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1.
Merriman, Claire.
Geometric and ergodic properties of certain classes of continued fractions.
Degree: PhD, Mathematics, 2019, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/105632 
► Continued fractions are systematically studied in number theory, dynamical systems, and ergodic theory, and appear frequently in other areas of mathematics. The first part of…
(more)
▼ Continued fractions are systematically studied in number theory, dynamical systems, and ergodic theory, and appear frequently in other areas of mathematics.
The first part of this thesis extends the connection between the dynamics of regular continued fractions and the geodesics on the modular surface \operatorname{PSL}(2, ℤ)\backslashℍ to the odd and grotesque continued fractions and the even continued fractions, as well as the Lehner and Farey expansions. I describe the natural extension of the corresponding Gauss maps as cross sections of the geodesic flow on modular surfaces. For the odd and grotesque continued fractions, Γ is an index two subgroup of \operatorname{PSL}(2,\Z); for the even continued fractions, Γ is an index three subgroup; and for the Lehner and Farey expansions, Γ=\operatorname{PSL}(2,\Z).
Nakada’s α-expansions interpolate between the regular continued fractions (α=1), Hurwitz singular continued fractions (α=\frac{√{5}-1}{2}), and nearest integer continued fractions (α=\frac{1}{2}). The second part of this thesis introduces a new version of α-expansions where all partial quotients are odd integers. I provide an explicit description of the natural extension of the corresponding Gauss map for \frac{√{5}-1}{2} ≤ α ≤ \frac{√{5}+1}{2}, and investigate several of the ergodic properties of these maps.
Advisors/Committee Members: Champaign%22%20%2Bcontributor%3A%28%22Boca%2C%20Florin%20P%22%29&pagesize-30">
Boca,
Florin P (advisor),
Champaign%22%20%2Bcontributor%3A%28%22Leininger%2C%20Christopher%22%29&pagesize-30">Leininger, Christopher (Committee Chair),
Champaign%22%20%2Bcontributor%3A%28%22Tyson%2C%20Jeremy%22%29&pagesize-30">Tyson, Jeremy (committee member),
Champaign%22%20%2Bcontributor%3A%28%22Zaharescu%2C%20Alexandru%22%29&pagesize-30">Zaharescu, Alexandru (committee member).
Subjects/Keywords: ergodic theory; continued fractions; cutting sequence
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APA (6th Edition):
Merriman, C. (2019). Geometric and ergodic properties of certain classes of continued fractions. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/105632
Chicago Manual of Style (16th Edition):
Merriman, Claire. “Geometric and ergodic properties of certain classes of continued fractions.” 2019. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed March 06, 2021.
http://hdl.handle.net/2142/105632.
MLA Handbook (7th Edition):
Merriman, Claire. “Geometric and ergodic properties of certain classes of continued fractions.” 2019. Web. 06 Mar 2021.
Vancouver:
Merriman C. Geometric and ergodic properties of certain classes of continued fractions. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2019. [cited 2021 Mar 06].
Available from: http://hdl.handle.net/2142/105632.
Council of Science Editors:
Merriman C. Geometric and ergodic properties of certain classes of continued fractions. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2019. Available from: http://hdl.handle.net/2142/105632
2.
Gao, Li.
On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.
Degree: PhD, Mathematics, 2018, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/101546
► Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study…
(more)
▼ Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces.
After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions.
In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus – the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting.
Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.
Advisors/Committee Members: Champaign%22%20%2Bcontributor%3A%28%22Junge%2C%20Marius%22%29&pagesize-30">Junge, Marius (advisor),
Champaign%22%20%2Bcontributor%3A%28%22Ruan%2C%20Zhong-Jin%22%29&pagesize-30">Ruan, Zhong-Jin (Committee Chair),
Champaign%22%20%2Bcontributor%3A%28%22Boca%2C%20Florin%20P.%22%29&pagesize-30">Boca, Florin P. (committee member),
Champaign%22%20%2Bcontributor%3A%28%22Oikhberg%2C%20Timur%22%29&pagesize-30">Oikhberg, Timur (committee member).
Subjects/Keywords: Noncommutative Euclidean spaces; Moyal Deformation; Pseudo-differential operators
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APA ·
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MLA ·
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CSE |
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APA (6th Edition):
Gao, L. (2018). On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/101546
Chicago Manual of Style (16th Edition):
Gao, Li. “On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.” 2018. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed March 06, 2021.
http://hdl.handle.net/2142/101546.
MLA Handbook (7th Edition):
Gao, Li. “On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.” 2018. Web. 06 Mar 2021.
Vancouver:
Gao L. On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2018. [cited 2021 Mar 06].
Available from: http://hdl.handle.net/2142/101546.
Council of Science Editors:
Gao L. On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2018. Available from: http://hdl.handle.net/2142/101546
3.
Ekvittayaniphon, Sakulbuth.
On sequences related to binary partition function and the Thue-Morse sequence.
Degree: PhD, Mathematics, 2018, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/101563
► In this dissertation, we discuss properties of the family of sequences \mathbf{ud} = {ud(n)}n ≥ 0 for positive integer d. We define them by letting…
(more)
▼ In this dissertation, we discuss properties of the family of sequences \mathbf{u
d} = {u
d(n)}
n ≥ 0 for positive integer d. We define them by letting u
d(n) be the coefficient of X
n in \prod
j=0∞ ≤ ft( 1 - X
2j)) ≤ ft( 1 - X
d∙ 2j))
-1. First, we discuss the binary partition function and its relationship with the sequence \mathbf{u
d}. We then give several intermediate results and identities. Afterward, we generalize the sequence with different initial values. We also look at the corresponding generating function. After this, we focus on its asymptotic behavior by illustrating the cases when d=3,5,9. Finally, we explain asymptotic behavior for general cases and establish conjectures based on numerical data.
Then, we investigate another family of sequences, \mathbf{x
k} = {x
k(n)}
n ≥ 0, defined by x
k(n) = |t
n+k - t
n| where bf{t} = {t
n}
n ≥ 0 is the Thue-Morse sequence. We give the frequency of 1's and 0's of each sequence \mathbf{x
k} and express them in terms of recurrence relations. We note the similarity with the Stern sequence, denoted by bf{s} = {s(n)}
n ≥ 0 . Further, we investigate the frequency of appearances of 00, 01, 10, and 11 of each sequence.
Finally, we define the correlation function related to the sequence \mathbf{x
k}, denoted by f(d), and the associated density function ~{f}(d). We present both recurrence relations, and closed formulas for values of d near powers of 2.
Advisors/Committee Members: Champaign%22%20%2Bcontributor%3A%28%22Reznick%2C%20Bruce%22%29&pagesize-30">Reznick, Bruce (advisor),
Champaign%22%20%2Bcontributor%3A%28%22Hildebrand%2C%20A.J.%22%29&pagesize-30">Hildebrand, A.J. (Committee Chair),
Champaign%22%20%2Bcontributor%3A%28%22Berndt%2C%20Bruce%20C%22%29&pagesize-30">Berndt, Bruce C (committee member),
Champaign%22%20%2Bcontributor%3A%28%22Boca%2C%20Florin%20P%22%29&pagesize-30">Boca, Florin P (committee member).
Subjects/Keywords: Binary Partition; Function; Thue-Morse Sequence
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Ekvittayaniphon, S. (2018). On sequences related to binary partition function and the Thue-Morse sequence. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/101563
Chicago Manual of Style (16th Edition):
Ekvittayaniphon, Sakulbuth. “On sequences related to binary partition function and the Thue-Morse sequence.” 2018. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed March 06, 2021.
http://hdl.handle.net/2142/101563.
MLA Handbook (7th Edition):
Ekvittayaniphon, Sakulbuth. “On sequences related to binary partition function and the Thue-Morse sequence.” 2018. Web. 06 Mar 2021.
Vancouver:
Ekvittayaniphon S. On sequences related to binary partition function and the Thue-Morse sequence. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2018. [cited 2021 Mar 06].
Available from: http://hdl.handle.net/2142/101563.
Council of Science Editors:
Ekvittayaniphon S. On sequences related to binary partition function and the Thue-Morse sequence. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2018. Available from: http://hdl.handle.net/2142/101563
4.
Liang, Jian.
Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences.
Degree: PhD, Mathematics, 2015, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/78377
► In this thesis, we will first follow Kirchberg’s categorical perspective to establish operator-valued WEP and QWEP. We develop similar properties as that in the classical…
(more)
▼ In this thesis, we will first follow Kirchberg’s categorical perspective to establish operator-valued WEP
and QWEP. We develop similar properties as that in the classical WEP and QWEP, and illustrate the
relations with the classical cases by some examples. Then we will discuss the notion of relative WEP in
the context of Hilbert correspondence and investigate the relations between relatively weak injectivity
and relative amenablity. Finally we will apply our discoveries to recent results on C∗ -norms, and
generically find a mechanism to construct a continuum number of C∗ -norms on some tensor products
which admit infinitely many copies.
Advisors/Committee Members: Champaign%22%20%2Bcontributor%3A%28%22Ruan%2C%20Zhong-Jin%22%29&pagesize-30">Ruan, Zhong-Jin (advisor),
Champaign%22%20%2Bcontributor%3A%28%22Boca%2C%20Florin%20P.%22%29&pagesize-30">Boca, Florin P. (Committee Chair),
Champaign%22%20%2Bcontributor%3A%28%22Junge%2C%20Marius%22%29&pagesize-30">Junge, Marius (committee member),
Champaign%22%20%2Bcontributor%3A%28%22Kavruk%2C%20Ali%20S.%22%29&pagesize-30">Kavruk, Ali S. (committee member).
Subjects/Keywords: Kirchberg; Module
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Record Details
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Liang, J. (2015). Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/78377
Chicago Manual of Style (16th Edition):
Liang, Jian. “Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences.” 2015. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed March 06, 2021.
http://hdl.handle.net/2142/78377.
MLA Handbook (7th Edition):
Liang, Jian. “Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences.” 2015. Web. 06 Mar 2021.
Vancouver:
Liang J. Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2015. [cited 2021 Mar 06].
Available from: http://hdl.handle.net/2142/78377.
Council of Science Editors:
Liang J. Operator-valued Kirchberg Theory and its connection to tensor norms and correspondences. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2015. Available from: http://hdl.handle.net/2142/78377
. 
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