2024-03-28T13:09:13Zhttps://kuscholarworks.ku.edu/oai/requestoai:kuscholarworks.ku.edu:1808/123092020-10-09T13:28:31Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Some application of Malliavin calculus to SPDE and convergence of densities
Lu, Fei
Hu, Yaozhong
Nualart, David
Duncan, Tyrone E.
Feng, Jin
Juhl, Ted
Tu, Xuemin
Mathematics
Central limit theorems on wiener chaos
Convergence of densities
Feynman-kac formula
Holder continuity of solutions to spdes
Malliavin calculus
Stochastic partial differential equatons
Some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation theory are studied in this dissertation. In Chapter 3, a Feynman-Kac formula is established for a stochastic heat equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H<1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral of the Gaussian noise. The existence of the Feynman-Kac integral then follows from the exponential integrability of this nonlinear stochastic integral. Then, techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic heat equation. In Chapter 4, the density formula in Malliavin calculus is used to study the joint H"{o}lder continuity of the solution to a nonlinear SPDE arising from the study of one dimensional super-processes. Dawson, Vaillancourt and Wang [Ann. Inst. Henri. Poincaré Probab. Stat., 36 (2000) 167-180] proved that the solution of this SPDE gives the density of the branching particles in a random environment. The time-space joint continuity of the density process was left as an open problem. Li, Wang, Xiong and Zhou [Probab. Theory Related Fields 153 (2012), no. 3-4, 441--469] proved that this solution is joint H"{o}lder continuous with exponent up to 1/10 in time and up to 1/2 in space. Using our new method of Malliavin calculus, we improve their result and obtain the optimal exponent 1/4 in time. In Chapter 5, we study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functional of a Gaussian process, in particular, the multiple integrals. They are assumed to be non-degenerate so that their probability densities exist. The tool we use is the Malliavin calculus, in particular, the density formula, the integration by parts formula and the Stein's method. Applications to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Ulenbeck is also considered. In Chapter 6, we apply an upper bound estimate from small deviation theory to prove the non-degeneracy of some functional of fractional Brownian motion.
2013-09-30T19:25:20Z
2013-09-30T19:25:20Z
2013-05-31
2013
Dissertation
http://dissertations.umi.com/ku:12688
http://hdl.handle.net/1808/12309
https://orcid.org/0000-0001-6842-7922
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/197342017-12-08T21:34:34Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Hyperbolic functions
Henry, Thomas B.
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2016-01-06T17:16:47Z
2016-01-06T17:16:47Z
1922
Thesis
http://hdl.handle.net/1808/19734
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/82342020-08-25T13:10:49Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On Some Physical Applications of Hamilton’s Operator
Owens, Frederick William
2011-10-18T17:08:44Z
2011-10-18T17:08:44Z
1902
Thesis
http://hdl.handle.net/1808/8234
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/209592017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
A study of the parabola having contact of the third order with a conic
Hout, Gus Jay
Thesis (M.A.)--University of Kansas, Mathematics, 1926.
2016-06-15T15:14:32Z
2016-06-15T15:14:32Z
1926
Thesis
http://hdl.handle.net/1808/20959
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/183502020-06-24T18:51:31Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Methods of generating plane cubic curves
Black, Florence
Thesis (M.A.)--University of Kansas, Mathematics, 1921. ; Includes bibliographical references.
2015-08-19T21:03:06Z
2015-08-19T21:03:06Z
1921
Thesis
http://hdl.handle.net/1808/18350
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/301042021-03-05T16:54:48Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves
Wang, Yao
Tu, Xuemin
Xu, Hongguo
Tsoflias, Georgios P
Mathematics
Geophysical engineering
back-scatter
surface waves
waveform inversion
An efficient subsurface imaging method employing back-scattered surface waves is developed to detect near-surface underground elastic-wave velocity anomalies, such as tunnels, sinkholes, fractures, faults, and abandoned manmade infrastructures. The back-scattered surface waves are generated by seismic waves impinging on the velocity anomalies and diffracting back toward the source. These wave events contain plentiful information of the subsurface velocity anomalies including spatial location, shape, size, and velocity of the interior medium. Studies have demonstrated that the back-scattered surface waves can be easily distinguished in the frequency-wavenumber (F-k) domain and have less interference by other wave modes. Based on these features, a near-surface velocity anomaly detection method by using waveform inversion of the back-scattered surface waves (BSWI) is proposed. The main objective of this thesis is to review the theoretical background and study the feasibility of the proposed BSWI method. The proposed BSWI method is tested with numerical and real-world examples. First, the numerical example uses the conventional full-waveform inversion (FWI) method as a benchmark to demonstrate the efficiency of BSWI method in detecting shallow velocity anomalies. Then, the BSWI method is tested with field data. In this study, 2D seismic data were acquired over a manmade concrete tunnel located on the main campus of the University of Kansas (KU). Different workflows including FWI method and BSWI method are applied to the acquired data and tested for imaging the known tunnel. The field example demonstrates that BSWI can accurately image the tunnel. Compared with FWI, BSWI is less demanding in data processing. Finally, this thesis concludes that the proposed BSWI method is capable of efficiently detecting a near-surface tunnel with the minimum amount of data processing which lends it as a method suitable for application in the field.
2020-03-21T18:52:33Z
2020-03-21T18:52:33Z
2019-05-31
2019
Thesis
http://dissertations.umi.com/ku:16526
http://hdl.handle.net/1808/30104
https://orcid.org/0000-0002-5077-8852
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/183692020-06-24T20:10:58Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Theory of reproducing kernels for Hilbert spaces of vector valued functions
Pedrick, George
Hilbert space
Vector valued functions
1 v. ; 29 cm. Includes bibliographical references.
The general theory of reproducing kernels developed by N. Aronszajn provides a unifying point of view for the study of an important class of Hilbert spaces of real or complex valued functions and for the application of the methods of Hilbert space theory to different problems in the theory of partial differential equations. With a view to applications to systems of such equations the form which the theory takes in the case of spaces of vector valued functions was investigated, initially for finite dimensional and Hilbert range spaces. It was found that the natural setting for such a generalization of the theory is that
in which the functions of the functional Hilbert space take their values in an arbitrary locally convex linear topological space, since all of the main results are essentially preserved in that setting and a more special case would restrict unduly the applications. The present study is confined to the exposition of the general theory with a few illustrations and undertakes to extend the basic notions of proper functional space, reproducing kernel and positive matrix and their proper ties as they occur in the paper of N. Aronszajn.
2015-08-28T15:03:08Z
2015-08-28T15:03:08Z
1957
Dissertation
http://hdl.handle.net/1808/18369
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/207472021-08-26T20:59:10Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On Fourier series
Switser, Lucretia Mae
Thesis (M.A.)--University of Kansas, Mathematics, 1923.
2016-05-03T17:00:25Z
2016-05-03T17:00:25Z
1923
Thesis
http://hdl.handle.net/1808/20747
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/214462017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
The classification of conics in modular geometries
Long, Maude
Thesis (M.A.)--University of Kansas, Mathematics, 1927.
2016-09-07T13:17:20Z
2016-09-07T13:17:20Z
1927
Thesis
http://hdl.handle.net/1808/21446
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/205202021-08-27T17:46:46Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
The irreducibility of certain sets of assumptions
Rosson, James Thomas
Thesis (M.A.)--University of Kansas, Mathematics, 1925.
2016-03-18T16:18:08Z
2016-03-18T16:18:08Z
1925
Thesis
http://hdl.handle.net/1808/20520
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/52582020-07-23T14:01:09Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Coarser connected topologies and non-normality points
Yengulalp, Lynne Christine
Fleissner, William
Agah, Arvin
Porter, Jack
Roitman, Judith
Torres, Rodolfo
Mathematics
We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is: When does a space have a coarser connected topology with a nice topological property? We will discuss some results when the property is Hausdorff and prove that if X is a non-compact metric space that has weight at least the cardinality of the continuum, then it has a coarser connected metrizable topology. The second topic is concerned with the following question: When is a point of the Stone-Cech remainder of a space a non-normality point of the remainder? We will discuss the question in the case that X is a discrete space and then when X is a metric space without isolated points. We show that under certain set-theoretic conditions, if X is a locally compact metric space without isolated points then every point in the Stone-Cech remainder is a non-normality point of the remainder.
2009-06-18T20:47:18Z
2009-06-18T20:47:18Z
2009-01-01
2009
Dissertation
http://dissertations.umi.com/ku:10301
http://hdl.handle.net/1808/5258
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/173162020-06-24T18:26:10Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Euclidean proofs of projective geometry theorems /
Farner, Elmer Franklin
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2015-04-06T16:50:45Z
2015-04-06T16:50:45Z
1922.
Thesis
http://hdl.handle.net/1808/17316
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/216962018-01-31T20:07:50Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Fractional Diffusion in Gaussian Noisy Environment
Hu, Guannan
Hu, Yaozhong
Nualart, David
Soo, Terry
Tu, Xuemin
Huan, Luke
Mathematics
chaos expansion
Fox's H-function;
fractional derivative
Green's functions
mild solution
multiple integral of the Itô type
Three types of stochastic partial differential equations are studied in this dissertation. We prove the existence and uniqueness of the solutions and obtain some properties of the solutions. Chapter 3 studies the linear stochastic partial differential equation of fractional orders both in time and space variables. We prove the existence and uniqueness of the solution and calculate the moment bounds of the solution when the noise has Reisz kernel as space covariance. Along the way, we obtain some new properties of the fundamental solutions. Chapter 4 studies the time-fractional diffusion with fractional Gaussian noise. We obtain conditions so that the square integrable solution exists uniquely. Chapter 5 studies the time-fractional diffusion where the Gaussian noise is general in time with space covariance given by fractional, Riesz and Bessel kernel.
2016-10-12T02:39:26Z
2016-10-12T02:39:26Z
2015-12-31
2015
Dissertation
http://dissertations.umi.com/ku:14276
http://hdl.handle.net/1808/21696
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/278762019-08-27T18:09:08Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Stability of Periodic Waves in Nonlocal Dispersive Equations
Claassen, Kyle Matthew
Johnson, Mathew A
Mantzavinos, Dionyssios
Miedlar, Agnieszka
Stanislavova, Milena
Jackson, Timothy
Mathematics
Bidirectional Whitham models
Dispersive Equations
Fractional Nonlinear Schrodinger Equation
In this work consisting of joint projects with my advisor, Dr. Mathew Johnson, we study the existence and stability of periodic waves in equations that possess nonlocal dispersion, i.e. equations in which the dispersion relation between the temporal frequency, omega, and wavenumber, k, of a plane wave is not polynomial in ik. In models that involve only classical derivative operators (known as local equations), the behavior of the system at a point is influenced solely by the behavior in an arbitrarily small neighborhood. In contrast, equations involving nonlocal operators incorporate long-range interactions as well. Such operators appear in numerous applications, including water wave theory and mathematical biology. Specifically, we establish the existence and nonlinear stability of a special class of periodic bound state solutions of the Fractional Nonlinear Schrodinger Equation, where the nonlocality of the fractional Laplacian presents formidable analytical challenges and elicits the development of functional-analytic tools to complement the absence of more-understood techniques commonly used to analyze local equations. Further, we use numerical methods to survey the existence and spectral stability of small- and large-amplitude periodic wavetrains in Bidirectional Whitham water wave models, which implement the exact (nonlocal) dispersion relation of the incompressible Euler equations and are thus expected to better capture high-frequency phenomena than the unidirectional Whitham and Korteweg-de Vries (KdV) equations.
2019-05-12T17:42:04Z
2019-05-12T17:42:04Z
2018-05-31
2018
Dissertation
http://dissertations.umi.com/ku:15929
http://hdl.handle.net/1808/27876
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/197082017-12-08T21:34:35Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Simple and complete K-points in a modular projective plane
Armstrong, Beulah May
Thesis (M.A.)--University of Kansas, Mathematics, 1918. ; Includes bibliographical references.
2016-01-06T17:16:14Z
2016-01-06T17:16:14Z
1918
Thesis
http://hdl.handle.net/1808/19708
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/83082020-08-26T13:35:13Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
An Analytical Determination of Certain Line Groups in the Plane
Pitcher, Arthur Dunn
2011-10-27T20:37:17Z
2011-10-27T20:37:17Z
1907
Thesis
http://hdl.handle.net/1808/8308
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/84912020-06-26T19:56:38Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
A Review of Jacobi’s Theoriae Functionum Ellipticarum
Fischer, Edward George
2011-11-22T21:47:40Z
2011-11-22T21:47:40Z
1912
Thesis
http://hdl.handle.net/1808/8491
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218482018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Generalized Fixed-Point Algebras for Twisted C*-Dynamical Systems
Huang, Leonard Tristan
Sheu, Albert J-L
Lerner, David E
Martin, Jeremy L
Torres, Rodolfo H
Ralston, John P
Mathematics
C*-Algebras
Generalized Fixed-Point Algebras
Morita-Rieffel Equivalence
Square-Integrability
Twisted C*-Dynamical Systems
Twisted Hilbert C*-Modules
In his paper “Generalized Fixed Point Algebras and Square-Integrable Group Actions,” Ralf Meyer showed how to construct generalized fixed-point algebras for C*-dynamical systems via their square-integrable representations on Hilbert C*-modules. His method extends Marc Rieffel’s construction of generalized fixed-point algebras from proper group actions. This dissertation seeks to generalize Meyer’s work to construct generalized fixed-point algebras for twisted C*-dynamical systems. To accomplish this, we must introduce some brand-new concepts, the foremost being that of a twisted Hilbert C*-module. A twisted Hilbert C*-module is basically a Hilbert C*-module equipped with a twisted group action that is compatible with the module’s right C*-algebra action and its C*-algebra-valued inner product. Twisted Hilbert C*-modules form a category, where morphisms are twisted-equivariant adjointable operators, and we will establish that Meyer’s bra-ket operators are morphisms between certain objects in this category. A by-product of our work is a twisted-equivariant version of Kasparov’s Stabilization Theorem, which states that every countably generated twisted Hilbert C*-module is isomorphic to an invariant orthogonal summand of the countable direct sum of a standard one if and only if the module is square-integrable. Given a twisted C*-dynamical system, we provide a definition of a relatively continuous subspace of a twisted Hilbert C*-module (inspired by Ruy Exel) and then prescribe a new method of constructing generalized fixed-point algebras that are Morita-Rieffel equivalent to an ideal of the corresponding reduced twisted crossed product. Our construction generalizes that of Meyer and, by extension, that of Rieffel. Our main result is the description of a classifying category for the class of all Hilbert modules over a reduced twisted crossed product. This implies that every Hilbert module over a $ d $-dimensional non-commutative torus can be constructed from a Hilbert space endowed with a twisted $ \mathbb{Z}^{d} $-action and a relatively continuous subspace.
2016-11-08T22:50:02Z
2016-11-08T22:50:02Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14575
http://hdl.handle.net/1808/21848
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/231472020-06-23T19:48:08Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Canonical expansions for the equations of curves
Allbritten, Pauline Elizabeth
Thesis (M.A.)--University of Kansas, Mathematics, 1932.
2017-02-14T19:29:42Z
2017-02-14T19:29:42Z
1932
Thesis
http://hdl.handle.net/1808/23147
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/178012020-06-24T19:55:59Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
The principle of duality in a projective space of N dimensions
Flinn, Ruby Vee
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2015-05-18T21:56:39Z
2015-05-18T21:56:39Z
1922.
Thesis
http://hdl.handle.net/1808/17801
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/41312020-07-20T14:10:46Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Steady-State Poisson-Nernst-Planck Systems: Asymptotic expansions and applications to ion channels
Abaid, Nicole Teresa
Liu, Weishi
Van Vleck, Erik
Oh, Myunghyun
Mathematics
Poisson-Nernst-Planck systems
Dynamical systems
Ion channels
Important properties of ion channels can be described by a steady state Poisson-Nernst-Plank system for electrodiffusion. The solution to the PNP system gives a relation between the current and electric potential of the ions in the channel, called the I-V curve. In this thesis, we will discuss the matched asymptotic expansions method of solving a singularly perturbed system and apply this method to find an approximate solution to the steady-state Poisson-Nernst-Planck system. In general, for nonlinear systems, it is impossible to obtain any representations of solutions. Due to the presence of a singular parameter in the PNP system, we can treat the system as a singularly perturbed problem. This system with specific nonlinearity has special structures that are crucial for the explicit higher order asymptotic expansions of the solutions. Although the ion channel problem considers only one cell, the I-V relation obtained in this thesis is consistent with the cubic-like assumption of the I-V relation in the FitzHugh-Nagumo model for action potential involving a population of ion channels. However, applications of the results of this thesis to ion channels are limited, since we considered a simplified model with two species of ions and a zero permanent charge in the channel.
2008-09-08
2008-09-08
2008-07-31
2008
Thesis
http://dissertations.umi.com/ku:2514
http://hdl.handle.net/1808/4131
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/210992017-12-08T21:42:12Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
F(x) and related functions
Eberhart, Paul
Thesis (M.A.)--University of Kansas, Mathematics, 1929.
2016-07-13T13:36:37Z
2016-07-13T13:36:37Z
1929
Thesis
http://hdl.handle.net/1808/21099
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/206542021-08-26T20:58:26Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Certain plane vectorial loci
Lynn, Nellie M.
Thesis (M.A.)--University of Kansas, Mathematics, 1923.
2016-04-08T13:44:46Z
2016-04-08T13:44:46Z
1923
Thesis
http://hdl.handle.net/1808/20654
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/246152017-12-08T21:43:44Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Parametric equations of plane curves
Johnson, Eula Mary
Thesis (M.A.)--University of Kansas, Mathematics, 1931.
2017-06-26T20:56:12Z
2017-06-26T20:56:12Z
1931
Thesis
http://hdl.handle.net/1808/24615
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/231642020-06-23T20:01:14Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Determinants as defined by differential equations
Stratton, Carol Lusetta
Thesis (M.A.)--University of Kansas, Mathematics, 1931.
2017-02-14T19:30:04Z
2017-02-14T19:30:04Z
1931
Thesis
http://hdl.handle.net/1808/23164
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/96112020-06-26T19:53:24Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
A Representation of Projective Space by the Points of a Line
Hess, George Wellman
2012-05-18T19:14:56Z
2012-05-18T19:14:56Z
1911-05
Thesis
http://hdl.handle.net/1808/9611
en_US
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/205892021-08-26T20:56:27Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Discussion of the function w²=z³-3tz²+a
Loewen, Otto Bismark
Thesis (M.A.)--University of Kansas, Mathematics, 1923.
2016-03-29T16:05:01Z
2016-03-29T16:05:01Z
1923
Thesis
http://hdl.handle.net/1808/20589
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/211552017-12-08T21:42:12Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Distribution of primes in certain number classes
Caldwell, Georgia Alberta Lee
Thesis (M.A.)--University of Kansas, Mathematics, 1929.
2016-07-21T15:09:01Z
2016-07-21T15:09:01Z
1929
Thesis
http://hdl.handle.net/1808/21155
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/78362020-08-12T13:22:36Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes
Song, Xiaoming
Hu, Yaozhong
Nualart, David
Feng, Jin
Koch, Paul D.
Tu, Xuemin
Mathematics
In this dissertation, I investigate two types of stochastic differential equations driven by
fractional Brownian motion and backward stochastic differential equations. Malliavin
calculus is a powerful tool in developing the main results in this dissertation.
This dissertation is organized as follows.
In Chapter 1, I introduce some notations and preliminaries on Malliavin Calculus
for both Brownian motion and fractional Brownian motion.
In Chapter 2, I study backward stochastic differential equations with general terminal
value and general random generator. In particular, the terminal value has not
necessary to be given by a forward diffusion equation. The randomness of the generator
does not need to be from a forward equation neither. Motivated from applications to
numerical simulations, first the Lp-H¨older continuity of the solution is obtained. Then,
several numerical approximation schemes for backward stochastic differential equations
are proposed and the rate of convergence of the schemes is established based on
the obtained Lp-H¨older continuity results.
Chapter 3 is concerned with a singular stochastic differential equation driven by
an additive one-dimensional fractional Brownian motion with Hurst parameter H > 1
2 .
Under some assumptions on the drift, we show that there is a unique solution, which
has moments of all orders. We also apply the techniques of Malliavin calculus to prove
that the solution has an absolutely continuous law at any time t > 0.
In Chapter 4, I am interested in some approximation solutions of a type of stochastic
differential equations driven by multi-dimensional fractional Brownian motion BH
with Hurst parameter H > 1
2 . In order to obtain an optimal rate of convergence, some
techniques are developed in the deterministic case. Some work in progress is contained
in this chapter.
The results obtained in Chapter 2 are accepted by the Annals of Applied Probability,
and the material contained in Chapter 3 has been published in Statistics and Probability
Letters 78 (2008) 2075-2085.
2011-08-02T14:15:01Z
2011-08-02T14:15:01Z
2011-04-18
2011
Dissertation
http://dissertations.umi.com/ku:11414
http://hdl.handle.net/1808/7836
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/263342018-04-26T19:11:22Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
QUANTUM FAMILIES OF MAPS
Kang, Su Chen
SHEU, Albert J.L.
PORTER, JACK
RALSTON, JOHN
SHAO, SHUANGLIN
STANISLAVOVA, MILENA
Mathematics
C*-algebras
compact quantum semigroup
non-commutative topology
quantum families of all maps
quantum space
quantum space of all maps
This dissertation investigates the theory of quantum families of maps, which formulates a non-commutative topological way to study quantum analogs of spaces of continuous mappings, classical objects of interest from general topology. The fundamental element of non-commutative topology is a C$^*$-algebra. In the theory of C$^*$-algebras, Gelfand Theorem says that every commutative C$^*$-algebra is C$^*$-isomorphic to $C_0(X)$, where $X$ is a locally compact Hausdorff space. Extending this Gelfand duality conceptually to all C$^*$-algebras (not only those commutative ones), the non-commutative or quantum topology views any C$^*$-algebra $A$ as the function algebra of a corresponding "virtual" space $\mathscr{QS}(A)$, called a quantum space. Piotr So{\l}tan defined a quantum concept of the family of all maps from a quantum space $\mathscr{QS}(M)$ to another quantum space $\mathscr{QS}(B)$, and established some general properties of related objects, extending classical results on such families of mappings. However, a lot of his results carry the assumption that $M$ is finite dimensional (and $B$ is finitely generated), only under which the quantum family of all maps was proved to exist (in a unique way). In this dissertation, we study the most fundamental and important question about the existence (and uniqueness) of the quantum space of all maps for infinite-dimensional cases, and solve it for the fundamental case of $M=C(\mathbb{N}\cup\{\bm\infty\})$ where $\mathbb{N}\cup\{\bm\infty\}$ is the one-point compactification of $\mathbb{N}$. We find that new structures outside purely C$^*$-algebraic framework are needed from the von Neumann algebra theory in order to handle such a new situation. This opens up a new direction of research in quantizing spaces of maps betweem more general quantum spaces.
2018-04-20T22:14:11Z
2018-04-20T22:14:11Z
2017-05-31
2017
Dissertation
http://dissertations.umi.com/ku:15335
http://hdl.handle.net/1808/26334
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/103222020-09-10T13:23:59Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Shearing Flows in Liquid Crystal Models
Dorn, Timothy
Liu, Weishi
Oh, Myunghyun
Stanislavova, Milena
Van Vleck, Erik
Shi, JiCong
Mathematics
Materials science
The liquid crystal phase is a phase of matter between the solid and liquid phase whose flow is characterized by a velocity field and a director field which describes locally the orientation of the liquid crystal. In this work we explore shearing flows in two related continuum models of liquid crystals. The first is a phenomenological model of frictional forces in a geological fault, which is motivated by the second model, the Leslie-Ericksen continuum theory of liquid crystals.
2012-10-28T17:16:28Z
2012-10-28T17:16:28Z
2012-05-31
2012
Dissertation
http://dissertations.umi.com/ku:12107
http://hdl.handle.net/1808/10322
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/197192017-12-08T21:31:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On the seminvariants of linear homogeneous differential equations of the third order
Green, Charles Francis
Thesis (M.A.)--University of Kansas, Mathematics, 1915. ; Includes bibliographical references.
2016-01-06T17:16:22Z
2016-01-06T17:16:22Z
1915
Thesis
http://hdl.handle.net/1808/19719
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218912018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Convergence Properties of Hausdorff Closed Spaces
Reynolds, John P.
Porter, Jack R
Bayer, Margaret
Dao, Hai Long
Martin, Jeremy
Nutting, Eileen
Mathematics
Convergence Spaces
General Topology
Hausdorff closed
Pretopologies
The purpose of this work is to study the topological property of Hausdorff closedness as a purely convergence theoretic property. It is the author's opinion that this perspective proves to be a natural one from which to study H-closedness. Chapter 1 provides a brief introduction to and history of the subject matter. Chapter 2 and the first section of Chapter 3 are mainly preliminary. Here the fundamental facts and definitions needed in the study of H-closed spaces, convergence spaces and especially pretopological spaces are given. In Chapter 2 most proofs are omitted for the sake of brevity, however in Section 3.1, many proofs are given in hopes of helping the reader gain an intuitive feel for pretopologies. Original work begins in Section 3.2, where a study of perfect maps between pretopological spaces is given. Chapters 4 and 5 make up the heart of this work. In Chapter 4, we take an in-depth look at the pretopology $\theta$. This convergence, which can be defined for any topological space, frames both H-closedness and the related property of being an H-set as convergence properties. Upon noting this fact, in Section 4.1, we immediately see the benefits of this framing. Of particular interest to those who have studied H-closed spaces are Theorems 4.1.5 and 4.1.8. Later in this chapter, so-called relatively $\theta$-compact filters, which are defined using the convergence $\theta$, are used to obtain a new characterization of countable Kat\v{e}tov spaces in terms of multifunctions. Chapter 5 provides an analogue of H-closedness which can be defined for any pretopological space. The definition of the so-called PHC spaces is due to the author. In Section 5.1, the PHC spaces are defined and their basic properties are investigated. In Section 5.2, we use the earlier work on perfect maps between pretopological spaces to generate new PHC spaces. Lastly, in Section 5.3, we study the PHC extensions of a pretopological space. In this section we have a construction which is analogous to the Kat\v{e}tov extension of a topological space. Theorem 5.3.6 points to an interesting difference between the usual Kat\v{e}tov extension and our pretopological version. We finish this work with an investigation of the cardinal invariants of pretopological spaces. We are particularly interested in obtaining cardinality bounds for compact Hausdorff pretopological spaces in terms of their cardinal invariants. Throughout the paper we seek to highlight results which distinguish pretopologies from topologies and this chapter features several results of this flavor.
2016-11-10T23:17:26Z
2016-11-10T23:17:26Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14645
http://hdl.handle.net/1808/21891
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/205212021-08-26T20:55:29Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On Jordan curves
Smith, Paul Althaus
Thesis (M.S.)--University of Kansas, Mathematics, 1923.
2016-03-18T16:18:08Z
2016-03-18T16:18:08Z
1923
Thesis
http://hdl.handle.net/1808/20521
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/215802017-12-08T21:40:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Envelopes
Penrod, Mabel Opal
Thesis (M.A.)--University of Kansas, Mathematics, 1928.
2016-09-30T14:39:14Z
2016-09-30T14:39:14Z
1928
Thesis
http://hdl.handle.net/1808/21580
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/122502020-10-14T13:50:45Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels
Zhang, Mingji
Liu, Weishi
Liu, Weishi
Stanislavova, Milena
Tu, Xuemin
Van Vleck, Erik
Shi, Jack
Mathematics
Biochemistry
Dynamics of Poisson-Nernst-Planck systems and its applications to ion channels are studied in this dissertation. The Poisson-Nernst-Planck systems serve as basic electro-diffusion equations modeling, for example, ion flow through membrane channels and transport of holes and electrons in semiconductors. The model can be derived from the more fundamental models of the Langevin-Poisson system and the Maxwell-Boltzmann equations, and from the energy variational analysis EnVarA. A brief description of the model is given in Chapter 2 including the physical meaning of each equation involved. Ion channels are cylindrical, hollow proteins that regulate the movement of ions ( mainly Na+, K+, Ca++ and Cl-;) through nearly all the membrane channels. When an initial potential is applied at one end of the channel, it will drive the ions through the channel, and the movement of these ions will produce the current which can be measured. Different initial potentials will result in different currents, and the collection of all those data will provide a relation, the so-called I-V (current-voltage) relation, which is an important characterization of two most relevant properties of a channel: permeation and selectivity. In Chapter 3, a classical Poisson-Nernst-Planck system is studied both analytically and numerically to investigate the cubic-like feature of the I-V relation. For the case of zero permanent charge, under electroneutrality boundary conditions at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion speciesis that the third order correction is cubic in the potential V , and furthermore, up to the third order, the cubic I-V relation has three distinct real roots (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Numerical simulations are performed and and they are consistent with our analytical results. In Chapter 4, we consider a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through membrane channels including ionic interaction modeled by a nonlocal hard-sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. Ion size effect on the I-V relations is investigated numerically. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I-V curves. This is accomplished through a numerical implement of the analytical strategy introduced in [46]. The second task is to numerically detect two critical potential values Vc and Vc. Our numerical detections are based on the defining properties of Vc and V c and are designed to use the numerical I-V curves directly. For the setting in the above mentioned reference, our numerical results agree well with the analytical predictions. In Chapter 5, a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through a membrane channel is analyzed, which includes a local hard-sphere potential that depends pointwise on ion concentrations to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singu- larly perturbed differential system. Based on the geometric singular perturbation theory, especially, on specific structures of this concrete model, the existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality (zero netcharge) boundary conditions, each of these two critical potentials separates the potential into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary con- centrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneutrality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations and critical potentials agree with those with a nonlocal hard-sphere potential examined in [46].
2013-09-29T16:37:28Z
2013-09-29T16:37:28Z
2013-08-31
2013
Dissertation
http://dissertations.umi.com/ku:12955
http://hdl.handle.net/1808/12250
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/186322018-11-20T18:10:32Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Solutions of Lattice Differential Equations over Inhomogeneous Media
Brucal-Hallare, Maila
Van Vleck, Erik S
Liu, Weishi
Stefanov, Atanas
Huang, Weizhang
Kieweg, Sarah
Mathematics
inhomogeneous medium
lattice differential equations
Nagumo equations
negative diffusion
This thesis investigates one-dimensional spatially-discrete reaction-diffusion equations with a diffusion term that involves nearest-neighbor coupling and with a reaction-term that is a smooth-cubic nonlinearity. Specifically, we consider two nontrivial examples of lattice differential equations (LDEs) on Z that are related to the (homogeneous) lattice Nagumo equation. The LDEs that we consider are used to model natural phenomena defined over an inhomogeneous medium, namely: (1) a lattice Nagumo equation with a negative diffusion coefficient. Such is still a well-posed problem in the LDE setting and has been shown to arise from a discrete model of phase transition for shape memory alloys. This thesis shows that the anti-diffusion lattice Nagumo equation has a period-2 traveling wavefront solution that is stable and unique. Utilizing the concrete expressions for the nonlinearities, we obtain criteria on the (d, a)-parameter plane that guarantee a display of bistable and monostable dynamics. Where there's bistable dynamics, we study the propagation failure phenomenon; where there's monostable dynamics, we compute a minimum wave speed for the traveling waves. (2) a lattice Nagumo equation that has a single diffusion-defect in the middle of Z, which may occur due to deviations in the diffusive property of the medium. This thesis shows that such an equation has a time-global solution which behaves as two fronts coming from the both sides of Z. A key idea for the existence proof is a characterization of the asymptotic behavior of the solutions for negative time in terms of an appropriate super-solution, sub-solution pair.
2015-10-12T22:22:37Z
2015-10-12T22:22:37Z
2012-12-31
2012
Dissertation
http://dissertations.umi.com/ku:12423
http://hdl.handle.net/1808/18632
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/83062020-08-26T13:32:31Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Certain Continuous Groups of Projective Transformations Treated Analytically
Mitchell, Ulysses Grant
2011-10-27T20:28:22Z
2011-10-27T20:28:22Z
1907
Thesis
http://hdl.handle.net/1808/8306
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/219142018-01-31T20:07:51Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Application of stochastic differential equations to option pricing
Wang, Peixin
Pasik-Duncan, Bozenna
Hu, Yaozhong
Talata, Zsolt
Mathematics
Applied mathematics
Black-Scholes model
BSDE
Mathematica
optimal cotrol
option pricing
stochastic differential equation
The financial world is a world of random things and unpredictable events. Along with the innovative development of diversity and complexity in modern financial market, there are more and more financial derivative emerged in the financial industry in order to gain higher yields as well as hedge the risk . As a result, to price the derivative , indeed the future uncertainty, become an interesting topic in the field of mathematical finance and financial quantitative analysis. In this thesis, I mainly focus on the application of stochastic differential equations to option pricing. Based on the arbitrage-free and risk-neutral assumption, I used the stochastic differential equations theory to solve the pricing problem for the European option of which underlying assets can be described by a geometric Brownian motion. The thesis explores the Black-Scholes model and forms an optimal control problem for the volatility that is an essential parameter in the Black-Scholes formula. Furthermore, the application of backward stochastic differential equations (BSDEs) has been discussed. I figured that BSDEs can model the pricing problem in a more clarifying and logical way. Also, based on the model discussed in the thesis, I provided a case study on pricing a Chinese option-like deposit product by using Mathematica, that shows the feasibility and applicability for the option pricing method based on stochastic differential equations.
2016-11-10T23:48:54Z
2016-11-10T23:48:54Z
2016-05-31
2016
Thesis
http://dissertations.umi.com/ku:14453
http://hdl.handle.net/1808/21914
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/298982021-03-05T16:53:01Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
On the Existence and Stability of Normalized Ground States of the Kawahara, Fourth Order NLS and the Ostrovsky Equations
Posukhovskyi, Iurii
Stefanov, Atanas
Stefanov, Atanas
Johnson, Mathew
Kong, Kyoungchul
Mantzavinos, Dionyssios
Torres, Rodolfo
Mathematics
Applied mathematics
Existence
Kawahara
NLS
Ostrovsky
Stability
In this dissertation we show the existence and stability of the normalized ground states for the Kawahara, fourth order nonlinear Schrödinger (NLS) and the generalized Ostrovsky equations. One of the starting points in our investigation were numerical stability results by S. Levandosky in [32], [31] which agree with our rigorous stability results. We show existence of the waves using variational techniques together with the concentration compactness argument. On the level of construction, we encounter certain obstacles in the form of new Gagliardo–Nirenberg–Sobolev type inequalities, which impose restrictions on the parameter space. We show stability utilizing spectral theory developed in the recent work by Z.Lin and C.Zeng in [35]. For the Kawahara model, our results provide a significant extension in the parameter space of the current rigorous results. In fact, our results rigorously establish the spectral stability for all acceptable values of the parameters. For the fourth order NLS models, we improve upon recent results on stability of, very special, explicit solutions in the one dimensional case. Our multidimensional results for the fourth order NLS equations seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second order derivatives vs pure Laplacian) has implications on the range of the existence and stability of the normalized waves. For the Ostrovsky equation, we show that all normalized waves we construct are spectrally stable. We also establish decay rates for the waves, extending the results in the paper by P. Zhang and Y. Liu [51].
2020-01-17T23:07:54Z
2020-01-17T23:07:54Z
2019-05-31
2019
Dissertation
http://dissertations.umi.com/ku:16539
http://hdl.handle.net/1808/29898
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/279972019-08-27T18:09:08Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Analytical studies of standing waves in three NLS models
Feng, Wen
Stanislavova, Milena
Stanislavova, Milena
Chen, Geng
Johnson, Mat
Kong, Man
Stefanov, Atanas
Mathematics
Existence
NLS
Spectral Stability
Standing Waves
In this work, we present analytical studies of standing waves in three NLS models. We first consider the spectral stability of ground states of fourth order semi-linear Schrödinger and Klein-Gordon equations and semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. We use Hamiltonian index counting theory, together with the information from a variational construction to develop sharp conditions for spectral stability for these waves. The second case is about the existence and the stability of the vortices for the NLS in higher dimensions. We extend the existence and stability results of Mizumachi from two-space dimensions to $n$ space dimensions. Finally, the third equation we consider is a nonlocal NLS which comes from modeling nonlinear waves in Parity-time symmetric systems. Here again, we investigate the spectral stability of standing waves of its $\mathcal{PT}$ symmetric solutions.
2019-05-18T19:24:12Z
2019-05-18T19:24:12Z
2018-08-31
2018
Dissertation
http://dissertations.umi.com/ku:16099
http://hdl.handle.net/1808/27997
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/204982021-08-26T21:59:59Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
An application of Lie's theory to the solution of differential equations
Fugate, Jessamine H.
Thesis (M.A.)--University of Kansas, Mathematics, 1924.
2016-03-08T20:24:58Z
2016-03-08T20:24:58Z
1924
Thesis
http://hdl.handle.net/1808/20498
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/186682018-01-31T20:07:49Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Graphs of Polytopes
Espenschied, William Joshua
Bayer, Margaret M
Martin, Jeremy L
Porter, Jack
Stahl, Saul
Kinnersley, Nancy
Mathematics
Anticlique
Gale diagram
Graph
Polytope
The graph of a polytope is the graph whose vertex set is the set of vertices of the polytope, and whose edge set is the set of edges of the polytope. Several problems concerning graphs of polytopes are discussed. The primary result is a set of bounds (Theorem 39) on the maximal size of an anticlique (sometimes called a coclique, stable set, or independent set) of the graph of a polytope based on its dimension and number of vertices. Two results concerning properties preserved by certain operations on polytopes are presented. The first is that the Gale diagram of a join of polytopes is the direct sum of the Gale diagrams of the polytopes and dually, that the Gale diagram of a direct sum of polytopes is the join of their Gale diagrams (Theorem 23). The second is that if two polytopes satisfy a weakened form of Gale's evenness condition, then so does their product (Theorem 32). It is shown, by other means, that, with only two exceptions, the complete bipartite graphs are never graphs of polytopes (Theorem 47). The techniques developed throughout are then used to show that the complete 3-partite graph K_{1,n,m} is the graph of a polytope if and only if K_{n,m} is the graph of a polytope (Theorem 49). It is then shown that K_{2,2,3} and K_{2,2,4} are never graphs of polytopes. A conjecture is then stated as to precisely when a complete multipartite graph is the graph of a polytope. Finally, a section is devoted to results concerning the dimensions for which the graph of a crosspolytope is the graph of a polytope.
2015-10-13T04:32:07Z
2015-10-13T04:32:07Z
2014-12-31
2014
Dissertation
http://dissertations.umi.com/ku:13679
http://hdl.handle.net/1808/18668
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/211392017-12-08T21:40:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Simple ratio coordinates
Jenison, John Richie
Thesis (M.A.)--University of Kansas, Mathematics, 1928.
2016-07-21T15:08:48Z
2016-07-21T15:08:48Z
1928
Thesis
http://hdl.handle.net/1808/21139
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/203682021-08-26T21:59:30Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Invariants of determinants with binary linear elements
Steininger, Edith
Thesis (M.A.)--University of Kansas, Mathematics, 1924.
2016-02-25T21:13:02Z
2016-02-25T21:13:02Z
1924
Thesis
http://hdl.handle.net/1808/20368
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/278862020-10-13T15:20:14Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Limit distributions for functionals of Gaussian processes
Jaramillo, Arturo
Nualart, David
Feng, Jin
Liu, Zhipeng
Soo, Terry
Zhang, Jianbo
Mathematics
Statistics
Theoretical mathematics
fracional Brownian motion
limit theorems
Local times
Malliavin calculus
random matrices
stochastic integration
This thesis is devoted to the study of the convergence in distribution of functionals of Gaussian processes. Most of the problems that we present are addressed by using an approach based on Malliavin calculus techniques. Our main contributions are the following: First, we study the asymptotic law of the approximate derivative of the self-intersection local time (SILT) in $[0,T]$ for the fractional Brownian motion. In order to do this, we describe the asymptotic behavior of the associated chaotic components and show that the first chaos approximates the SILT in $L^2$. Secondly, we examine the asymptotic law of the approximate self-intersection local time process for the fractional Brownian motion. We achieve this in two steps: the first part consists on proving the convergence of the finite dimensional distributions by using the `multidimensional fourth moment theorem'. The second part consists on proving the tightness property, for which we follow an approach based on Malliavin calculus techniques. The third problem consists on proving a non-central limit theorem for the process of weak symmetric Riemann sums for a wide variety of self-similar Gaussian processes. We address this problem by using the so-called small blocks-big blocks methodology and a central limit theorem for the power variations of self-similar Gaussian processes. Finally, we address the problem of determining conditions under which the eigenvalues of an Hermitian matrix-valued Gaussian process collide with positive probability.
2019-05-12T17:57:15Z
2019-05-12T17:57:15Z
2018-05-31
2018
Dissertation
http://dissertations.umi.com/ku:15911
http://hdl.handle.net/1808/27886
https://orcid.org/0000-0002-7650-4235
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/122212020-10-15T14:41:21Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Simulation Methods Comparison and Parameter Estimation for a Fractional Stochastic Volatility Model with Application in Stock Price Analysis
Wu, Fengmei
Nualart, David
Hu, Yaozhong
Pasik-Duncan, Bozenna
Applied mathematics
This paper studies continuous-time stock pricing models with stochastic volatility driven by fractional Brownian motion. We compare two ways for simulating the paths of stochastic volatility and stock price when the Hurst parameter of fractional Brown motion is between 0.5 and 1. The first approach, is to use truncated fractional Brownian motion to approximate the fractional Brownian motion and estimate the volatility by Monte Carlo integral and symbolic integral. In the second one, Euler method is employed in simulation, without truncating the fractional Brownian process. Simulating the fractional Brownian motion in the second approach, we use spectral representation. Simulation results show that the latter is more efficient than using the symbolic integral and Monte Carlo integral is the worst. The application of the stochastic model is illustrated through real financial data.
2013-09-29T15:18:53Z
2013-09-29T15:18:53Z
2013-08-31
2013
Thesis
http://dissertations.umi.com/ku:13016
http://hdl.handle.net/1808/12221
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/207682017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Partial derivative operators applied to the derivation of relations among sums of determinants
Smith, Ronald Gibson
Thesis (M.A.)--University of Kansas, Mathematics, 1926.
2016-05-12T17:08:15Z
2016-05-12T17:08:15Z
1926
Thesis
http://hdl.handle.net/1808/20768
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/184632020-06-24T18:34:12Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Historical development of the number concept
Marm, Anna
Thesis (M.A.)--University of Kansas, Mathematics, 1918. ; Includes bibliographical references.
2015-09-21T18:30:20Z
2015-09-21T18:30:20Z
1918
Thesis
http://hdl.handle.net/1808/18463
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/300712021-03-05T16:53:56Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing
Yu, Yufei
Huang, Weizhang
Huang, Weizhang
Keshmiri, Shawn
Liu, Weishi
Tu, Xuemin
Van Vleck, Erik
Applied mathematics
Moving sharp fronts are an important feature of many mathematical models from physical sciences and cause challenges in numerical computation. In order to obtain accurate solutions, a high resolution of mesh is necessary, which results in high computational cost if a fixed mesh is used. As a solution to this issue, an adaptive mesh method, which is called the moving mesh partial differential equation (MMPDE) method, is described in this work. The MMPDE method has the advantage of adaptively relocating the mesh points to increase the densities around sharp layers of the solutions, without increasing the mesh size. Moreover, this strategy can generate a nonsingular mesh even on non-convex and non-simply connected domains, given that the initial mesh is nonsingular. The focus of this thesis is on the application of the MMPDE method to mathematical models from physical sciences and image segmentation. In particular, this thesis includes the selection of the regularization parameter for the Ambrosio-Tortorelli functional, a simulation of the contact sets in the evolution of the micro-electro mechanical systems, and a numerical study of the flux selectivity in the Poisson-Nernst-Planck model. Sharp interfaces take place in all these three models, bringing interesting features and rich phenomena to study.
2020-03-16T20:28:14Z
2020-03-16T20:28:14Z
2019-05-31
2019
Dissertation
http://dissertations.umi.com/ku:16485
http://hdl.handle.net/1808/30071
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/122652020-10-15T13:32:07Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
The Rebate Value Process with Some Applications
Welch, Nathan M.
Pasik-Duncan, Bozenna
Pasik-Duncan, Bozenna
Duncan, Tyrone E.
Katz, Daniel
Mathematics
Finance
Credit derivatives
Enlargement of filtration
Hazard process
Mathematical finance
Mathematics of credit
Recovery
In the pricing of credit derivatives default is modelled as a stopping time and prices are typically determined by separation of cash-flows before and at default. In a general risk-neutral valuation setting, this technique suggests the decomposition of an asset which holds even if the asset is not credit-sensitive. The rebate value process is introduced and related to the price of an asset before and after default. The financial interpretation of this process is different depending on the type of asset decomposed. An interpretation of recovery is illustrated by pricing several standard credit-sensitive assets including a risky coupon bond and a credit default swap (CDS). An interpretation of insurance is illustrated by pricing the complements of the credit ``building blocks'' with respect to the stopping time. Several applications of these complements are presented including a risky interest rate swap and a full-recovery CDS.
2013-09-29T17:01:03Z
2013-09-29T17:01:03Z
2013-08-31
2013
Thesis
http://dissertations.umi.com/ku:12863
http://hdl.handle.net/1808/12265
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/269022020-10-13T14:05:15Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
A Comparison of Some Dimension Reduction Techniques with Varied Parameters
Oderio, Nicholas
Pasik-Duncan, Bozenna
Duncan, Tyrone
Van Vleck, Erik
Applied mathematics
Mathematics
application
Dimensionality reduction
principal component analysis
This paper presents and explains several methods of dimensionality reduction of data sets, beginning with the well known PCA and moving onto techniques that deal with data on a nonlinear manifold. Methods for handling data whose underlying structure is a nonlinear manifold are separated by whether or not sparse matrices are involved in the computation. Additionally, the methods discussed are demonstrated and compared by running them on data sets whose underlying structure is known. Results from same methods with different values for input parameters are also examined. Finally, some results on a small set of Persyst EEG data collected as a part of the Epilepsy Bioinformatics Study for Antiepileptogenic Therapy from the Laboratory of Neuro Imaging at USC Stevens Institute of Neuroimaging and Informatics in the Keck School of Medicine of USC is analyzed using some of these methods.
2018-10-22T16:01:29Z
2018-10-22T16:01:29Z
2017-13-31
2017
Thesis
http://dissertations.umi.com/ku:15662
http://hdl.handle.net/1808/26902
https://orcid.org/0000-0002-3949-2511
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/217272017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Analytical line geometry of the plane
Dougherty, Lucy Taft
Thesis (M.A.)--University of Kansas, Mathematics, 1927.
2016-10-13T18:43:57Z
2016-10-13T18:43:57Z
1927
Thesis
http://hdl.handle.net/1808/21727
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218952018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Depth and Associated Primes of Modules over a Ring
Se, Tony
Dao, Hailong
Jiang, Yunfeng
Katz, Daniel
Lang, Jeffrey
Nutting, Eileen
Mathematics
Cohen-Macaulay
coherent functors
divisor class group
F -regularity
Frobenius endomorphism
semigroup rings
This thesis consists of three main topics. In the first topic, we let $R$ be a commutative Noetherian ring, $I,J$ ideals of $R$, $M$ a finitely generated $R$-module and $F$ an $R$-linear covariant functor. We ask whether the sets $\operatorname{Ass}_R F(M/I^n M)$ and the values $\operatorname{depth}_J F(M/I^n M)$ become independent of $n$ for large $n$. In the second topic, we consider rings of the form $R = k[x^a,x^{p_1}y^{q_1}, \ldots,x^{p_t}y^{q_t},y^b]$, where $k$ is a field and $x,y$ are indeterminates over $k$. We will try to formulate simple criteria to determine whether or not $R$ is Cohen-Macaulay. Finally, in the third topic we introduce and study basic properties of two types of modules over a commutative Noetherian ring $R$ of positive prime characteristic. The first is the category of modules of finite $F$-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call $F$-abundant modules. These include, for example, the ring $R$ itself and the canonical module when $R$ has positive splitting dimension. We prove many facts about these two categories and how they are related. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest, such as complete intersections and invariant subrings.
2016-11-10T23:22:38Z
2016-11-10T23:22:38Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14512
http://hdl.handle.net/1808/21895
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/53542020-07-21T16:11:18Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
SOME NEW DEVELOPMENTS ON TWO SEPARATE TOPICS: STATISTICAL CROSS VALIDATION AND FLOODPLAIN MAPPING
Kastens, Jude Heathcliff
Lerner, David
Church, James
Cobb, Ben
Egbert, Stephen
Huggins, Don
Van Vleck, Erik
Mathematics
This dissertation describes two unrelated threads of research. The first is a study of cross validation (CV), which is a data resampling method. CV is used for model ranking in model selection and for estimating expected prediction error of a model. A review of three resampling methods is provided in Chapter 1. Chapter 2 contains results from simulations that examine various properties of CV, in particular the use of CV for model selection in small sample settings as well as the expected value of the delete-d cross validation statistic. The second research thread is described in Chapter 3, where a new, physically-based computational model (called FLDPLN, or "Floodplain") for mapping potential inundation extents (floodplains) using gridded topographic data is introduced. Due to the parametric economy of FLDPLN, this model has significant advantages over existing methods such as hydrodynamic models. The model is validated using imagery from an actual flood event.
2009-08-06T17:31:53Z
2009-08-06T17:31:53Z
2008-08-06
2008
Dissertation
http://dissertations2.umi.com/ku:2647
http://hdl.handle.net/1808/5354
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/213072017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Some configurations and projective properties of a complete five-point
Wrestler, Ferna E.
Thesis (M.A.)--University of Kansas, Mathematics, 1926.
2016-08-11T15:05:48Z
2016-08-11T15:05:48Z
1926
Thesis
http://hdl.handle.net/1808/21307
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/280422019-08-27T18:09:09Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Serre's Condition and Depth of Stanley-Reisner Rings
Holmes, Brent
Dao, Hailong
Witt, Emily
Katz, Daniel
Martin, Jeremy
Agah, Arvin
Mathematics
Combinatorics
Commutative Algebra
Depth
Hirsch Conjecture
Homological Algebra
Serre's Condition
The aim of this work is to garner a deeper understanding of the relationship between depth of a ring and connectivity properties of the spectrum of that ring. We examine with particular interest the case where our ring is a Stanley- Reisner ring. In this circumstance, we consider the simplicial complex that corresponds to the spectrum of R. We examine properties of simplicial complexes whose Stanley-Reisner rings satisfy depth conditions such as Cohen-Macaulay and Serre's condition (S_l). We leverage these properties to use algebraic tools to examine combinatorial problems. For example, the gluing lemma in (Hol18) allows us to construct bounds on the diameter of a class of graphs acting as a generalization of the 1-skeleton of polytopes. Throughout, we give special consideration to Serre's condition (S_l). We create a generalized Serre's condition (S_l^j) and prove equivalent homological, topological, and combinatorial properties for this condition. We generalize many well-known results pertaining to (S_l) to apply to (S_l^j). This work also explores a generalization of the nerve complex and considers the correlation between the homologies of the nerve complex of a Stanley-Reisner ring and depth properties of that ring. Finally we explore rank selection theorems for simplicial complexes. We prove many results on depth properties of simplicial complexes. In particular, we prove that rank selected subcomplexes of balanced (S_l) simplicial complexes retain (S_l). The primary focus of this work is on Stanley-Reisner rings, however, other commutative, Noetherian rings are also considered.
2019-05-19T01:57:36Z
2019-05-19T01:57:36Z
2018-12-31
2018
Dissertation
http://dissertations.umi.com/ku:16270
http://hdl.handle.net/1808/28042
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/234612017-12-08T21:43:44Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Number scales
Venard, Winnona S.
Thesis (M.A.)--University of Kansas, Mathematics, 1931.
2017-03-21T14:33:49Z
2017-03-21T14:33:49Z
1931
Thesis
http://hdl.handle.net/1808/23461
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/197352017-12-08T21:31:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Some properties of inscribed quadrilaterals
Harnley, Paul Whitmore
Thesis (M.A.)--University of Kansas, Mathematics, 1916. ; Includes bibliographical references.
2016-01-06T17:16:48Z
2016-01-06T17:16:48Z
1916
Thesis
http://hdl.handle.net/1808/19735
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/55282020-07-27T15:03:23Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
A Study of Vandermonde-like Matrix Systems With Emphasis on Preconditioning and Krylov Matrix Connection.
Saraswat, Jyoti
Xu, Dr. Hongguo
Huang, Weizhang
Vleck, Dr. Erik Van
Mathematics
Krylov matrix
Preconditioner
Vandermonde matrix
The study focuses primarily on Vandermonde-like matrix systems. The idea is to express Vandermonde and Vandermonde-like matrix systems as the problems related to Krylov Matrices. The connection provides a different angle to view the Vandermonde-like systems. Krylov subspace methods are strongly related to polynomial spaces, hence a nice connection can be established using LU factorization as proposed by Bjorck and Pereyra and QR factorization by Reichel. Further an algorithm to generate a preconditioner is incorporated in GR algorithm given by Reichel . This generates a preconditioner for Vandermonde-like matrices consisting of polynomials which obey a three term recurrence relation. This general preconditioner works effectively for Vandermonde matrices as well. The preconditioner is then tested on various distinct nodes. Based on results obtained, it is established that the condition number of Vandermonde -like matrices can be lowered significantly by application of the preconditioner, for some cases.
2009-10-13T04:25:22Z
2009-10-13T04:25:22Z
2009-06-11
2009
Thesis
http://dissertations.umi.com/ku:10431
http://hdl.handle.net/1808/5528
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/149672017-12-08T21:46:53Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Coordinate Systems in One and Two Dimensions
Wood, Frank Edwin
Presented to the Faculty of the Graduate School of The University of Kansas In Partial Fulfillment of the Requirements for the Degree of Master of Arts.
2014-08-27T18:32:06Z
2014-08-27T18:32:06Z
1914-06-01
Thesis
Wood, Frank Edwin. "Coordinate Systems in One and Two Dimensions." University of Kansas. June 1914.
http://hdl.handle.net/1808/14967
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/84842020-08-27T13:37:02Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
The History of the Teaching of Arithmetic in the United States Prior to 1860
Bouse, Thornton Lynn
2011-11-22T21:20:27Z
2011-11-22T21:20:27Z
1912-06
Thesis
http://hdl.handle.net/1808/8484
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/53202020-07-23T14:08:10Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Some Properties of Realcompact Subspaces and Coarser Normal Spaces
Niknejad, Jila
Porter, Jack R
Fleissner, William
Himmelberg, Charles
Kinnersley, Nancy
Roitman, Judith
Torres, Rodolfo
Mathematics
In this work we obtain results in two areas of topology, normal condensations of products and size of realcompact subspaces of a space. In 2000 Swardson proved that every uncountable compact space has a realcompact subspace of the same cardinality as the first uncountable cardinal. In the first chapter the work of Swardson is continued to prove that realcompact spaces with pseudocharacter no greater than the first uncountable cardinal have realcompact subspaces of the same size as the first uncountable cardinal. Under continuum hypothesis, a consequence is that every uncountable realcompact space has a realcompact subspace of the same size as the first uncountable cardinal. We also prove that every realcompact right-separated set of size larger than continuum has a realcompact subspace of size of any cardinal less or equal to continuum. A corollary is that every compact set of size bigger or equal to continuum has a realcompact subset of size less or equal to continuum, answering a question by Professor William Fleissner. In 1997 Buzjakova proved that for a pseudocompact space X, there exists an ordinal such that the product of X and that ordinal condenses onto a normal space if and only if X condenses onto a compact space. In the third chapter, we extend Buzjakovas's method to prove that for a Tychonoff space X, there exists an ordinal such that if the product of X and that ordinal condenses onto a normal space, then X condenses onto a countably paracompact space.
2009-07-30T04:13:10Z
2009-07-30T04:13:10Z
2009-04-23
2009
Dissertation
http://dissertations.umi.com/ku:10297
http://hdl.handle.net/1808/5320
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/188132017-12-08T21:31:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Hessians and Steinerians of plane quartic curves
Nelson, Cyril Arthur
Thesis (M.A.)--University of Kansas, Mathematics, 1916. ; Includes bibliographical references.
2015-11-03T14:49:08Z
2015-11-03T14:49:08Z
1916
Thesis
http://hdl.handle.net/1808/18813
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/191292017-12-08T21:34:34Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Inversion of integrals of the first kind for hyperelliptic curves
Bamberger, Bernice Fay
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2015-12-04T15:10:05Z
2015-12-04T15:10:05Z
1922
Thesis
http://hdl.handle.net/1808/19129
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/267492021-04-21T21:03:17Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
A reduced system of differential equations for the invariants of ternary forms
Black, Florence
Dissertation (Ph.D.)--University of Kansas, Mathematics, 1926.
2018-09-26T15:27:56Z
2018-09-26T15:27:56Z
1926
Dissertation
http://hdl.handle.net/1808/26749
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/185592017-12-08T21:31:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On the projective differential geometry of cubic ruled surfaces
Dueker, Ottilia Wilhelmina
Thesis (M.A.)--University of Kansas, Mathematics, 1915. ; Includes bibliographical references.
2015-10-06T17:57:58Z
2015-10-06T17:57:58Z
1915
Thesis
http://hdl.handle.net/1808/18559
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/108082020-09-22T14:49:12Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Super-Stretched and Countable Cohen-Macaulay Type
Stone, Branden
Huneke, Craig
Katz, Daniel
Dao, Hailong
Purnaprajna, Bangere
Agah, Arvin
Mathematics
Countable type
Maximal cohen macaulay
Super-stretched
This dissertation defines what it means for a Cohen-Macaulay ring to to be super-stretched. In particular, a super-stretched Cohen-Macaulay ring of positive dimension has h-vector (1), (1,n), or (1,n,1). It is shown that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In Chapter 1, some background material is given along with some preliminary definitions. This chapter defines what it means to be stretched and super-stretched. The chapter ends by showing a couple of scenarios when these two notions coincide. Chapter 2 deals with super-stretched rings that are standard graded. We begin the chapter by exploring the graded category and defining what it means to be graded countable Cohen-Macaulay type. Equivalent characterizations of super-stretched are discovered and it is shown that rings of graded countable Cohen-Macaulay type are super-stretched. The chapter ends by analyzing standard graded rings that are super-stretched with minimal multiplicity. In Chapter 3, we examine what it means for a local ring to be super-stretched. Finally, Chapter 4 uses the previous results to give a partial answer to the following question: Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, it is shown there is a positive answer when the ring is not Gorenstein. Throughout the chapter, many different cases of graded countable Cohen-Macaulay type are classified. Further, the Gorenstein case is studied is shown to be helpful in giving support to the following folklore conjecture: a Gorenstein ring of countable Cohen-Macaulay representation type is a hypersurface. It is shown that the conjecture holds for one dimensional standard graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type.
2013-02-17T15:42:24Z
2013-02-17T15:42:24Z
2012-08-12
2012
Dissertation
http://dissertations.umi.com/ku:12279
http://hdl.handle.net/1808/10808
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/213722017-12-08T21:40:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
The function H(z)
Hattan, Corinne
Thesis (M.A.)--University of Kansas, Mathematics, 1928.
2016-08-23T17:10:53Z
2016-08-23T17:10:53Z
1928
Thesis
http://hdl.handle.net/1808/21372
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/162022017-12-08T21:46:53Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
The Arbelos
Welch, Gertrude
Submitted to the Department of Mathematics and the Faculty of the Graduate School of the University of Kansas in partial fulfillment of the requirements for the degree of Master of Arts.
2015-01-07T18:54:13Z
2015-01-07T18:54:13Z
1949-08-01
Thesis
Welch, Gertrude. "The Arbelos" University of Kansas. August, 1949.
http://hdl.handle.net/1808/16202
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/298862021-03-05T16:53:56Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Surface and bulk moving mesh methods based on equidistribution and alignment
Kolasinski, Avary Justice
Huang, Weizhang
Gavosto, Estela
Miedlar, Agnieszka
Shontz, Suzanne
Van Vleck, Erik
Applied mathematics
Mathematics
Discretization
Mesh adaptation
Numerical analysis
Partial differential equations
Scientific computing
Surface moving mesh methods
In this dissertation, we first present a new functional for variational mesh generation and adaptation that is formulated by combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive which, when employed with the moving mesh partial differential equation method, allows various theoretical properties to be proved. Numerical examples for bulk meshes demonstrate that the new functional performs comparably to a similar existing functional that is known to work well but contains an additional parameter. Variational mesh adaptation for bulk meshes has been well developed however, surface moving mesh methods are limited. Here, we present a surface moving mesh method for general surfaces with or without explicit parameterization. The development starts with formulating the equidistribution and alignment conditions for surface meshes from which, we establish a meshing energy functional. The moving mesh equation is then defined as the gradient system of the energy functional, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. A selection of two-dimensional and three-dimensional examples are presented.
2020-01-17T22:37:35Z
2020-01-17T22:37:35Z
2019-05-31
2019
Dissertation
http://dissertations.umi.com/ku:16491
http://hdl.handle.net/1808/29886
https://orcid.org/0000-0002-7968-2635
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/111662020-09-29T13:35:42Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Quotients of Metric Spaces
Herman, Robert A.
C. J. Himmelberg
The question of when a given property of a topological space is preserved under mappings is one of the most familiar problems of general topology. Among the properties of greatest interest for general spaces is, without doubt, that of metrizability; metrizability always implies, in particular, a number of important special topological properties of the space in question (normality, regularity, paracompactness, etc.). To determine in general the conditions for preservation of metrizability under mappings appears to be a difficult problem. It may well be that in the class of arbitrary continuous mappings the problem has no meaningful solution. The purpose of this paper is to obtain conditions for the preservation of metrizability under mappings appears to be a difficult problem. It may well be that in the class of arbitrary continuous mappings the problem has no meaningful solution. The purpose of this paper is to obtain conditions for the preservation of metrizability by quotient mappings and to study the properties of quotient spaces of metric spaces. We will use "iff" as an abbreviation for "if and only if". If f is a function from X onto Y, we will write f: X --->> Y.
2013-05-20T18:21:14Z
2013-05-20T18:21:14Z
1968
Thesis
http://hdl.handle.net/1808/11166
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
The University of Kansas
oai:kuscholarworks.ku.edu:1808/53322020-06-30T00:44:54Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
A STRUCTURED METHOD FOR THE REAL QUADRATIC EIGENVALUE PROBLEM FOR SPECIFIC GYROSCOPIC SYSTEMS
Rush, Wade Drury
Xu, Hongguo
Van Vleck, Erik
Huang, Weizhang
Mathematics
Cholesky
Eigenvalue
Givens
Gyroscopic
Quadratic
Skew-symmetric
This study examines a specific numerical approach that computes the eigenvalues (normal modes) of a Quadratic Eigenvalue Problem (QEP) of the form (&lambda^2 & middotI + &lambda · B + C)· x = 0 where B is constrained to a real skew-symmetric matrix and C is constrained to a real symmetric positive definite matrix. A widely used linearization of this QEP is the companion matrix A which is an 2n-by-2n matrix such that (1,1) block is a n-by-n skew symmetric matrix, the (1,2) block is an n-by-n symmetric positive definite matrix, (2,1) block is the Identity matrix and finally the (2,2) zero block.. The goal is to find an algorithm method which diagonalizes matrix A without contaminating the (2,2) zero block. Once this algorithm is developed, the study measures the eigenvalue error bounds and compare its efficiency to the standard symmetric QR workhorse. Also, this approach preserves the structure of the error matrix in the same form as the QEP. In ensuring that the error matrix structure is a QEP, this algorithm provides fertile ground for future analysis in sensitivity and perturbation errors in the algorithm's eigenvalues. This study concludes that the algorithm appears to have a reasonable error bound; and it is more cost efficient in finding the eigenvalues then the symmetric QR algorithm.
2009-07-30T19:48:28Z
2009-07-30T19:48:28Z
2008-12-15
2008
Thesis
http://dissertations.umi.com/ku:10073
http://hdl.handle.net/1808/5332
https://orcid.org/0000-0002-1839-8921
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/190252018-01-31T20:07:50Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control
Clifton, Cody Edward
Pasik-Duncan, Bozenna
Duncan, Tyrone E.
Evans, Joseph B.
Talata, Zsolt
Tu, Xuemin
Mathematics
Applied mathematics
Adaptive Control
PageRank
Parameter Estimation
Stochastic System
A key feature of modern web search engines is the ability to display relevant and reputable pages near the top of the list of query results. The PageRank algorithm provides one way of achieving such a useful hierarchical indexing by assigning a measure of relative importance, called the PageRank value, to each webpage. PageRank is motivated by the inherently hypertextual structure of the World Wide Web; specifically, the idea that pages with more incoming hyperlinks should be considered more popular and that popular pages should rank highly in search results, all other factors being equal. We begin by overviewing the original PageRank algorithm and discussing subsequent developments in the mathematical theory of PageRank. We focus on important contributions to improving the quality of rankings via topic-dependent or "personalized" PageRank, as well as techniques for improving the efficiency of PageRank computation based on Monte Carlo methods, extrapolation and adaptive methods, and aggregation methods We next present a model for PageRank whose dynamics are described by a controlled stochastic system that depends on an unknown parameter. The fact that the value of the parameter is unknown implies that the system is unknown. We establish strong consistency of a least squares estimator for the parameter. Furthermore, motivated by recent work on distributed randomized methods for PageRank computation, we show that the least squares estimator remains strongly consistent within a distributed framework. Finally, we consider the problem of controlling the stochastic system model for PageRank. Under various cost criteria, we use the least squares estimates of the unknown parameter to iteratively construct an adaptive control policy whose performance, according to the long-run average cost, is equivalent to the optimal stationary control that would be used if we had knowledge of the true value of the parameter. This research lays a foundation for future work in a number of areas, including testing the estimation and control procedures on real data or larger scale simulation models, considering more general parameter estimation methods such as weighted least squares, and introducing other types of control policies.
2015-12-03T00:03:30Z
2015-12-03T00:03:30Z
2015-05-31
2015
Dissertation
http://dissertations.umi.com/ku:14020
http://hdl.handle.net/1808/19025
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/84462020-08-26T14:43:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Foundations of Arithmetic
Frizell, Arthur Bowes
2011-11-22T17:28:59Z
2011-11-22T17:28:59Z
1910
Thesis
http://hdl.handle.net/1808/8446
en_US
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/270182018-10-25T20:40:51Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Riemannian Geometry on Some Noncommutative Spaces
Chen, Wei-Da
Sheu, Albert J.-L.
Kong, Man Cheong
Shao, Shuanglin
Stefanov, Atanas G.
Torres, Rodolfo H.
Mathematics
Chern-Gauß-Bonnet Theorem
Levi-Civita Connections
Noncommutative Tori
Quantum Discs
Quantum Spheres
Riemann Curvatures
This dissertation enquires into how the theory and mechanism of Riemannian geometry can be introduced into and integrated with the existent ones in noncommutative geometry, a branch of mathematics inspired by the development of quantum physics that concentrates on C*-algebras and related research. In conformity with the Gelfand duality, a cornerstone theorem in noncommutative geometry that establishes a one-to-one correspondence between commutative C*-algebras and locally compact Hausdorff spaces, it is suggested that a noncommutative C*-algebra notionally be deemed a "virtual noncommutative space". Based on this ideology are some forms of Riemannian geometry anticipated to reincarnate on C*-algebras. J. Rosenberg demonstrated such a reincarnation on noncommutative tori. Especially, a corresponding adaptation of the Fundamental Theorem of Riemannian Geometry was attained. Moreover, based on this adaptation, he established a variant of the Gauß-Bonnet Theorem for noncommutative 2-tori. M. A. Peterka and A. J.-L. Sheu subsequently presented extensions and generalisations to the framework developed by Rosenberg. Specifically, an enhanced Gauß-Bonnet Theorem was substantiated for noncommutative 2-tori. In this dissertation, we shall first tender results that are closely related to the aforementioned work on noncommutative tori, proposing several extensions of the two Gauß-Bonnet Theorems already obtained for noncommutative 2-tori and exhibiting extensions of the theorem for two special cases on noncommutative 4-tori. Thereafter, we shall transcribe Rosenberg's framework and results for quantum discs and 2-spheres with a version of the Fundamental Theorem proved. Finally, an asymptotic behaviour of the total curvature will be demonstrated for quantum complex projective lines as an illustrative example.
2018-10-24T22:35:15Z
2018-10-24T22:35:15Z
2017-12-31
2017
Dissertation
http://dissertations.umi.com/ku:15660
http://hdl.handle.net/1808/27018
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/262752018-03-31T08:01:40Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
A study of analytic surfaces by means of a projective theory of envelopes
Greer, Edison
Dissertation (Ph.D.)--University of Kansas, Mathematics, 1946.
2018-03-30T17:02:25Z
2018-03-30T17:02:25Z
1946
Dissertation
http://hdl.handle.net/1808/26275
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/299012021-03-05T16:53:01Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Dynamics of Essentially Unstable Nonlinear Waves
Smith, Connor Yoshio
Johnson, Mathew
Stanislavova, Milena
Liu, Weishi
Mantzavinos, Dionyssios
Lamb, Jonathan P
Mathematics
ad-hoc periodic wave train
Bloch transform
modified Kuramoto-Sivashinsky
St. Venant equations
Unstable essential spectrum
In this thesis we primarily consider the stability of traveling wave solutions to a modified Kuramoto-Sivashinsky Equation equation modeling nanoscale pattern formation and the St. Venant equations modeling shallow water flow down an inclined plane. Numerical evidence suggests that these equations have no unstable spectrum other than λ =0, however they both have unstable essential spectrum. This unstable essential spectrum manifests as a convecting, oscillating perturbation which grows to a certain size independent on the initial perturbation — precluding stability in the regular L^2(R) space. Exponentially weighted spaces are typically used to handle such instabilities, and in Theorem 5.7 we prove asymptotic orbital linear stability in such an exponentially weighted space. We also discuss difficulties with extending this to a nonlinear stability result. In Section 5.5 we discuss another way of obtaining stability, through ad-hoc periodic wave trains. Chapter 6 concerns the general problem of creating a spectral projection to project away unstable essential spectrum. We consider this problem in the context of spatially periodic-coefficient PDE by proposing a candidate spectral projection defined via the Bloch transform and showing that initial perturbations which activate a sufficiently unstable part of the essential spectrum lead to solutions which are not Lyapunov stable. We also extend these results to dissipative systems of conservation laws. Additional chapters of interest are Chapter 3 where we address finding the spectrum and Chapter 4 where we discuss the numerics which lead to many of the figures in this thesis.
2020-01-17T23:14:54Z
2020-01-17T23:14:54Z
2019-05-31
2019
Dissertation
http://dissertations.umi.com/ku:16520
http://hdl.handle.net/1808/29901
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218002018-01-31T20:07:51Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
The Lattice of Compactifications of a Locally Compact Space
Eichelberger, Luke Alan
Porter, Jack
2017/10/04: The ETD release form is attached to this record as a license file.
Gavosto, Estela A
Katz, Daniel
Mathematics
Beta-Families
Compactifications
Hausdorff Compactifications
Locally Compact
Topology
This is an expanded version of [5] by Magill. The results of [5] are proven with greater detail and any result stated in [5] but not proven is proven here. Let K (X) and K (Y) be used to indicate the lattice of Hausdorff compactifications of locally compact, non-compact spaces X and Y with X and Y Tychonoff. This paper primarily concerns how a lattice isomorphism between K (X) and K (Y) exists if and only if a homeomorphism between particular extensions of X and Y exists with specified properties. On the way to proving the main results, we prove several lemmas about β − families of compact extensions of Tychonoff spaces. Some of the Lemmas slightly generalize corresponding lemmas in [5]. Efforts are made to make this paper self- contained.
2016-11-03T23:01:04Z
2016-11-03T23:01:04Z
2016-05-31
2016
Thesis
http://dissertations.umi.com/ku:14653
http://hdl.handle.net/1808/21800
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/313182021-03-05T16:53:01Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models
Hadadifard, Fazel
Stefanov, Atanas
Stefanov, Atanas
Torres, Rodolfo
Stanislavova, Milena
Johnson, Mathew
Ackley, Brian
Applied mathematics
Mathematics
Asymptotic profile
Boussinesq system
Long time behavior
Quasi-geostrophic
Reaction diffusion model
Through this dissertation we present the sharp time decay rates for three equations, namely quasi--geostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reaction-diffusion models. In addition, in each case, we provide the dominant part of the solution which leads to the long term asymptotic profiles of each model. The first two equations, arising in fluid dynamics, model some aspect of the shallow waters with horizontal and vertical structures. Indeed, quasi--geostrophis equation models the horizontal inertia forces of a flow. As a result of that, atmospheric and oceanographic flows which take place over horizontal length scales, which are very large compare to their vertical length scales, are studied by SQG equation. On the other hand BSQ system models some vertical aspect of the flow, namely the speed, pressure and the temperature of the flow. In coastal engineering, BSQ type equations have a vast application in computer modeling. Lastly, a plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector. In order to study these equations, we made some developments in the "scaling variable" methods, so that it fits over models. In particular, we now have a good understanding of this method when it is applied to the equations with fractional dissipations.
2021-02-02T16:57:42Z
2021-02-02T16:57:42Z
2019-5-31
2019
Dissertation
http://dissertations.umi.com/ku:16686
http://hdl.handle.net/1808/31318
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/268842018-10-20T08:03:01Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Generalized vectors and the matrix equation x([K])=B
Foreman, William Calvin
Dissertation (Ph.D.)--University of Kansas, Mathematics, 1952.
2018-10-19T19:14:58Z
2018-10-19T19:14:58Z
1952
Dissertation
http://hdl.handle.net/1808/26884
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/210922020-06-30T01:04:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
A study of a Riemann surface
Shoemaker, Violet Elizabeth
Thesis (M.A.)--University of Kansas, Mathematics, 1926.
2016-07-13T13:36:30Z
2016-07-13T13:36:30Z
1926
Thesis
http://hdl.handle.net/1808/21092
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/210882017-12-08T21:40:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Projective differential geometry of space curves
Senor, Muriel Elizabeth
Thesis (M.A.)--University of Kansas, Mathematics, 1928.
2016-07-13T13:36:24Z
2016-07-13T13:36:24Z
1928
Thesis
http://hdl.handle.net/1808/21088
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/210372019-04-11T19:30:49Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Monomial groups
Crouch, Ralph
Dissertation (Ph. D.)--University of Kansas, Mathematics, 1954.
2016-06-29T17:17:04Z
2016-06-29T17:17:04Z
1954
Dissertation
http://hdl.handle.net/1808/21037
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218662018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Numerical solutions of rough differential equations and stochastic differential equations
Liu, Yanghui
Nualart, David
Hu, Yaozhong
Nualart, David
Hu, Yaozhong
Feng, Jin
Tu, Xuemin
Zhang, Jianbo
Soo, Terry
Mathematics
fourth moment theorem
fractional Brownian motions
Numerical solutions
rough differential equations
stochastic differential equations
In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations (SDE) driven by fractional Brownian motions (fBm). The dissertation is organized as follows. In Chapter 1, we introduce the basic settings and define time-discrete numerical approximation schemes. In Chapter 2, we consider the Euler scheme for SDEs driven by fBms. For a SDE driven by a fBm with Hurst parameter $H> \frac12$ it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H \rightarrow \frac12$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate of the naive Euler scheme deteriorates for $H \rightarrow \frac12$. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has the rate of convergence $\gamma_n^{-1}$, where $ \gamma_n=n^{ 2H-\frac12}$ when $H \frac12$ it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H \rightarrow \frac12$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate of the naive Euler scheme deteriorates for $H \rightarrow \frac12$. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has the rate of convergence $\gamma_n^{-1}$, where $ \gamma_n=n^{ 2H-\frac12}$ when $H \frac34$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t, 0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n, 0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $ \gamma_n (X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in ( \frac12, \frac34]$. In the case $H \frac 34$, we show the $L^p$ convergence of $n(X^n_t-X_t)$ and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme. In Chapter 3, we consider the Crank-Nicolson method for a SDE driven by a $m$-dimensional fBm. We consider the Crank-Nicolson method in three cases: (i) $m1$; (ii) $m=1$ and and the drift term is equal to non-zero; and (iii) $m=1$ and the drift term is equal to zero. We will show that the convergence rate of the Crank-Nicolson method is $n^{ 1/2-2H}$, $n^{-1/2-H}$ and $n^{-2H}$, respectively, in these three cases, and these convergence rates are exact in the sense that the error process for the Crank-Nicolson method converges to the solution of a linear SDE. Our main tools are the fractional calculus and the fourth moment theorem. In Chapter 4, we study two variations of the time-discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the H\"older exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Our explicit expressions of the convergence rates allow us to compare different incomplete Taylor schemes, and then help us construct the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.
2016-11-10T22:42:56Z
2016-11-10T22:42:56Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14642
http://hdl.handle.net/1808/21866
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/220772020-06-23T20:20:53Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
On summation of finite series
Mallonee, Pauline Virginia
Thesis (M.A.)--University of Kansas, Mathematics, 1929.
2016-11-30T14:14:42Z
2016-11-30T14:14:42Z
1929
Thesis
http://hdl.handle.net/1808/22077
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218982018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Some Studies on Parameter Estimations
Su, Chen
Hu, Yaozhong
Tu, Xuemin
Nualart, David
Soo, Terry
Zhang, Jianbo
Mathematics
Bayesian methods
implicit sampling
inverse problems
maximum likelihood estimator
stochastic differential equations
Parameter estimation has wide applications in such fields as finance, biological science, weather prediction, oil deposit detection, etc. Researchers are particularly interested in reconstructing some unknown parameters from the observed data set which may be sparse and noisy. This is a typical inverse problem which tends to be ill-conditioned in many cases. A plethora of literature has been devoted to this area and there has been a concrete progress recently in the design of more efficient estimation techniques. Depending on the model (usually an equation or a system of equations) we choose, we divide the estimation into two categories: stochastic and deterministic parameter estimations. The former involves a stochastic system, usually a stochastic differential equation (SDE) or system of SDEs where unknown parameters are present. A standard estimator to use for stochastic parameter estimation problems is the maximum likelihood estimator (MLE), since it, in many situations, enjoys desirable properties such as consistency and asymptotic normality. One major obstacle in obtaining the MLE is that the transition density of SDE, which is essential for deriving MLE, is often not available. To address this issue, various approximations of transition density was introduced in the past decades. In chapter 1, we will present several popular density approximation schemes including Euler-Maruyama methods and Hermite expansions. We will also introduce the parametrix approximation in which we derive a point-wise approximation of the transition density that is uniform in the parameter. As a consequence, the approximated MLE from parametrix method will eventually converge to the true MLE so that those desired properties of MLE can be preserved. We will see some applications of the parametrix approximation and some necessary preliminaries regarding the ergodicity of SDEs and the consistency of the estimators will also be presented. The deterministic parameter estimation involves a partial differential equation (PDE) or a system of PDEs. A key feature for this type of estimation is the high level of uncertainty for recovering the parameters, i.e. different choices of parameters may all yield reasonable explanation of the data. This is a typical feature for many ill-conditioned inverse problems. The Bayesian inference formulation provides a systematic way to characterize this uncertainty. It incorporates a prior, which is from the historical data before any experiment is done, and a likelihood, which measures how likely the data will be provided that certain parameter value is chosen, to form a posterior density. It generates a neat solution which takes the form of a posterior probability density. However, how to interpret this posterior density is a non-trivial task since the forward model may be very expensive and the discretized parameter field may result in a high dimensional density. As a consequence, efficient sampling techniques are called for to better characterize the posterior. In chapter 2, we will introduce some traditional sampling methods such as Gaussian approximations, MCMC and importance sampling. We also introduce our implicit sampling methods together with its sequential implementation. We will apply these methods to a seismic wave inversion problem where a detailed comparison among other methods demonstrates a clear superiority of our implicit sampling method. Finally in chapter 3, we will give some concluding remarks and point out possible future work.
2016-11-10T23:27:28Z
2016-11-10T23:27:28Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14630
http://hdl.handle.net/1808/21898
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/208832017-12-08T21:38:01Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Distribution of prime points in certain modular spaces
Reagan, Charles Arthur
Thesis (M.A.)--University of Kansas, Mathematics, 1926.
2016-05-31T13:42:41Z
2016-05-31T13:42:41Z
1926
Thesis
http://hdl.handle.net/1808/20883
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/183452020-06-24T20:04:17Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
The Riemann surface for the function W²=Z³+3tz+2
Jacobs, Jessie Marie
Thesis (M.A.)--University of Kansas, Mathematics, 1916. ; Includes bibliographical references.
2015-08-19T21:02:55Z
2015-08-19T21:02:55Z
1916
Thesis
http://hdl.handle.net/1808/18345
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/191762018-10-29T15:55:12Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Nonlinear Integrals, Diffusion in Random Environments and Stochastic Partial Differential Equations
Le, Khoa Nguyen
Hu, Yaozhong
Feng, Jin
Nualart, David
Stefanov, Atanas
Zhang, Jianbo
Mathematics
Feynman-Kac formula
Garsia-Rodemich-Rumsey inequality
random environment
stochastic partial differential equation
transport differential equation
young integration
In this dissertation, we investigate various problems in the analysis of stochastic (partial) differential equations. A part of the dissertation introduces several notions of nonlinear integrations. Some differential equations associated with nonlinear integrations are investigated. Examples include transport differential equations in space-time random fields and parabolic equations with potentials of the type $\partial_t W$, where $W$ is continuous in time variable and smooth in the spatial variables. Another part of the dissertation studies nonlinear stochastic convolution equations driven by a multiplicative Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ in the spatial variable. The other part of the dissertation gives rigorous meaning to the Brox differential equation $X(t)=\cB(t)-\frac12\int_0^t \dot{W}(X(s))ds$ where $\cB$ and $W$ are independent Brownian motions. Furthermore, it is shown that the Brox differential equation has a unique strong solution which is a time-changed spatial transformation of a Brownian motion. Along the way, some appropriate tools are developed in order to solve these problems. In particular, we establish a multiparameter version of Garsia-Rodemich-Rumsey inequality which allows one to control rectangular increments in any dimensions of multivariate functions, definitions and compact criteria for some new functions spaces are developed. The methodologies employed form a combination of stochastic analysis, Malliavin calculus and functional analytic tools. Several parts of the dissertation are joint work of the author with Yaozhong Hu, Jingyu Huang, David Nualart, Leonid Mytnik and Samy Tindel.
2015-12-11T23:09:04Z
2015-12-11T23:09:04Z
2015-08-31
2015
Dissertation
http://dissertations.umi.com/ku:14132
http://hdl.handle.net/1808/19176
https://orcid.org/0000-0002-7654-7139
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/123292020-10-06T13:37:25Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Bilinear Littlewood-Paley Square Functions and Singular Integrals
Hart, Jarod Victor
Torres, Rodolfo
Torres, Rodolfo
Gavosto, Estela
Orr, James
Shao, Shuanglin
Stefenov, Atanas
Mathematics
Bilinear
Calder´on- Zygmund theory
Harmonic analysis
Littlewood-paley
Singular integral
In this dissertation we further develop the bilinear theory of vector valued Calderoón-Zygmund operators, Littlewood-Paley square functions, and singu- lar integral operators. These areas of harmonic analysis are motivated by po- tential theory, boundary value problems in partial differential equations, har- monic and analytic extension problems in complex analysis, and many other classical problems in analysis. Multilinear operator theory addresses difficul- ties that arise from product type operations in harmonic analysis. We first introduce Banach valued Calderoón-Zygmund operators in a bilinear setting, and prove weak endpoint estimates and interpolation results for them. By viewing Littlewood-Paley square functions as Calderoón-Zygmund operators taking values in a particular Banach space, we are able to obtain bounds of the square functions on product Lebesgue spaces for a complete set of in- dices. We give an in depth analysis of Littlewood-Paley square functions, which includes estimates on some products of smooth function spaces as well as the estimates on product Lebesgue spaces that are needed to apply the vec- tor valued Calderoón-Zygmund results. Finally, we prove boundedness criteria for a certain class of bilinear singular integral operators on product Lebesgue spaces using Littlewood-Paley square function techniques. We provide a new proof of the bilinear T1 theorem that does not rely on the linear version of the result. We also prove a bilinear Tb theorem, a result missing in the theory so far. The Littlewood-Paley square function techniques developed in this work are a powerful tool has potential to solve problems in areas like oscillatory integral operator theory, multiparameter operator theory, Fourier restriction, and non-linear partial differential equations.
2013-09-30T20:08:34Z
2013-09-30T20:08:34Z
2013-05-31
2013
Dissertation
http://dissertations.umi.com/ku:12720
http://hdl.handle.net/1808/12329
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/218022018-01-31T20:07:47Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Three Dimensional Jacobian Derivations And Divisor Class Groups
Alkarni, Shalan
Lang, Jeffrey
Mandal, Satyagopal
Purnaprajna, Bangere
Jiang, Yunfeng
Brinton, Jacquelene
Mathematics
Algebra
Algebraic Geometry
Class Groups
Commutative Algebra
Divisors
Group of Logarithmic Derivatives
In this thesis, we use P. Samuel's purely inseparable descent methods to investigate the divisor class groups of the intersections of pairs of hypersurfaces of the form $w_1^p=f$, $w_2^p=g$ in affine $5$-space with $f$, $g$ in $A=k[x,y,z]$; $k$ is an algebraically closed field of characteristic $p$ $$ $0$. This corresponds to studying the divisor class group of the kernels of three dimensional Jacobian derivations on $A$ that are regular in codimension one. Our computations focus primarily on pairs where $f$, $g$ are quadratic forms. We find results concerning the order and the type of these groups. We show that the divisor class group is a direct sum of up to three copies of $\mathbb{Z}_p$, is never trivial, and is generated by those hyperplane sections whose forms are factors of linear combinations of $f$ and $g$.
2016-11-03T23:08:42Z
2016-11-03T23:08:42Z
2016-05-31
2016
Dissertation
http://dissertations.umi.com/ku:14635
http://hdl.handle.net/1808/21802
en
openAccess
Copyright held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/188232017-12-08T21:34:34Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
Linear complexes and congruences in modular geometry
Strickler, Lillian Ruth
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2015-11-03T14:49:22Z
2015-11-03T14:49:22Z
1922
Thesis
http://hdl.handle.net/1808/18823
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/78302020-08-12T13:44:02Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Identification and Adaptive Control Methods for Some Stochastic Systems
Zachariou, Ioannis
Pasik-Duncan, Bozenna
Duncan, Tyrone E.
Porter, Jack
Talata, Zsolt
Agah, Arvin
Mathematics
This dissertation is focused on the identification and adaptive control of some stochastic systems. Initially a survey of some adaptive control problems for both discrete and continuous time stochastic systems is provided. Discrete time branching processes are described and some results on parameter estimation and adaptive control for these processes are reviewed. Then continuous time branching processes are introduced and the main results in this dissertation concerning estimation and adaptive control are given. The family of estimators is shown to be strongly consistent and the optimal rate of convergence of this family of estimators is obtained. Furthermore some other asymptotic properties of these estimators are verified. An adaptive control is given that posses self-tuning property. It is shown that it does not achieve the optimal asymptotic cost for the known system. Finally some computational methods and simulations are given for a variety of stochastic differential equations driven by a Brownian motion or an arbitrary fractional Brownian motion and computational properties of the parameter estimates for the branching processes are given.
2011-08-02T12:36:25Z
2011-08-02T12:36:25Z
2011-02-21
2011
Dissertation
http://dissertations.umi.com/ku:11316
http://hdl.handle.net/1808/7830
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/181212020-06-24T16:17:05Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
A special Riemann surface with application to the hyperelliptic case
Conwell, Herman Henry
Thesis (M.A.)--University of Kansas, Mathematics, 1915. ; Includes bibliographical references.
2015-06-18T19:54:39Z
2015-06-18T19:54:39Z
1915
Thesis
http://hdl.handle.net/1808/18121
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/214512017-12-08T21:40:50Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Duality in Euclidean geometry
Reagan, Lewis Martin
Thesis (M.A.)--University of Kansas, Mathematics, 1928.
2016-09-07T13:17:23Z
2016-09-07T13:17:23Z
1928
Thesis
http://hdl.handle.net/1808/21451
openAccess
This work is in the public domain and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/129772020-10-21T13:22:08Zcom_1808_263com_1808_1260col_1808_14149col_1808_1951
Finding Eigenvalues of Unitary Matrices
Gu, Peidi
Xu, Hongguo
Xu, Hongguo
Tu, Xuemin
Van Vleck, Erik
Mathematics
Eigenvalue
Givens matrix
Householder reflector
Iteration
Schur parameter form
Unitary matrix
The study introduces methods of finding eigenvalues for unitary matrices and pencils. Bunse-Gerstner and Elsner ([2]) proposed an algorithm of using the Schur parameter pencil to solve eigenproblems for unitary matrices and pencils. This thesis reviews the Schur parameter pencil algorithm. The method is divided into two phases: Reducing a unitary pencil to a Schur parameter form and QR-type shifted iteration. The algorithm is proved to be backward stable and more efficient than the standard QR/QZ algorithm. However, during the process of reduction, norms of vectors are frequently compared for numerical stability, which causes a lot of extra work for computations. Based on the idea in [8], we introduce a modified Schur parameter algorithm to avoid such frequent comparison. The modified algorithm is still divided into two phases similar to the one in [2]. A detailed reduction process and shifted iteration are described in this thesis.
2014-02-05T16:33:25Z
2014-02-05T16:33:25Z
2013-12-31
2013
Thesis
http://dissertations.umi.com/ku:13159
http://hdl.handle.net/1808/12977
en
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
oai:kuscholarworks.ku.edu:1808/211802017-12-08T21:45:28Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
The continuity of integral transformations with positive kernels between L(p) spaces with mixed norms
Taylor, Howard L.
Dissertation (Ph.D.)--University of Kansas, Mathematics, 1968.
2016-07-22T19:44:41Z
2016-07-22T19:44:41Z
1968
Dissertation
http://hdl.handle.net/1808/21180
openAccess
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University of Kansas
oai:kuscholarworks.ku.edu:1808/183582020-06-24T18:53:47Zcom_1808_263com_1808_1260col_1808_14149col_1808_7158
On coaxial minors of determinants
Babcock, Wealthy Consuelo
Thesis (M.A.)--University of Kansas, Mathematics, 1922. ; Includes bibliographical references.
2015-08-19T21:03:13Z
2015-08-19T21:03:13Z
1922
Thesis
http://hdl.handle.net/1808/18358
openAccess
This work is in the public domain according to U.S. copyright law and is available for users to copy, use, and redistribute in part or in whole. No known restrictions apply to the work.
University of Kansas
oai:kuscholarworks.ku.edu:1808/56502020-07-28T12:26:37Zcom_1808_1260com_1808_263col_1808_1952col_1808_14149
Approximating Artinian Rings by Gorenstein Rings and 3-Standardness of the Maximal Ideal
Hariharan, Ananthnarayan
Huneke, Craig
Bayer, Margaret
Bricke, John
Katz, Daniel
Mandal, Satyagopal
Mathematics
Gorenstein colength
N-standardness
We study two different problems in this dissertation. In the first part, we wish to understand how one can approximate an Artinian local ring by a Gorenstein Artin local ring. We make this notion precise in Chapter 2, by introducing a number associated to an Artin local ring, called its Gorenstein colength . We study the basic properties and give bounds on this number in this chapter. We extend results due to W. Teter, C.Huneke and A. Vraciu by studying the relation of Gorenstein colength with self-dual ideals. In particular, we also answer the question as to when the Gorenstein colength is at most two. In Chapter 3, we show that there is a natural upper bound for Gorenstein colength of some special rings. We compute the Gorenstein colengths of these rings by constructing some Gorenstein Artin rings. We further show that the Gorenstein colength of Artinian quotients of two-dimensional regular local rings are also bounded above by the same upper bound by using a formula due to Hoskin and Deligne. Given two Gorenstein Artin local rings, L. Avramov and W. F. Moore construct another Gorenstein Artin local ring called a connected sum. We use this to improve a result of C. Huneke and A. Vraciu in Chapter 4. We also define the notion of a connected sum more generally and apply it to give bounds on the Gorenstein colengths of fibre products of Artinian local rings. In the second part of the thesis, we study a notion called n-standardness of ideals primary to the maximal ideal in a Cohen-Macaulay local ring. We first prove the equivalence of n-standardness to the vanishing of a certain Koszul homology module up to a certain degree. We go over the properties of Koszul complexes and homology needed for this purpose in Chapter 5. In Chapter 6, we study conditions under which the maximal ideal is 3-standard. We first prove results when the residue field is of prime characteristic and use the method of reduction to prime characteristic to extend the results to the characteristic zero case. As an application, we see that this helps us extend a result due to T. Puthenpurakal in which he shows that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.
2010-01-07T17:55:44Z
2010-01-07T17:55:44Z
2009-07-23
2009
Dissertation
http://dissertations.umi.com/ku:10489
http://hdl.handle.net/1808/5650
EN
openAccess
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
University of Kansas
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