Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2009. / Includes bibliographical references (p. 85-89).

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--Hanʼguk Oegugŏ Taehakkyo, 1990. / "August 1990." "8"--Spine. Abstract in Korean. Includes bibliographical references (p. 167-173).

Quantum systems in a d-dimensional Hilbert space are considered, where the phase spase is Z(d) x Z(d). An analytic representation in a cell S in the complex plane using Theta functions, is defined. The analytic functions have exactly d zeros in a cell S. The reproducing kernel plays a central role in this formalism. Wigner and Weyl functions are also studied. Quantum systems with positions in a circle S and momenta in Z are also studied. An analytic representation in a strip A in the complex plane is also defined. Coherent states on a circle are studied. The reproducing kernel is given. Wigner and Weyl functions are considered.

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’s analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.

Quantum systems in a d-dimensional Hilbert space are considered, where the phase spase is Z(d) x Z(d). An analytic representation in a cell S in the complex plane using Theta functions, is defined. The analytic functions have exactly d zeros in a cell S. The reproducing kernel plays a central role in this formalism. Wigner and Weyl functions are also studied. Quantum systems with positions in a circle S and momenta in Z are also studied. An analytic representation in a strip A in the complex plane is also defined. Coherent states on a circle are studied. The reproducing kernel is given. Wigner and Weyl functions are considered.

Neurofeedback (NF) is an important treatment for attention deficit/hyperactivity disorder (ADHD). In ADHD, cognitive control deficits pose considerable problems to patients. However, NF protocols are not yet optimized to enhance cognitive control alongside with clinical symptoms, partly because they are not driven by basic cognitive neuroscience. In this study, we evaluated different EEG theta and/or beta frequency band NF protocols designed to enhance cognitive control. Participants were n = 157 children and adolescents, n = 129 of them were patients with ADHD (n = 28 typically developing (TD) controls). Patients with ADHD were divided into five groups in the order of referral, with four of them taking part in different NF protocols systematically varying theta and beta power. The fifth ADHD group and the TD group did not undergo NF. All NF protocols resulted in reductions of ADHD symptoms. Importantly, only when beta frequencies were enhanced during NF (without any theta regulation or in combination with theta upregulation), consistent enhancing effects in both response inhibition and conflict control were achieved. The theta/beta NF protocol most widely used in clinical settings revealed comparatively limited effects. Enhancements in beta band activity are key when aiming to improve cognitive control functions in ADHD. This calls for a change in the use of theta/beta NF protocols and shows that protocols differing from the current clinical standard are effective in enhancing important facets of cognitive control in ADHD. Further studies need to examine regulation data within the neurofeedback sessions to provide more information about the mechanisms underlying the observed effects.