi (Committee Chair).
Subjects/Keywords: stability; compressed sensing; Kaczmarz Algorithm; randomization; null space property; dictionary
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APA ·
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MLA ·
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CSE |
Export
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APA (6th Edition):
Chen, X. (2012). Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12443
Chicago Manual of Style (16th Edition):
Chen, Xuemei. “Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm.” 2012. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12443.
MLA Handbook (7th Edition):
Chen, Xuemei. “Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm.” 2012. Web. 23 Jan 2021.
Vancouver:
Chen X. Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm. [Internet] [Doctoral dissertation]. Vanderbilt University; 2012. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12443.
Council of Science Editors:
Chen X. Stability of compressed sensing for dictionaries and almost sure convergence rate for the Kaczmarz algorithm. [Doctoral Dissertation]. Vanderbilt University; 2012. Available from: http://hdl.handle.net/1803/12443
Vanderbilt University
5.
Leshen, Sara.
Balian-Low Type Results for Gabor Schauder Bases.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/10973
► The uncertainty principle implies that a function and its Fourier transform cannot both be well-localized. The Balian-Low theorem is a version of the uncertainty principle…
(more)
▼ The uncertainty principle implies that a function and its Fourier transform cannot both be well-localized. The Balian-Low theorem is a version of the uncertainty principle for generators of Gabor orthonormal bases. This dissertation proves a new Balian-Low type theorem for compactly supported generators of Gabor Schauder bases. Moreover, we show that the classical Balian-Low theorem for orthonormal bases does not hold for Schauder bases.
Advisors/Committee Members: Akram Aldroubi (committee member), Doug Hardin (committee member), Gieri Simonett (committee member), David Smith (committee member), Alexander Powell (Committee Chair).
Subjects/Keywords: Schauder bases; uncertainty principle; time-frequency analysis
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APA (6th Edition):
Leshen, S. (2019). Balian-Low Type Results for Gabor Schauder Bases. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/10973
Chicago Manual of Style (16th Edition):
Leshen, Sara. “Balian-Low Type Results for Gabor Schauder Bases.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/10973.
MLA Handbook (7th Edition):
Leshen, Sara. “Balian-Low Type Results for Gabor Schauder Bases.” 2019. Web. 23 Jan 2021.
Vancouver:
Leshen S. Balian-Low Type Results for Gabor Schauder Bases. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/10973.
Council of Science Editors:
Leshen S. Balian-Low Type Results for Gabor Schauder Bases. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/10973
Vanderbilt University
6.
Das, Sayan.
Poisson boundaries of finite von Neumann algebras.
Degree: PhD, Mathematics, 2017, Vanderbilt University
URL: http://hdl.handle.net/1803/13350
► Poisson boundaries of groups plays a major role in the study of group actions on measure spaces. In this work, we study noncommmutative Poisson boundaries…
(more)
▼ Poisson boundaries of groups plays a major role in the study of group actions on measure spaces. In this work, we study noncommmutative Poisson boundaries of finite von Neumannalgebras. We prove a noncommutative analogue of the double ergodicity theorem due to V.Kaimanovich and give applications to the study of derivations on a finite von Neumann algebra,and the similarity problem. We also prove a boundary rigidity theorem, using double ergodicity. We also define and study the notions of noncommutative Avez entropy, and noncommutative Fustenberg entropy, and show an entropy gap theorem for von Neumann algebras with property(T).
Advisors/Committee Members: Robert J. Scherrer (committee member), Akram Aldroubi (committee member), Dietmar Bisch (committee member), Jesse D. Peterson (Committee Chair), Vaughan Jones (Committee Chair).
Subjects/Keywords: property (T); amenability; hyperstate; entropy; rigidity; operator systems
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APA (6th Edition):
Das, S. (2017). Poisson boundaries of finite von Neumann algebras. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/13350
Chicago Manual of Style (16th Edition):
Das, Sayan. “Poisson boundaries of finite von Neumann algebras.” 2017. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/13350.
MLA Handbook (7th Edition):
Das, Sayan. “Poisson boundaries of finite von Neumann algebras.” 2017. Web. 23 Jan 2021.
Vancouver:
Das S. Poisson boundaries of finite von Neumann algebras. [Internet] [Doctoral dissertation]. Vanderbilt University; 2017. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/13350.
Council of Science Editors:
Das S. Poisson boundaries of finite von Neumann algebras. [Doctoral Dissertation]. Vanderbilt University; 2017. Available from: http://hdl.handle.net/1803/13350
Vanderbilt University
7.
LeCrone, Jeremy.
On the axisymmetric surface diffusion flow.
Degree: PhD, Mathematics, 2012, Vanderbilt University
URL: http://hdl.handle.net/1803/12360
► In this thesis, we establish analytic results for the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In the first part of the…
(more)
▼ In this thesis, we establish analytic results for the axisymmetric surface
diffusion flow (ASD), a fourth-order geometric evolution law. In the first part
of the work, we develop a general theory establishing maximal regularity results
for a broad class of abstract, higher-order elliptic operators, in the setting
of periodic little-Hölder spaces. These results are then applied, in the
second part of the thesis, to prove well-posedness results for ASD.
In particular, we prove that ASD generates a real analytic semiflow in the space of
(2 + alpha)-little-Hölder regular surfaces of revolution embedded in R
3.
Further, we give conditions for global existence of solutions and we prove that
solutions are real analytic in time and space.
We also investigate the dynamic properties of solutions to ASD in the second part of the thesis.
Utilizing a connection to axisymmetric surfaces with constant mean curvature,
we characterize the equilibria of ASD. We focus on the family of cylinders
as equilibria of ASD and we establish results regarding the stability, instability
and bifurcation behavior of cylinders with the radius acting as
a bifurcation parameter for the problem.
Advisors/Committee Members: Emmanuele DiBenedetto (committee member), Alexander Powell (committee member), Akram Aldroubi (committee member), Zhaohua Ding (committee member), Gieri Simonett (Committee Chair).
Subjects/Keywords: fourier multipliers; nonlinear stability; maximal regularity; periodic boundary conditions; surface diffusion
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APA (6th Edition):
LeCrone, J. (2012). On the axisymmetric surface diffusion flow. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12360
Chicago Manual of Style (16th Edition):
LeCrone, Jeremy. “On the axisymmetric surface diffusion flow.” 2012. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12360.
MLA Handbook (7th Edition):
LeCrone, Jeremy. “On the axisymmetric surface diffusion flow.” 2012. Web. 23 Jan 2021.
Vancouver:
LeCrone J. On the axisymmetric surface diffusion flow. [Internet] [Doctoral dissertation]. Vanderbilt University; 2012. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12360.
Council of Science Editors:
LeCrone J. On the axisymmetric surface diffusion flow. [Doctoral Dissertation]. Vanderbilt University; 2012. Available from: http://hdl.handle.net/1803/12360
Vanderbilt University
8.
Gui, Bin.
A unitary tensor product theory for unitary vertex operator algebra modules.
Degree: PhD, Mathematics, 2018, Vanderbilt University
URL: http://hdl.handle.net/1803/12719
► Let V be a unitary vertex operator algebra (VOA) satisfying the following conditions: (1) V is of CFT type. (2) Every N-gradable weak V -module…
(more)
▼ Let V be a unitary vertex operator algebra (VOA) satisfying the following conditions: (1) V is of CFT type. (2) Every N-gradable weak V -module is completely reducible. (3) V is C2-cofinite. Let Rep(V)be the category of unitary V -modules, and let C be a subcategory of Rep(V) whose objects are closed under taking tensor product. Then C is a ribbon fusion category. For any objects Wi; Wj of C, we define a sesquilinear form on the
tensor product Wi bWj. We show that if these sesquilinear forms are positive definite (i.e.,
when they are inner products), then the ribbon category C is unitary. We show that if the unitary V -modules and a generating set of intertwining operators in C satisfy certain energy bounds, then these sesquilinear forms are positive definite. Our result can be applied to the modular tensor categories associated to unitary Virasoro VOAs, and unitary affine VOAs
of type An; Dn; G2, and more.
Advisors/Committee Members: Thomas Weiler (committee member), Jesse Peterson (committee member), Dietmar Bisch (committee member), Akram Aldroubi (committee member), Vaughan Jones (Committee Chair).
Subjects/Keywords: algebraic quantum field theory; tensor category; conformal field theory; Vertex operator algebra; unitary modular tensor category
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APA ·
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APA (6th Edition):
Gui, B. (2018). A unitary tensor product theory for unitary vertex operator algebra modules. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12719
Chicago Manual of Style (16th Edition):
Gui, Bin. “A unitary tensor product theory for unitary vertex operator algebra modules.” 2018. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12719.
MLA Handbook (7th Edition):
Gui, Bin. “A unitary tensor product theory for unitary vertex operator algebra modules.” 2018. Web. 23 Jan 2021.
Vancouver:
Gui B. A unitary tensor product theory for unitary vertex operator algebra modules. [Internet] [Doctoral dissertation]. Vanderbilt University; 2018. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12719.
Council of Science Editors:
Gui B. A unitary tensor product theory for unitary vertex operator algebra modules. [Doctoral Dissertation]. Vanderbilt University; 2018. Available from: http://hdl.handle.net/1803/12719
Vanderbilt University
9.
Li, Shiying.
Adaptive Methods and Collocation by Splines for Solving Differential Equations.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/12932
► Splines have been used to approximate the solutions of differential equations for a while. In the first part of this thesis, adaptive algorithms based on…
(more)
▼ Splines have been used to approximate the solutions of differential equations for a while. In the first part of this thesis, adaptive algorithms based on the finite element method and splines on triangulations with hanging vertices are introduced and tested. In the second part, spline-based collocation methods are investigated: ordinary collocation, a generalized collocation model in 1D and 2D, and least-squares collocation with splines on triangulations. In particular, existence, uniqueness and error bounds of the (generalized) collocation solutions in the cubic case are presented. An error bound for the least-squares collocation on triangulations in approximating the solutions of the Possion equation is also given. Numerical examples are provided in all of the mentioned cases.
Advisors/Committee Members: Akram Aldroubi (committee member), Alexander Powell (committee member), Caglar Oskay (committee member), Larry Schumaker (Committee Chair), Marian Neamtu (Committee Chair).
Subjects/Keywords: Numerical Methods for Differential Equations; Collocation Methods; Splines; Approximation Theory; Adaptive Methods
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APA ·
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APA (6th Edition):
Li, S. (2019). Adaptive Methods and Collocation by Splines for Solving Differential Equations. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12932
Chicago Manual of Style (16th Edition):
Li, Shiying. “Adaptive Methods and Collocation by Splines for Solving Differential Equations.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12932.
MLA Handbook (7th Edition):
Li, Shiying. “Adaptive Methods and Collocation by Splines for Solving Differential Equations.” 2019. Web. 23 Jan 2021.
Vancouver:
Li S. Adaptive Methods and Collocation by Splines for Solving Differential Equations. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12932.
Council of Science Editors:
Li S. Adaptive Methods and Collocation by Splines for Solving Differential Equations. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/12932
Vanderbilt University
10.
Tang, Sui.
Dynamical Sampling.
Degree: PhD, Mathematics, 2016, Vanderbilt University
URL: http://hdl.handle.net/1803/12430
► Let f ∈ l2(I) be a signal at time t = 0 of an evolution process controlled by a bounded linear operator A that produces…
(more)
▼ Let f ∈ l
2(I) be a signal at time t = 0 of an evolution process controlled by a
bounded linear operator A that produces the signals A f , A
2 f , · · · at times t = 1, 2, · · · . Let Y = { f (i), Af (i), · · · , A^(l
i )f (i) : i ∈ Ω ⊂ I} be the spatio-temporal samples taken at various time levels. The problem under consideration is to find necessary and sufficient conditions on A, Ω, l
i in order to recover any f ∈ l
2(I) from the measurements Y . This is the so called Dynamical Sampling Problem in which we seek to recover a signal f by combining coarse samples of f and its futures states A
lf . Various versions of dynamical sampling problems exhibit features that are similar to many fundamental problems: deconvolution, filter banks, super-resolution, compressed sensing etc. In this dissertation, we will study these problems.
Advisors/Committee Members: Ed Saff (committee member), Alex Powell (committee member), Doug Hardin (committee member), Akos Ledeczi (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: Sampling theory; Frame theory; channel estimation
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APA (6th Edition):
Tang, S. (2016). Dynamical Sampling. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12430
Chicago Manual of Style (16th Edition):
Tang, Sui. “Dynamical Sampling.” 2016. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12430.
MLA Handbook (7th Edition):
Tang, Sui. “Dynamical Sampling.” 2016. Web. 23 Jan 2021.
Vancouver:
Tang S. Dynamical Sampling. [Internet] [Doctoral dissertation]. Vanderbilt University; 2016. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12430.
Council of Science Editors:
Tang S. Dynamical Sampling. [Doctoral Dissertation]. Vanderbilt University; 2016. Available from: http://hdl.handle.net/1803/12430
Vanderbilt University
11.
Davis, Jacqueline Teresa Haines.
Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems.
Degree: PhD, Mathematics, 2014, Vanderbilt University
URL: http://hdl.handle.net/1803/13080
► Dynamical sampling is a new type of sampling problem that results from sampling an evolving signal at various times and asks the question: when do…
(more)
▼ Dynamical sampling is a new type of sampling problem that results from sampling an evolving signal at various times and asks the question: when do coarse samplings taken at varying times contain the same information as a finer sampling taken at the earliest time? In other words, under what conditions on an evolving system, can time samples be traded for spatial samples?
In this dissertation, this problem is answered for evolution rules given by convolution and sampling sets that are quasi-uniform. The problem is studied in finite dimensions, infinite dimensions, and shift-invariant spaces.
Advisors/Committee Members: Jerry Spinrad (committee member), Larry Schumaker (committee member), Ed Saff (committee member), Doug Hardin (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: spatio-temporal trade-off; sampling theory
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APA (6th Edition):
Davis, J. T. H. (2014). Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/13080
Chicago Manual of Style (16th Edition):
Davis, Jacqueline Teresa Haines. “Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems.” 2014. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/13080.
MLA Handbook (7th Edition):
Davis, Jacqueline Teresa Haines. “Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems.” 2014. Web. 23 Jan 2021.
Vancouver:
Davis JTH. Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems. [Internet] [Doctoral dissertation]. Vanderbilt University; 2014. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/13080.
Council of Science Editors:
Davis JTH. Spatio-temporal trade-off for quasi-uniform sampling of signals in evolutionary systems. [Doctoral Dissertation]. Vanderbilt University; 2014. Available from: http://hdl.handle.net/1803/13080
Vanderbilt University
12.
Lee, Chang Hsin.
Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints.
Degree: PhD, Mathematics, 2017, Vanderbilt University
URL: http://hdl.handle.net/1803/12587
► A fundamental problem in signal processing called signal reconstruction, or signal recovery, is the determination of a signal from a sequence of samples obtained from…
(more)
▼ A fundamental problem in signal processing called signal reconstruction, or signal recovery, is the determination of a signal from a sequence of samples obtained
from the signal. The sampling process can be viewed as obtaining measurements from
a set of measurement vectors in an N-dimensional space. Studies on the reconstruction problem have resulted
in major breakthroughs in technology in the past century, and practical solutions to the
problem are still essential in the advancement of fields such as image processing and speech
recognition.
Consistent reconstruction and Rangan-Goyal algorithm are two algorithms that produce
estimates of a signal from consistency constraints when the measurements are
corrupted with i.i.d uniformly distributed noises. Under the assumption that the
measurements are taken with i.i.d. unit-norm random vectors, the second error moments
of both algorithms are known to converge with a rate of O(N
2). In this work, we showed
that the general p-th error moments of both algorithms converge with a rate of O(N
p)
under general admissibility conditions on the sampling distribution that no longer require
the measurement vectors to be unit-norm.
Advisors/Committee Members: Mike Neamtu (committee member), Doug Hardin (committee member), Brett Byram (committee member), Akram Aldroubi (committee member), Alexander Powell (Committee Chair).
Subjects/Keywords: digital signal processing; consistent reconstruction; Rangan-Goyal algorithm; signal reconstruction
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APA (6th Edition):
Lee, C. H. (2017). Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12587
Chicago Manual of Style (16th Edition):
Lee, Chang Hsin. “Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints.” 2017. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12587.
MLA Handbook (7th Edition):
Lee, Chang Hsin. “Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints.” 2017. Web. 23 Jan 2021.
Vancouver:
Lee CH. Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints. [Internet] [Doctoral dissertation]. Vanderbilt University; 2017. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12587.
Council of Science Editors:
Lee CH. Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints. [Doctoral Dissertation]. Vanderbilt University; 2017. Available from: http://hdl.handle.net/1803/12587
Vanderbilt University
13.
Bosuwan, Nattapong.
Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations.
Degree: PhD, Mathematics, 2013, Vanderbilt University
URL: http://hdl.handle.net/1803/12774
► We investigate two subjects in asymptotic analysis. The first focuses on a class of rational functions called Padé-orthogonal approximants. We study the relation of the…
(more)
▼ We investigate two subjects in asymptotic analysis.
The first focuses on a class of rational functions called Padé-orthogonal approximants. We study the relation of the convergence of poles of row sequences of Padé-orthogonal approximants and the singularities of the approximated function. We prove both direct and inverse results for these row sequences. Thereby, we obtain analogues of the theorems of R. de Montessus de Ballore and E. Fabry.
The second concerns the so-called maximal and minimal N-point Riesz s-polarization constants and associated configurations.
First, we investigate basic asymptotic properties when N fixed and s varying of these constants and configurations.
Next, we prove a conjecture of T. Erdélyi and E.B. Saff, concerning the dominant term as N → ∞ of the maximal N-point Riesz d-polarization constant of an infinite compact subset A of a d-dimensional C
1-manifold embedded in ℝ
m. Moreover, if we assume further that \mathcal{H}
d(A)>0, we show that the maximal N-point Riesz d-polarization configurations of A distribute asymptotically uniformly on A with respect to \mathcal H
d|
A. These results also hold for finite unions of such sets A provided that their pairwise intersections have \mathcal H
d-measure zero. Finally, we determine the maximal and minimal N-point Riesz s-polarization configurations of the unit sphere 𝕊
m in ℝ
m+1 for certain values s.
Advisors/Committee Members: Douglas P. Hardin (committee member), Akram Aldroubi (committee member), Marcus H. Mendenhall (committee member), Edward B. Saff (Committee Chair).
Subjects/Keywords: rational approximation; Pade approximants; Fabrys Theorem; Riesz polarization; Chebyshev constants Montessus de Ballore s Theorem; Riesz energy; rational approximation; Pade approximants; Riesz energy; Chebyshev constants Montessus de Ballore s Theorem; Riesz polarization; Fabrys Theorem
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APA (6th Edition):
Bosuwan, N. (2013). Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12774
Chicago Manual of Style (16th Edition):
Bosuwan, Nattapong. “Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations.” 2013. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12774.
MLA Handbook (7th Edition):
Bosuwan, Nattapong. “Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations.” 2013. Web. 23 Jan 2021.
Vancouver:
Bosuwan N. Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations. [Internet] [Doctoral dissertation]. Vanderbilt University; 2013. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12774.
Council of Science Editors:
Bosuwan N. Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations. [Doctoral Dissertation]. Vanderbilt University; 2013. Available from: http://hdl.handle.net/1803/12774
Vanderbilt University
14.
Wang, Lujun.
Trivariate polynomial splines on 3D T-meshes.
Degree: PhD, Mathematics, 2012, Vanderbilt University
URL: http://hdl.handle.net/1803/12315
► Trivariate polynomial spline spaces defined on three-dimensional (3D) T-meshes are useful tools for the finite element method. In addition to dimension formulae, explicit basis functions…
(more)
▼ Trivariate polynomial spline spaces defined on three-dimensional (3D) T-meshes are useful tools for the finite element method. In addition to dimension formulae, explicit basis functions are constructed. The approach uses Bernstein- Bezier methods to get precise conditions on the geometry of the meshes which lead to local and stable bases. Hermite interpolation using polynomial splines on 3D T-meshes is also discussed in detail, leading to an error bound for interpolation of smooth functions.
Advisors/Committee Members: Marian Neamtu (committee member), Douglas P. Hardin (committee member), Robert E. Bodenheimer (committee member), Akram Aldroubi (committee member), Larry L. Schumaker (Committee Chair).
Subjects/Keywords: 3D T-meshes; Hermite interpolation; splines; hanging vertices
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MLA ·
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APA (6th Edition):
Wang, L. (2012). Trivariate polynomial splines on 3D T-meshes. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12315
Chicago Manual of Style (16th Edition):
Wang, Lujun. “Trivariate polynomial splines on 3D T-meshes.” 2012. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12315.
MLA Handbook (7th Edition):
Wang, Lujun. “Trivariate polynomial splines on 3D T-meshes.” 2012. Web. 23 Jan 2021.
Vancouver:
Wang L. Trivariate polynomial splines on 3D T-meshes. [Internet] [Doctoral dissertation]. Vanderbilt University; 2012. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12315.
Council of Science Editors:
Wang L. Trivariate polynomial splines on 3D T-meshes. [Doctoral Dissertation]. Vanderbilt University; 2012. Available from: http://hdl.handle.net/1803/12315
Vanderbilt University
15.
Petrosyan, Armenak.
Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.
Degree: PhD, Mathematics, 2017, Vanderbilt University
URL: http://hdl.handle.net/1803/12331
► The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset W of its domain. However, taking…
(more)
▼ The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset W of its domain. However, taking samples on an appropriate sampling set W is not always practical or even possible - for example, associated measuring devices may be too expensive or scarce.
In the dynamical sampling problem, it is assumed that the sparseness of the sampling locations can be compensated by involving dynamics. For example, when f is the initial state of a physical process (say, change of temperature or air pollution), we can sample its values at the same sampling locations as time progresses, and try to recover f from the combination of these spatio-temporal samples.
In our recent work, we have taken a more operator-theoretic approach to the dynamical sampling problem. We assume that the unknown function f is a vector in some Hilbert space H and A is a bounded linear operator on H. The samples are given in the form (A
nf,g) for 0 £ n<L(g) and g Î G,
where G is a countable (finite or infinite) set of vectors in H, and the function L:G® {1,2,...,¥} represents the "sampling level." Then the main problem becomes to recover the unknown vector f Î H from the above measurements.
The dynamical sampling problem has potential applications in plenacoustic sampling, on-chip sensing, data center temperature sensing, neuron-imaging, and satellite remote sensing, and more generally to Wireless Sensor Networks (WSN). It also has connections and applications to other areas of mathematics including C
*-algebras, spectral theory of normal operators, and frame theory.
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Advisors/Committee Members: David Smith (committee member), Gieri Simonett (committee member), Alexander Powell (committee member), Douglas Hardin (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: Feichtinger conjecture; Reconstruction; Shift-Invariant Spaces; Sub-Sampling; Frames; Dynamical Sampling; Sampling Theory
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Export
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APA (6th Edition):
Petrosyan, A. (2017). Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12331
Chicago Manual of Style (16th Edition):
Petrosyan, Armenak. “Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.” 2017. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12331.
MLA Handbook (7th Edition):
Petrosyan, Armenak. “Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.” 2017. Web. 23 Jan 2021.
Vancouver:
Petrosyan A. Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. [Internet] [Doctoral dissertation]. Vanderbilt University; 2017. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12331.
Council of Science Editors:
Petrosyan A. Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. [Doctoral Dissertation]. Vanderbilt University; 2017. Available from: http://hdl.handle.net/1803/12331
Vanderbilt University
16.
Solava, Ryan William.
On the fine structure of graphs avoiding certain complete bipartite minors.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/13941
► Avoiding complete bipartite graphs as minors, and in particular K2,t as a minor, has been used to give sufficient conditions for Hamiltonicity. For this reason…
(more)
▼ Avoiding complete bipartite graphs as minors, and in particular K
2,t as a minor, has been used to give sufficient conditions for Hamiltonicity. For this reason and others, the classes of K
2,t-minor-free graphs are of interest. Ding gave a rough description of the K
2,t-minor-free graphs that gives necessary conditions for a graph to be K
2,t-minor-free. We refine this description and characterize the K
2,t-minor-free graphs in the 3- and 4-connected cases, giving necessary and sufficient conditions. We also give a program for characterizing the 3-connected K
2,5-minor-free graphs. As part of this program, we prove a result on fan expansions that is of independent interest.
Advisors/Committee Members: Akram Aldroubi (committee member), Paul Edelman (committee member), Michael Mihalik (committee member), Jerry Spinrad (committee member), Mark Ellingham (Committee Chair).
Subjects/Keywords: Graph theory; Graph minors
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APA ·
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MLA ·
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Export
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APA (6th Edition):
Solava, R. W. (2019). On the fine structure of graphs avoiding certain complete bipartite minors. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/13941
Chicago Manual of Style (16th Edition):
Solava, Ryan William. “On the fine structure of graphs avoiding certain complete bipartite minors.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/13941.
MLA Handbook (7th Edition):
Solava, Ryan William. “On the fine structure of graphs avoiding certain complete bipartite minors.” 2019. Web. 23 Jan 2021.
Vancouver:
Solava RW. On the fine structure of graphs avoiding certain complete bipartite minors. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/13941.
Council of Science Editors:
Solava RW. On the fine structure of graphs avoiding certain complete bipartite minors. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/13941
Vanderbilt University
17.
Spaeth, Anneliese Heidi.
A Determination of the Existence of Various Types of Positive Systems in L^p.
Degree: PhD, Mathematics, 2013, Vanderbilt University
URL: http://hdl.handle.net/1803/12950
► We consider various types of generalized bases in spaces of the type Lp(T), where T=[0,1]. More specifically, we determine whether there exists a system {fn}n,…
(more)
▼ We consider various types of generalized bases in spaces of the type L
p(T), where T=[0,1]. More specifically, we determine whether there exists a system {f
n}
n, of the type under consideration, with the property f
n(t)>=0 almost everywhere, for each n in the natural numbers. We refer to a system with the property of almost everywhere non-negativity, as a positive system.
In the spaces with 1<= p < infinity, we determine that there do not exist positive unconditional Schauder bases, and positive unconditional quasibases. In the aforementioned spaces, we determine that there do exist positive conditional quasibases, positive conditional pseudobases, and positive exact systems. In the spaces with 1< p < infinity, we determine that there do not exist positive monotone bases, and that there do exist positive exact systems with exact dual systems. In L
2(T), we demonstrate that there do not exist positive frames, positive orthonormal bases, and positive Riesz bases. Finally, in the spaces with 0< p <= infinity, we show that there do exist positive Hamel bases. Secondary considerations explore product systems on the spaces L
p(T
2).
Advisors/Committee Members: Akram Aldroubi (committee member), Doug Hardin (committee member), Alan Peters (committee member), Glenn Webb (committee member), Alexander Powell (Committee Chair).
Subjects/Keywords: quasibasis; Schauder basis; pseudobasis; Walsh basis
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MLA ·
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APA (6th Edition):
Spaeth, A. H. (2013). A Determination of the Existence of Various Types of Positive Systems in L^p. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12950
Chicago Manual of Style (16th Edition):
Spaeth, Anneliese Heidi. “A Determination of the Existence of Various Types of Positive Systems in L^p.” 2013. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12950.
MLA Handbook (7th Edition):
Spaeth, Anneliese Heidi. “A Determination of the Existence of Various Types of Positive Systems in L^p.” 2013. Web. 23 Jan 2021.
Vancouver:
Spaeth AH. A Determination of the Existence of Various Types of Positive Systems in L^p. [Internet] [Doctoral dissertation]. Vanderbilt University; 2013. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12950.
Council of Science Editors:
Spaeth AH. A Determination of the Existence of Various Types of Positive Systems in L^p. [Doctoral Dissertation]. Vanderbilt University; 2013. Available from: http://hdl.handle.net/1803/12950
Vanderbilt University
18.
Michaels, Timothy Joseph.
Node Generation on Surfaces and Bounds on Minimal Riesz Energy.
Degree: PhD, Mathematics, 2017, Vanderbilt University
URL: http://hdl.handle.net/1803/14461
► Discretizing a manifold is a far reaching subject throughout the mathematical and physical sciences. This thesis has two principal foci. We present and analyze a…
(more)
▼ Discretizing a manifold is a far reaching subject throughout the mathematical and physical sciences. This thesis has two principal foci. We present and analyze a variety of algorithms for generating point configurations on d-dimensional sphere and the torus, as well introduce a generic strategy for generating locally quasi-uniform points of variable density on any full dimensional subset of Euclidean space. The methods and algorithms are concentrated on construction and computation, though we also prove some properties of distribution and mesh ratio. For the variable density nodes, we consider the particular application to atmospheric modeling with radial basis functions. We implement a parallelizable algorithm to initialize good starting configurations for efficient modeling.
Secondly, we prove a lower bound on the asymptotic Riesz minimal energy in the hypersingular case based off of the linear programming method. This general framework for obtaining lower bounds for minimal energy configurations on the d-dimensional sphere was developed by Yudin and based on the Delsart-Goethals-Seidel bounds on spherical designs. Combining these methods with Levenshtein's work on maximal spherical codes, explicit universal lower bounds are established depending only on the potential function for any monotone potential. We extend this to the asymptotic case as N approaches infinity. In addition, we apply this method to infinite Gaussian potentials on Euclidean space.
Advisors/Committee Members: Alex Powell (committee member), Akram Aldroubi (committee member), Eric Barth (committee member), Ed Saff (Committee Chair), Doug Hardin (Committee Chair).
Subjects/Keywords: spherical configurations; atmospheric modeling; minimal energy
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APA (6th Edition):
Michaels, T. J. (2017). Node Generation on Surfaces and Bounds on Minimal Riesz Energy. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/14461
Chicago Manual of Style (16th Edition):
Michaels, Timothy Joseph. “Node Generation on Surfaces and Bounds on Minimal Riesz Energy.” 2017. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/14461.
MLA Handbook (7th Edition):
Michaels, Timothy Joseph. “Node Generation on Surfaces and Bounds on Minimal Riesz Energy.” 2017. Web. 23 Jan 2021.
Vancouver:
Michaels TJ. Node Generation on Surfaces and Bounds on Minimal Riesz Energy. [Internet] [Doctoral dissertation]. Vanderbilt University; 2017. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/14461.
Council of Science Editors:
Michaels TJ. Node Generation on Surfaces and Bounds on Minimal Riesz Energy. [Doctoral Dissertation]. Vanderbilt University; 2017. Available from: http://hdl.handle.net/1803/14461
Vanderbilt University
19.
Su, Yujian.
Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2.
Degree: PhD, Mathematics, 2015, Vanderbilt University
URL: http://hdl.handle.net/1803/14500
► Let Λ be a lattice in Rd with positive co-volume. Among Λ-periodic N-point configurations, we consider the minimal renormalized Riesz s-energy mathcal{E}s,Lambda(N). While the dominant…
(more)
▼ Let Λ be a lattice in R
d with positive co-volume. Among Λ-periodic N-point configurations, we consider the minimal renormalized Riesz s-energy mathcal{E}
s,Lambda(N). While the dominant term in the asymptotic expansion of mathcal{E}
s,Lambda(N) as N goes to infinity in the long range case that 0<s<d (or s=log) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for s>0 they are of the form
C
s,d|Lambda|
-s/dN
1+s/d and -frac{2}{d}Nlog N+left(C
log,d-2zeta'_Λ(0)
ight)N where we show that the constant C
s,d is independent of the lattice Λ.
We also solve the 4-point maximal polarization problem on S
2. We prove that the vertices of a regular tetrahedron on S
2 maximize the minimum of discrete potentials on S
2 whenever the potential is of the form
sumlimits
k=14f(|x-x
k|
2), where f:[0,4]
ightarrow[0,infty] is non-increasing and strictly convex with f(0)=limlimits
x o 0+f(x).
Advisors/Committee Members: Kirill Bolotin (committee member), Akram Aldroubi (committee member), Edward Saff (committee member), Alexander Powell (committee member), Douglas Hardin (Committee Chair).
Subjects/Keywords: polarization; max-min problems; Epstein Hurwitz Zeta function; Ewald summation; periodic energy
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APA ·
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MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Su, Y. (2015). Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/14500
Chicago Manual of Style (16th Edition):
Su, Yujian. “Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2.” 2015. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/14500.
MLA Handbook (7th Edition):
Su, Yujian. “Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2.” 2015. Web. 23 Jan 2021.
Vancouver:
Su Y. Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2. [Internet] [Doctoral dissertation]. Vanderbilt University; 2015. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/14500.
Council of Science Editors:
Su Y. Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2. [Doctoral Dissertation]. Vanderbilt University; 2015. Available from: http://hdl.handle.net/1803/14500
Vanderbilt University
20.
Khorasgani, Hamed Ghazavi.
Model- and data-driven approaches to fault detection and isolation in complex systems.
Degree: PhD, Electrical Engineering, 2018, Vanderbilt University
URL: http://hdl.handle.net/1803/10449
► Complex engineering systems pervade every aspect of our daily lives. The need for increased performance, safety, and reliability of these systems provide the motivation for…
(more)
▼ Complex engineering systems pervade every aspect of our daily lives. The need for increased performance, safety, and reliability of these systems provide the motivation for developing system health monitoring methodologies for these systems. It is important to develop diagnosis for nonlinear systems that are robust to model uncertainties and noisy measurements. For greater applicability, it becomes essential to extend these diagnosis methods to cover hybrid behaviors, and to apply in a distributed manner to complex systems. On the other hand, the lack of sufficiently rich and complete models for these complex systems, but the availability of data collected from system operations may indicate the need to move toward data-driven diagnosis approaches. The contributions of this thesis research are primarily categorized into: 1) model-based, and 2) data-driven diagnosis.
For robust model-based diagnosis, we have developed sensitivity analysis methods to quantify the performance of residuals generated for fault diagnosis in the presence of noise and uncertainty. We combine the robustness analysis with efficient residual selection algorithms to find a residual set that meets pre-specified diagnostic criteria. A second contribution of this thesis in the model-based diagnosis domain is developing two general approaches for distributed fault detection and isolation. The first method guarantees minimum communication of measurement values among subsystems. The second algorithm is computationally more efficient and it does not use the global system model for designing the local (distributed) diagnosers. However, it does not guarantee the minimum communication of measurement values. Our next contribution is developing model based methods that combine mode detection and diagnosis of faults in hybrid systems. Our approach does not require pre-enumeration of all possible modes and, therefore, it is feasible for hybrid systems with large number of switching elements.
Finally, towards our contributions to data-driven diagnosis, we combine the use of unsupervised learning techniques with expert knowledge to develop an anomaly detection method to discover previously unknown faults in a complex system. Our clustering algorithm helps determine regions of nominal behavior, and by extension outliers and anomalous groups that are well-separated from the nominal clusters. We derive significant features associate for each outlier group, and then consult experts to identify and characterize these anomalies as faults in system behavior.
Advisors/Committee Members: Gabor Karsai (committee member), Xenofon Koutsoukos (committee member), Akram Aldroubi (committee member), Don Wilkes (committee member), Gautam Biswas (Committee Chair).
Subjects/Keywords: Fault diagnosis; Fault detection and isolation; Unsupervised learning; Data-driven monitoring; Complex systems; Anomaly detection
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Manager
APA (6th Edition):
Khorasgani, H. G. (2018). Model- and data-driven approaches to fault detection and isolation in complex systems. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/10449
Chicago Manual of Style (16th Edition):
Khorasgani, Hamed Ghazavi. “Model- and data-driven approaches to fault detection and isolation in complex systems.” 2018. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/10449.
MLA Handbook (7th Edition):
Khorasgani, Hamed Ghazavi. “Model- and data-driven approaches to fault detection and isolation in complex systems.” 2018. Web. 23 Jan 2021.
Vancouver:
Khorasgani HG. Model- and data-driven approaches to fault detection and isolation in complex systems. [Internet] [Doctoral dissertation]. Vanderbilt University; 2018. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/10449.
Council of Science Editors:
Khorasgani HG. Model- and data-driven approaches to fault detection and isolation in complex systems. [Doctoral Dissertation]. Vanderbilt University; 2018. Available from: http://hdl.handle.net/1803/10449
Vanderbilt University
21.
Northington, Michael Carr V.
Balian-Low Type Theorems for Shift-Invariant Spaces.
Degree: PhD, Mathematics, 2016, Vanderbilt University
URL: http://hdl.handle.net/1803/11348
► Shift-invariant spaces and Gabor systems are frequently used in approximation theory and signal processing. In these settings, it is advantageous for the generators of such…
(more)
▼ Shift-invariant spaces and Gabor systems are frequently used in approximation theory and signal processing. In these settings, it is advantageous for the generators of such spaces to be localized and for the spaces to be representative of a large class of functions. For Gabor systems, the celebrated Balian-Low Theorem shows that if the integer translations and modulations of a function in L
2(R) form an orthonormal basis for L
2(R), then either the function or its fourier transform must be poorly localized. The present work shows that similar results hold in certain shift-invariant spaces. In particular, if the integer translates of a well-localized function in L
2(R
d) form a frame for the shift-invariant space, V, generated by the function then V cannot be invariant under any non-integer shift. Similar results are proven under a variety of different basis properties, for finitely many generating functions, and with the extra-invariance property replaced with redundancy in the translates of the generator. Examples are given showing the sharpness of these results.
Advisors/Committee Members: Doug Hardin (committee member), Akram Aldroubi (committee member), Gieri Simonett (committee member), Yevgeniy Vorobeychik (committee member), Alexander Powell (Committee Chair).
Subjects/Keywords: sobolev spaces; time-frequency analysis; shift-invariant spaces
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APA (6th Edition):
Northington, M. C. V. (2016). Balian-Low Type Theorems for Shift-Invariant Spaces. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11348
Chicago Manual of Style (16th Edition):
Northington, Michael Carr V. “Balian-Low Type Theorems for Shift-Invariant Spaces.” 2016. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11348.
MLA Handbook (7th Edition):
Northington, Michael Carr V. “Balian-Low Type Theorems for Shift-Invariant Spaces.” 2016. Web. 23 Jan 2021.
Vancouver:
Northington MCV. Balian-Low Type Theorems for Shift-Invariant Spaces. [Internet] [Doctoral dissertation]. Vanderbilt University; 2016. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11348.
Council of Science Editors:
Northington MCV. Balian-Low Type Theorems for Shift-Invariant Spaces. [Doctoral Dissertation]. Vanderbilt University; 2016. Available from: http://hdl.handle.net/1803/11348
Vanderbilt University
22.
Villalobos Guillén, Cristóbal.
A Measure Theoretic Approach for the Recovery of Remanent Magnetizations.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/11475
► This work is motivated by the problem of recovering the magnetization M of a rock sample from a given set of measurements for the magnetic…
(more)
▼ This work is motivated by the problem of recovering the magnetization M of a rock sample from
a given set of measurements for the magnetic field it generates. Modeling the magnetization by an
R 3 -valued measure, we focus on the study of inverse problems for the Poisson equation with source
term the divergence of M; that is,
∆Φ = divM,
where Φ denotes the Magnetic Scalar Potential whose gradient is assumed to be known on a set
disjoint from the support of the measure M. We develop methods for recovering M based on total
variation regularization of measures. We provide sufficient conditions for the unique recovery of a
magnetization in cases where it is uni-directional or when the magnetization has a support which
is sparse in the sense that it is purely 1-unrectifiable.
In the last chapter we work on the ideal case where the magnetized sample is contained in a
subset of the horizontal plane. For this case we show that all magnetizations which do not generate
a magnetic field can be decomposed as a superposition of loops. The findings presented in this
chapter rely on the theory of functions of Bounded Variation and sets of finite perimeter and give
a characterization for magnetizations that do not generate a magnetic field.
Numerical examples are provided to illustrate the main theoretical results.
Advisors/Committee Members: Marcelo M. Disconzi (committee member), Guilherme A.R. Gualda (committee member), Edward B. Saff (committee member), Akram Aldroubi (committee member), Douglas P. Hardin (Committee Chair).
Subjects/Keywords: geometric measure theory; BV fucntions; Inverse problems; Inverse problems in electromagnetism
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MLA ·
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CSE |
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APA (6th Edition):
Villalobos Guillén, C. (2019). A Measure Theoretic Approach for the Recovery of Remanent Magnetizations. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11475
Chicago Manual of Style (16th Edition):
Villalobos Guillén, Cristóbal. “A Measure Theoretic Approach for the Recovery of Remanent Magnetizations.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11475.
MLA Handbook (7th Edition):
Villalobos Guillén, Cristóbal. “A Measure Theoretic Approach for the Recovery of Remanent Magnetizations.” 2019. Web. 23 Jan 2021.
Vancouver:
Villalobos Guillén C. A Measure Theoretic Approach for the Recovery of Remanent Magnetizations. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11475.
Council of Science Editors:
Villalobos Guillén C. A Measure Theoretic Approach for the Recovery of Remanent Magnetizations. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/11475
Vanderbilt University
23.
Huang, Longxiu.
Dynamical Sampling and its Applications.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/10916
► Dynamical sampling is a new area in sampling theory that deals with signals that evolve over time under the action of a linear operator. There…
(more)
▼ Dynamical sampling is a new area in sampling theory that deals with signals that evolve over time under the action of a linear operator. There are lots of studies on various aspects of the dynamical sampling problem. However, they all focus on uniform discrete time-sets mathcal{T} subset ℕ. In our first paper, we concentrate on the case mathcal{T} = [0, L]. The goal of the present work is to study the frame property of the systems {A
tg : g in𝓖, t in [0, L]}. To this end, we also characterize the completeness and Besselness properties of these systems. In our second paper, we consider dynamical sampling when the samples are corrupted by additive noises. The purpose of the second paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and study the impact of additive noise. The algorithms are implemented and tested on synthetic and real data sets, and denoising techniques are integrated to mitigate the effect of the noise. We also develop theoretical and numerical results that validate the algorithm for recovering the driving operators, which are defined via a real symmetric convolution.
Advisors/Committee Members: Alexander M. Powell (committee member), Larry L. Schumaker (committee member), David S. Smith (committee member), Douglas P. Hardin (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: Cadzow denoising algorithm; numerical linear algebra; continuous frames; dynamical sampling
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CSE |
Export
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Manager
APA (6th Edition):
Huang, L. (2019). Dynamical Sampling and its Applications. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/10916
Chicago Manual of Style (16th Edition):
Huang, Longxiu. “Dynamical Sampling and its Applications.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/10916.
MLA Handbook (7th Edition):
Huang, Longxiu. “Dynamical Sampling and its Applications.” 2019. Web. 23 Jan 2021.
Vancouver:
Huang L. Dynamical Sampling and its Applications. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/10916.
Council of Science Editors:
Huang L. Dynamical Sampling and its Applications. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/10916
Vanderbilt University
24.
Feigenbaum, Ahram Samuel.
Applications of Modular Forms to Geometry and Interpolation Problems.
Degree: PhD, Mathematics, 2019, Vanderbilt University
URL: http://hdl.handle.net/1803/14333
► The sphere packing problem asks for the densest collection of non-overlapping con- gruent spheres in Rn. In 2016, Viazovska proved that the E8 lattice is…
(more)
▼ The sphere packing problem asks for the densest collection of non-overlapping con-
gruent spheres in Rn. In 2016, Viazovska proved that the E8 lattice is optimal for n = 8.
Subsequently, she with Cohn, Kumar, Miller, and Radchenko that showed the Leech lat-
tice was optimal for n = 24. Their proofs relied on the theory of weakly holomorphic
and quasi-modular forms to construct Fourier eigenfunctions with prescribed zeros at
distances in the E8 and Leech lattices. Similar ideas were applied by Radchenko and
Viazovska to obtain interpolation formulas for real Schwartz functions and by Cohn and
Gon¸calves to study uncertainty principles in harmonic analysis. In this thesis, we de-
velop a unified approach to the construction of such functions. We show that the weakly
holomorphic and weakly quasi-modular forms behind them are uniquely defined by the
conditions that they be eigenfunctions of the Fourier transform belonging to the Schwartz
class. We construct the Fourier eigenfunctions for all n divisible by 4. We also show an
extension of the interpolation formula given by Radchenko and Viazovska in R to radial
functions in R2 and R
Advisors/Committee Members: D. Mitch Wilkes (committee member), Ed Saff (committee member), Akram Aldroubi (committee member), Alex Powell (committee member), Larry Rolen (committee member), Doug Hardin (Committee Chair).
Subjects/Keywords: Modular Forms; Fourier Transform
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APA (6th Edition):
Feigenbaum, A. S. (2019). Applications of Modular Forms to Geometry and Interpolation Problems. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/14333
Chicago Manual of Style (16th Edition):
Feigenbaum, Ahram Samuel. “Applications of Modular Forms to Geometry and Interpolation Problems.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/14333.
MLA Handbook (7th Edition):
Feigenbaum, Ahram Samuel. “Applications of Modular Forms to Geometry and Interpolation Problems.” 2019. Web. 23 Jan 2021.
Vancouver:
Feigenbaum AS. Applications of Modular Forms to Geometry and Interpolation Problems. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/14333.
Council of Science Editors:
Feigenbaum AS. Applications of Modular Forms to Geometry and Interpolation Problems. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://hdl.handle.net/1803/14333
Vanderbilt University
25.
Yattselev, Maxim L.
Non-Hermitian Orthogonality and Meromorphic Approximation.
Degree: PhD, Mathematics, 2007, Vanderbilt University
URL: http://hdl.handle.net/1803/13602
► In this thesis we consider the distribution of poles of meromorphic and multipoint Padé approximants to the sum of a rational function and a Cauchy…
(more)
▼ In this thesis we consider the distribution of poles of meromorphic and multipoint Padé approximants to the sum of a rational function and a Cauchy transform of a complex measure.
Advisors/Committee Members: Yanqin Fan (committee member), Doug Hardin (committee member), Akram Aldroubi (committee member), Edward B. Saff (Committee Chair).
Subjects/Keywords: meromorphic; rational; Padé approximation; orthogonal polynomials
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Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
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APA (6th Edition):
Yattselev, M. L. (2007). Non-Hermitian Orthogonality and Meromorphic Approximation. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/13602
Chicago Manual of Style (16th Edition):
Yattselev, Maxim L. “Non-Hermitian Orthogonality and Meromorphic Approximation.” 2007. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/13602.
MLA Handbook (7th Edition):
Yattselev, Maxim L. “Non-Hermitian Orthogonality and Meromorphic Approximation.” 2007. Web. 23 Jan 2021.
Vancouver:
Yattselev ML. Non-Hermitian Orthogonality and Meromorphic Approximation. [Internet] [Doctoral dissertation]. Vanderbilt University; 2007. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/13602.
Council of Science Editors:
Yattselev ML. Non-Hermitian Orthogonality and Meromorphic Approximation. [Doctoral Dissertation]. Vanderbilt University; 2007. Available from: http://hdl.handle.net/1803/13602
Vanderbilt University
26.
Calef, Matthew Thomas.
Theoretical and Computational Investigations of Minimal Energy Problems.
Degree: PhD, Mathematics, 2009, Vanderbilt University
URL: http://hdl.handle.net/1803/13017
► Let A be a d-dimensional compact subset of p-dimensional Euclidean space. For s in (0,d) the Riesz s-equilibrium measure is the unique Borel probability measure…
(more)
▼ Let A be a d-dimensional compact subset of p-dimensional Euclidean space. For s in (0,d) the Riesz s-equilibrium measure is the unique Borel probability measure that minimizes the Riesz s-energy over the set of all Borel probability measures supported on A. In this paper we show that if A is a strictly self-similar d-fractal or a strongly Hausdorff d-rectifiable set, then the s-equilibrium measures converge in the weak-star topology to normalized Hausdorff measure restricted to A as s approaches d from below.
Additionally we describe numerical experiments on the 2-sphere involving discrete energies mediated by the Riesz s-kernel. These experiments provide approximate discrete minimal energies for N=20,...,200 and s=0,...,3 where, in the case s=0, the Riesz kernel becomes the logarithmic kernel. These experiments corroborate several conjectures regarding the asymptotic expansion as N goes go infinity of the minimal N-point energies. Further, the number of stable configurations observed as a function of N and s is reported. Finally, two algorithms used in this experiment are presented. The first minimizes the effect of roundoff error when computing sums of many terms, the second uses graph theory to speed the identification of isometries between collections of on the 2-sphere.
Advisors/Committee Members: Ed Saff (committee member), Akram Aldroubi (committee member), Mark Ellingham (committee member), Marcus Mendenhall (committee member), Douglas Hardin (Committee Chair).
Subjects/Keywords: potential theory; normalized d-energy; density
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APA (6th Edition):
Calef, M. T. (2009). Theoretical and Computational Investigations of Minimal Energy Problems. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/13017
Chicago Manual of Style (16th Edition):
Calef, Matthew Thomas. “Theoretical and Computational Investigations of Minimal Energy Problems.” 2009. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/13017.
MLA Handbook (7th Edition):
Calef, Matthew Thomas. “Theoretical and Computational Investigations of Minimal Energy Problems.” 2009. Web. 23 Jan 2021.
Vancouver:
Calef MT. Theoretical and Computational Investigations of Minimal Energy Problems. [Internet] [Doctoral dissertation]. Vanderbilt University; 2009. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/13017.
Council of Science Editors:
Calef MT. Theoretical and Computational Investigations of Minimal Energy Problems. [Doctoral Dissertation]. Vanderbilt University; 2009. Available from: http://hdl.handle.net/1803/13017
Vanderbilt University
27.
Borodachov, Sergiy Volodymyrovych.
Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets.
Degree: PhD, Mathematics, 2006, Vanderbilt University
URL: http://hdl.handle.net/1803/12648
► This work studies the behavior of the minimal discrete Riesz s-energy and best-packing distance on rectifiable sets as the cardinality N of point configurations gets…
(more)
▼ This work studies the behavior of the minimal discrete Riesz s-energy and best-packing distance on rectifiable sets as the cardinality N of point configurations gets large.
We extend known asymptotic results for the minimal s-energy on d-dimensional rectifiable manifolds (s>d, d is an integer) to the case of d-dimensional countably rectifiable sets whose d-dimensional Minkowski content equals the d-dimensional Hausdorff measure, and show that these results fail for countably rectifiable sets not satisfying this condition and s sufficiently large.
We also show that the asymptotic behavior of the best-packing distance on a d-dimensional countably rectifiable set A is the same as on a d-dimensional cube of the same d-dimensional Hausdorff measure if and only if this measure and d-dimensional Minkowski content of A are equal.
Uniformity of the asymptotic distribution of optimal points is proved on d-rectifiable sets both for the minimum s-energy problem (s>d) and best-packing.
For smooth Jordan curves we get the next order term of the minimal Riesz s-energy for s>1.
Known lower estimate of the minimal pairwise distance between minimum s-energy points on d-dimensional rectifiable manifolds (s>d) is extended to arbitrary compact sets of positive d-dimensional Hausdorff measure.
We also consider the problem of minimizing energy of particles interacting via the Riesz potential multiplied by a weight depending on positions of both points. For s>d, closed d-rectifiable sets and a bounded weight which is continuous and positive on the diagonal, we obtain asymptotic behavior of the minimal energy and limit distribution of optimal configurations. We also prove separation estimates for the minimal weighted energy problem.
Advisors/Committee Members: Akram Aldroubi (committee member), Doug Hardin (committee member), Michael Mihalik (committee member), Sokrates Pantelides (committee member), Edward Saff (Committee Chair).
Subjects/Keywords: best-packing; minimum energy; Riesz potential; rectifiable set
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Export
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APA (6th Edition):
Borodachov, S. V. (2006). Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/12648
Chicago Manual of Style (16th Edition):
Borodachov, Sergiy Volodymyrovych. “Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets.” 2006. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/12648.
MLA Handbook (7th Edition):
Borodachov, Sergiy Volodymyrovych. “Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets.” 2006. Web. 23 Jan 2021.
Vancouver:
Borodachov SV. Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets. [Internet] [Doctoral dissertation]. Vanderbilt University; 2006. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/12648.
Council of Science Editors:
Borodachov SV. Asymptotic Results for the minimum energy
and Best-Packing
Problems on Rectifiable Sets. [Doctoral Dissertation]. Vanderbilt University; 2006. Available from: http://hdl.handle.net/1803/12648
Vanderbilt University
28.
Acosta Reyes, Ernesto.
Non-linear optimal signal models and stability of sampling-reconstruction.
Degree: PhD, Mathematics, 2009, Vanderbilt University
URL: http://hdl.handle.net/1803/11687
► This dissertation has two main goals: To study the stability of sampling-reconstruction models, and to study the existence of optimal non-linear signal models. In the…
(more)
▼ This dissertation has two main goals: To study the stability of sampling-reconstruction
models, and to study the existence of optimal non-linear signal models. In the ¯rst part,
we describe and quantify admissible perturbation of the sampling set X(Jitter), or the
measuring devices (the way the sampling is performed, that is, we average a signal by
¯nite Borel measures), or the generator of the shift-invariant space (class to which belongs
the signal to be sampled and reconstructed from its samples on X). We also study the
reconstruction of a signal belonging to a shift-invariant space from its samples using an
iterative algorithm, and we show that the sequence de¯ned by the algorithm converges to
the original signal geometrically fast. Furthermore, we show that the reconstruction error
due to perturbations we describe is controlled continuously by the perturbation errors.
In the second part, we guarantee the existence of a signal model M =
S
i2I Ci from
observed data F = ff1; : : : ; fmg ½ H that minimize the quantity e(F; fC1; : : : ;Clg) =
Pm
j=1 min1·i·l d2(fj ;Ci), where H is a separable Hilbert space (Problem 1). Su±cient
conditions are given over the class C = fCigi2I in terms of the weak operator topology
in order to guarantee the existence of a minimizer to Problem 1 for any set of data F.
Moreover, we consider the problem when the class C is de¯ned in terms of a collection
of unitary operators applied to a convex subset of H, and we obtain an algorithm for
constructing collections of closed subspaces for which we a priori know that Problem 1 can
be solved. As a consequence, we obtain the well-known qualitative version of the Eckart-
Young Theorem.
Advisors/Committee Members: Professor Gieri Simonett (committee member), Professor Nilanjan Sarkar (committee member), Professor Douglas Hardin (committee member), Professor Larry Schumaker (committee member), Professor Akram Aldroubi (Committee Chair).
Subjects/Keywords: optimal non-linear signal models; Stability of sampling-reconstruction
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APA (6th Edition):
Acosta Reyes, E. (2009). Non-linear optimal signal models and stability of sampling-reconstruction. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11687
Chicago Manual of Style (16th Edition):
Acosta Reyes, Ernesto. “Non-linear optimal signal models and stability of sampling-reconstruction.” 2009. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11687.
MLA Handbook (7th Edition):
Acosta Reyes, Ernesto. “Non-linear optimal signal models and stability of sampling-reconstruction.” 2009. Web. 23 Jan 2021.
Vancouver:
Acosta Reyes E. Non-linear optimal signal models and stability of sampling-reconstruction. [Internet] [Doctoral dissertation]. Vanderbilt University; 2009. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11687.
Council of Science Editors:
Acosta Reyes E. Non-linear optimal signal models and stability of sampling-reconstruction. [Doctoral Dissertation]. Vanderbilt University; 2009. Available from: http://hdl.handle.net/1803/11687
Vanderbilt University
29.
Leonetti, Casey Clark.
Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces.
Degree: PhD, Mathematics, 2007, Vanderbilt University
URL: http://hdl.handle.net/1803/11682
► In the following chapters we provide error estimates for signals reconstructed from corrupt data. Two different types of error are considered. First, we address the…
(more)
▼ In the following chapters we provide error estimates for signals reconstructed from corrupt data. Two different types of error are considered. First, we address the problem of reconstructing a continuous function defined on <b>R</b>
d from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean 0 and variance σ
2. We sample the continuous function <i>f</i> on the uniform lattice (1/m)<b>Z</b>
d and show for large enough m that the variance of the error between the frame reconstruction <i>f</i>
ε from noisy samples of <i>f</i> and the function <i>f</i> satisfy var(<i>f</i>
ε (<i>x</i>)-<i>f</i>(<i>x</i>))≈ (σ
2/m
d)<i>C</i>
x. Second, we address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the set X={<i>x
j: j ∈ J</i>} on which a signal <i>f</i> is sampled is not precisely known. Two main questions are considered. First, if sampling a function <i>f</i> on the countable set X leads to unique and stable reconstruction of <i>f</i>, then when does sampling on the set X'={<i>x
j</i>+δ<sub><i>j</i></sub>: <i>j</i> ∈ <i>J</i>} also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function <i> f</i> using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original <i>f</i>?
Advisors/Committee Members: Benoit Dawant (committee member), Guoliang Yu (committee member), Larry Schumaker (committee member), Douglas P. Hardin (committee member), Akram Aldroubi (Committee Chair).
Subjects/Keywords: jitter; noise; sampling; shift-invariant space; frame; fourier
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MLA ·
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CSE |
Export
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Manager
APA (6th Edition):
Leonetti, C. C. (2007). Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/11682
Chicago Manual of Style (16th Edition):
Leonetti, Casey Clark. “Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces.” 2007. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/11682.
MLA Handbook (7th Edition):
Leonetti, Casey Clark. “Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces.” 2007. Web. 23 Jan 2021.
Vancouver:
Leonetti CC. Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces. [Internet] [Doctoral dissertation]. Vanderbilt University; 2007. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/11682.
Council of Science Editors:
Leonetti CC. Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces. [Doctoral Dissertation]. Vanderbilt University; 2007. Available from: http://hdl.handle.net/1803/11682
Vanderbilt University
30.
Halder, Bibhrajit.
Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems.
Degree: PhD, Mechanical Engineering, 2006, Vanderbilt University
URL: http://hdl.handle.net/1803/14338
► The demand for automation in modern society has significantly increased during the last few decades. Robotic systems play an important role in automation industries that…
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▼ The demand for automation in modern society has significantly increased during the last few decades. Robotic systems play an important role in automation industries that include manufacturing, assembly, and biotechnology among others. In addition, there is a growing need for unmanned operation in different services and research sectors such as search and rescue operation, nuclear waste clean-up, and planetary exploration. Robots can perform repetitive tasks efficiently and can function in a harsh and unsafe environment. However, robots are susceptible to system faults. Faults may result in mission failures that are costly in mission critical enterprises. Therefore fault detection and isolation (FDI) is important for reliable and safe robot operations.
In this dissertation, we present a new approach, called the robust nonlinear analytic redundancy (RNLAR) technique, to sensor and actuator FDI for input-affine nonlinear multivariable dynamic systems in the presence of model-plant-mismatch and process disturbances. Robust FDI is important because of the universal existence of model uncertainties and process disturbances in most systems. The new approach is based on analytic redundancy relation, which has primarily been used in the linear domain. The proposed RNLAR technique extends the current state-of-the-art in analytic redundancy relation-based FDI into the nonlinear domain. The RNLAR technique is used to design primary residual vectors (PRV) to detect actuator and sensor faults. Primary residual vectors are designed in such a manner that they are highly sensitive to the faults and less sensitive to model-plant-mismatch and process disturbances. The proposed methodology is applied to the actuator and sensor fault detection of a wheeled mobile robot as well as a robotic manipulator.
The order of redundancy relation is used to characterize the robustness of the RNLAR technique. It is proved that an increase in the order of redundancy relation increases the robustness of the RNLAR technique. This result extends the existing relationship between the order of redundancy relation and robustness from the linear domain to the nonlinear domain.
Finally, a robust fault isolation technique is presented in this work. The PRVs are transformed into a set of structured residual vectors (SRV) for fault isolation. Experimental results on a Pioneer 3-DX mobile robot and a PUMA 560 robotic manipulator are presented to justify the effectiveness of the RNLAR technique.
Advisors/Committee Members: Eric Barth (committee member), Michael Goldfarb (committee member), George E. Cook (committee member), Akram Aldroubi (committee member), Nilanjan Sarkar (Committee Chair).
Subjects/Keywords: Fault location (Engineering); mobile robots; nonlinear systems; PUMA; order of redundancy; Fault detection; analytical redundancy; robustness; Robots – Error detection and recovery
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APA (6th Edition):
Halder, B. (2006). Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://hdl.handle.net/1803/14338
Chicago Manual of Style (16th Edition):
Halder, Bibhrajit. “Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems.” 2006. Doctoral Dissertation, Vanderbilt University. Accessed January 23, 2021.
http://hdl.handle.net/1803/14338.
MLA Handbook (7th Edition):
Halder, Bibhrajit. “Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems.” 2006. Web. 23 Jan 2021.
Vancouver:
Halder B. Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems. [Internet] [Doctoral dissertation]. Vanderbilt University; 2006. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/1803/14338.
Council of Science Editors:
Halder B. Robust nonlinear analytic redundancy for fault detection and isolation of robotic systems. [Doctoral Dissertation]. Vanderbilt University; 2006. Available from: http://hdl.handle.net/1803/14338
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