在這篇論文中，我們將會首先討論什麼概率測度μ 上的L²空間會存在指數型正交基(exponential orthonormal basis) ，而μ 則稱為一個譜測度若指數型正交基存在。這個問題源於1974年的Fuglede猜想和Jorgensen與Pedersen對分形譜測度存在性的研究。我們有興趣理解怎麼樣的測度會是譜測度，而對於沒有指數型正交基的測度，我們則探討它們會否存在更廣義且在Fourier分析中常用的指數型基，如Riesz基或Fourier框架(Fourier frame) 。 / 我們知道一個測度可以唯一分解成離散、奇異和絶對連續三部份。我們首先証明譜測度肯定是純型(pure type) 。若測度是絶對連續，我們對有Fourier框架的測度的密度給出一個完全的刻直到。這個結果對研究Gabor框架的問題都有幫助。對於離散且只有有限個非零質量原子的測度，我們証明它們全都都有Riesz基。在最困難作出一般討論的奇異測度中，我們透過譜測度與離散測度的卷積找出了有Riesz基但沒有指數型正交基的奇異測度。我們進而探討自彷函數送代系統(affine IFSs) ，我們証明到如果一個自彷函數送代系統是測度分離且有Fourier框架，那麼它在每一個函數的概率權都是一樣的。我們亦証明了Laba-Wang猜想在絶對連續的自相似測度上是正確的。這些結果都表示了一個有Fourier框架的測度都應該在其支撐上有一定的均勻性。 / 在論文的第二部分我們會探討自彷tile其Fourier變換的零點集。在自彷tile的研究中，其中一個基本問題就是刻劃其數字集(digit set)使得那自彷函數送代系統的不變集能以平移密鋪空間。透過Kenyon條件，我們可將這個問題轉化成理解Fourier變換的零點集。男一方面，指數型正交基的存在性亦需要我們探討Fourier變換的零點集，而自彷tile 的Fourier變換是可以明確寫出來的。這使自彷tile成為一個很好去研究tilings和譜測度相互關係的好例子。 / 我們利用了分圓多項式(cyclotomic polynomials)對一維自彷tile的零點集進行了一個詳細的研究。從這裡我們把tile的數字集寫成某些分圓多項式的乘積。這個乘積亦可以一個樹上的切集(blocking)表示出來。這個表示亦把tile數字集的乘積形式(product-forms) 一般化成高階乘積形式。我們証明了在任何維數的tile數字集都是整數tile(即它們能平移密鋪整數集Z) 。這個結果讓我們能使用Coven和Meyerowitz所提出的整數tile分解方法，來使tile數字集完整刻劃成高階模乘積形式(high order modulo product-forms) 當數字集的數目為p[superscript α]q而p，q則是質數。由於我們對零點集亦完全清楚，這對在自彷tile上尋找完備的指數型正交基提供了一個新的方向。 / In this thesis, we will first consider when a probability measure μ admits an exponential orthonormal basis on its L² space (μ is called spectral measures).This problem originates from the conjecture of Fuglede in 1974, and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthonormal bases, but some do not. It generates a lot of interest in understand- ing what kind of measures are spectral measures. For those measures failing to have exponential orthonormal bases, it is interesting to know whether such mea- sures still have Riesz bases and Fourier frames, which are generalized concepts of orthonormal bases with wide range of uses in Fourier analysis. / It is well-known that a measure has a unique decomposition as the discrete, singular and absolutely continuous parts. We first show that spectral measures must be of pure type. If the measure is absolutely continuous, we completely classify the class of densities of the measures with Fourier frames. This result has new applications to topics in applied harmonic analysis, like the Gabor analysis. For the discrete measures with finite number of atoms, we show that they all have Riesz bases. For the case of singular measure, which is the most difficult one, we show that there exist measures with Riesz bases but not orthonormal bases by considering convolution between spectral measures and discrete measures. We then investigate affine iterated function systems (IFSs), we show that if an IFS has measure disjoint condition and admits a Fourier frame, then the probability weights are all equal. Moreover, we also show that the Łaba-Wang conjecture is true if the self-similar measure is absolutely continuous. These results indicate that measures with Fourier frames must have certain kind of uniformity on the support. / In the second part of the thesis we study the zero sets of Fourier transform of self-affine tiles. One of the fundamental problems in self-affine tiles is to classify the digit sets so that the attractors form tiles. This problem can be turned to study the zeros of the Fourier transform via the Kenyon criterion. On the other hand, existence of exponential orthonormal bases requires us to know the zero sets of the Fourier transform. Self-affine tiles are translational tiles arising from IFSs with its Fourier transform written explicitly. It therefore serves as an ideal place to investigate the relation of tilings and spectral measures. / We carry out a detail study in the zero sets of the one-dimensional tiles using cyclotomic polynomials. From this we characterize the tile digit sets through some product of cyclotomic polynomials represented in terms of a blocking in a tree, which is a generalization of the product-form to higher order. We show that tile digit sets in any dimension are integer tiles. This result allows us to use the decomposition method of integer tiles by Coven and Meyerowitz to provide the explicit classification of the tile digit sets in terms of the higher order modulo product-forms when number of the digits is p[superscript α]q, p, q are primes. Since the zero sets are completely known, this provides us a new approach to study the existence of complete orthogonal exponentials in the self-affine tiles on R¹. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lai, Chun Kit. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 128-135). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.9 / Chapter 1.1 --- Background and Motivations --- p.9 / Chapter 1.2 --- Results on spectral measures --- p.13 / Chapter 1.3 --- Results on self-affine tiles --- p.16 / Chapter 2 --- Fourier Frames: Absolutely Continuous Measures --- p.21 / Chapter 2.1 --- Beurling densities --- p.22 / Chapter 2.2 --- Law of pure type --- p.28 / Chapter 2.3 --- Absolutely continuous F-spectral measures --- p.31 / Chapter 2.3.1 --- Proof by Beurling densities --- p.32 / Chapter 2.3.2 --- Proof by translational absolute continuity --- p.35 / Chapter 2.4 --- Applications to applied harmonic analysis --- p.40 / Chapter 2.5 --- Remarks and open questions --- p.42 / Chapter 3 --- Fourier Frames: Discrete and Singular Measures --- p.45 / Chapter 3.1 --- Discrete measures --- p.46 / Chapter 3.2 --- Convolutions with discrete measures --- p.50 / Chapter 3.3 --- Self-affine measures --- p.56 / Chapter 3.4 --- Iterated function systems on R¹ --- p.65 / Chapter 3.5 --- Concluding remarks --- p.70 / Chapter 4 --- Spectral structure of tile digit sets --- p.74 / Chapter 4.1 --- Preliminaries --- p.76 / Chapter 4.2 --- Modulo product-forms --- p.81 / Chapter 4.3 --- Higher order product-forms --- p.86 / Chapter 4.4 --- Φ-tree, blocking and kernel polynomials --- p.90 / Chapter 5 --- Classifications of tile digit sets --- p.101 / Chapter 5.1 --- Tile digit sets --- p.101 / Chapter 5.2 --- The p[superscript α]q[superscript β] integer tiles --- p.105 / Chapter 5.3 --- Tile digit sets for b = p[superscript α]q --- p.112 / Chapter 5.4 --- Self-similar measures: Absolute continuity --- p.122 / Chapter 5.5 --- Remarks --- p.126

The main object of study in this thesis is branching Brownian motion, in which each particle moves like a Brownian motion and gives birth to new particles at some rate. In particular we are interested in where particles are located in this model at large times T : so, for a function f up to time T , we want to know how many particles have paths that look like f. Additive spine martingales are central to the study, and we also investigate some simple general properties of changes of measure related to such martingales.

by Li Jian Lin. / "November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 85-90) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Lo Yiu Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-90). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Main Idea --- p.5 / Chapter 1.2. --- Organization --- p.8 / Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9 / Chapter 2.1. --- A Review on Measure Theory --- p.9 / Chapter 2.2. --- Submanifolds in Rn --- p.11 / Chapter 2.3. --- Several Results from PDEs --- p.17 / Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20 / Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24 / Chapter 3.1. --- Some Preliminaries --- p.24 / Chapter 3.2. --- Approximate Graphical Decomposition --- p.30 / Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41 / Chapter 4.1. --- Willmore Functional and Area --- p.41 / Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45 / Chapter 4.3. --- Higher Regularity at Good Points --- p.54 / Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62 / Chapter 4.5. --- Regularity near Bad Points --- p.64 / Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83 / References --- p.89

Patients who are treated empathically tend to be more satisfied and have better health outcomes. Many instruments for measuring empathy in different health contexts have been developed for postgraduate and to a lesser extent undergraduate health professional use. This thesis presents the results of three studies. The first is a systematic review of the literature describing empathy measures for use in the health professional context. The review was conducted to find the best measure specifically designed for and evaluated in the clinical context that assessed empathy reliably and preferably from a patient perspective. The Consultation and Relational Empathy (CARE) measure was selected from the 12 potential empathy measures available. This instrument was developed and tested in general practice consultations in Scotland, and its usefulness for assessing physicians’ empathy has been demonstrated. The second study, which forms the main research component of the thesis, is an evaluation of the utility of the CARE measure in assessing medical undergraduates on general practice placements. This study aimed to establish whether the CARE measure offers a reliable, valid, and practical means of assessing certain aspects of senior clinical medical students’ consultation skills. Fifth-year students who performed a two- or three-month GP attachment in academic years 2009–2010, 2010–2011, and 2011– 2012 were invited to participate. In 2011 participation in the study became a course requirement. Consulted patients were invited to complete a modified CARE questionnaire ‘on the student’. The form initially included 11 questions answered on a five -point scale; from the academic year 2010–2011, the form also included a free-text comment option. Patients completed questionnaires anonymously. Forms were collated xvi and forwarded to the principle investigator. A minimum of 25 forms were required for each student in order to minimise potential selection bias in the analysis. In the first year of the study the response rate was 56 %. During the next two academic years, this rate improved to over 95%. The total number of the CARE measure forms returned was 2,145, with 1,808 adequately completed. The average score of the students recruited was 4.57 (standard deviation = 0.19), which is higher than average scores for general practitioners. Generalisability theory calculations showed that the CARE measure had high internal consistency, with 25 patients per student delivering an overall reliability coefficient of 0.74. Using the CARE measure with students significantly correlated at a moderate level with relevant outcome assessments such as the mini-Clinical Evaluation Exercise (mini-CEX), Objective Structured Clinical Examination (OSCE) at the end of the GP block, and tutor scores (r = 0.38, r = 0.33, r = 0.33, respectively; P < 0.05). The findings suggest that the CARE measure is a useful tool for assessing senior medical students’ consultation skills, at least in the context of extended general practice attachments. In addition, the measure provides students with valuable and authentic patient feedback for their training. The third study was a ‘mapping exercise’ designed to establish the role of current assessment tools at the University of Dundee in relation to Tomorrow’s Doctors outcomes, and specifically where/how CARE might fit in. Dundee medical faculty staff, block leaders, undergraduate medical educators, and others who engaged directly with assessments and with experience of the existing exams were invited to ‘map’ the value of each aspect of the medical school’s 11 assessment tools against General Medical Council (GMC) attributes. Data was collected electronically from 17 staff xvii members. The results of the exercise revealed both gaps in assessment and over-testing of specific GMC attributes. These findings offer Dundee Medical School an opportunity to optimise their approach to assessment and achieve a broader coverage of GMC outcomes. Thus, the CARE measure offers a logical extension to aspects of the OSCE assessment. The results show that the CARE measure is a feasible, acceptable, reliable, and valid measure for the assessment of students. In addition, as it incorporates authentic patient feedback into training, this measure can offer unique utility to the medical school’s curriculum and assessment as it incorporates the rarely employed patient voice.

Lai, Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 83-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Spectral sets in Rd --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Fundamental domains and convex sets in Rd --- p.15 / Chapter 2.3 --- Finite union of cubes --- p.20 / Chapter 3 --- Spectral theory on discrete groups --- p.27 / Chapter 3.1 --- Finite groups and Zd --- p.28 / Chapter 3.2 --- Rational spectrums and tiling sets --- p.32 / Chapter 3.3 --- Fuglede´ةs Problem in R1 --- p.37 / Chapter 3.4 --- "Failure of Fuglede´ةs Conjecture in Rd, d >3" --- p.42 / Chapter 4 --- Self-similar tiles in R1 --- p.49 / Chapter 4.1 --- Basics of self-similar tiles --- p.49 / Chapter 4.2 --- Self-similar tile digit sets and spectral problem --- p.52 / Chapter 4.3 --- Kenyon criterion --- p.55 / Chapter 5 --- Spectral self-similar measures --- p.66 / Chapter 5.1 --- Spectral self-similar measures --- p.66 / Chapter 5.2 --- One-dimensional self-similar measures --- p.72 / Chapter 5.3 --- General properties of spectral measures --- p.80 / Bibliography --- p.83

In this thesis we compute an explicit Plancherel fromula for PGL_2(F) where F is a non-archimedean local field. Let G be connected reductive group over a non-archimedean local field F. We show that we can obtain types and covers as defined by Kutzko and Bushnell for G/Z coming from types and covers of G in a very explicit way. We then compute those types and covers for GL_2(F ) which give rise to all types and covers for PGL_2(F) that are in the principal series. The Hecke algebra is a Hilbert algebra and has a measure associated to it called Plancherel measure of the Hecke algebra. We have that computing the Plancherel measure for PGL_2(F) essentially reduces to computing the Plancherel measure for the Hecke algebra for every type. We get that the Hacke algebras come in two flavors; they are either the group ring of the integers or they are a free algebra in two generators s_1, s_2 subject to the relations s_1^2=1 and s_2^2=(q^{-1/2}-q^{-1/2})s_2+1, where q is the order of the residue field. The Plancherel measure for both algebras are known, as a result we obtain the Plancherel measure for PGL_2(F).

An attitude is a mental state that influences the way an individual responds to relevant objects and situations. Attitudes toward the elderly have a potential impact on all Americans, as the population is growing older. These attitudes constitute an important area of study because their nature is unclear and their impact extensive. The Kogan Attitudes Toward Old People Scale (OP) was examined to establish reliability and validity estimates. A questionnaire, consisting of this scale and four others, was administered to a sample of adults. Factor analysis of the OP revealed two factors, which were somewhat ambiguous. Internal consistency estimates for these factors and the total OP ranged from .72 to .92. Validity estimates were in the expected direction and ranged from -.13 to .36. Gender, age, education level, and degree of tolerance for others immerged as predictive variables for reported attitudes toward the elderly. It was concluded that the use of the OP is of questionable utility. If one must use it, the total OP score or only the items from Factors 1 and 2 should be utilized, as long as the discussed weaknesses, such as the sampling and validation procedures, are taken into account

For my senior thesis, I am planning to blend my mathematical studies with my second field of study, psychology. In particular, to develop and test various scoring methods for a multidimensional, psychometric measure of social skills/competence. I would work with the Social Skills Inventory (see below) and an existing data set, using statistical modelling to design a more representative total score measure. The current total score measure does not appear to take into account balances and value weights of the six inventory items.

Research indicates that relational aggression, social aggression, and indirect aggression are important predictors and outcomes of social development (Archer & Coyne, 2005). Socially, indirectly, and relationally aggressive behaviors are utilized in order to harm an individual's social status, relationships, and/or social resources (Archer, & Coyne, 2005), but scholars disagree about the extent of the similarities and differences between these subtypes. Previous efforts to understand the distinction between these subtypes of aggression have been limited by how these behaviors have been operationalized and studied. The primary aim of the current study was to develop a self-report measure of these aggressive behaviors for emerging adults by utilizing factor analytic techniques to examine existing and newly created items. A series of five stages was used to code all items into existing theoretical categories of behavior (e.g., social aggression), establish the factor structure of the items, select the best items to measure each factor, test measurement invariance across subgroups (e.g., men and women), ensure strong psychometric properties, and relate the final factor structure to relevant developmental correlates (e.g., depressive symptoms). Three independent samples of emerging adults aged 18 - 29 years (49.51% - 52.33% women; Mage= 25.71 - 26.26) were recruited online through Amazon's Mechanical Turk (sample 1 N = 299; sample 2 N = 299; sample 3 N = 119). Indirect, social, and relational aggression items were selected and adapted from existing self-report measures of these constructs for adults and several new items were created from qualitative interviews with emerging adults. Through a rigorous theoretical, methodological, and statistical approach, the Relational/Social Aggression in Adulthood Measure (RSAAM) was developed. The final factor structure consisted of three factors: Ignoring, Gossip, and Relational Manipulation. The three factors demonstrated measurement invariance across gender and educational groups and strong internal consistency and test-retest reliability. Purely relationally manipulative behaviors were distinct from other, related behaviors (i.e., ignoring, gossip) and were also differentially related to developmental correlates. Findings suggest that it may be advantageous for researchers to move beyond broad theoretical definitions of relational and social aggression and instead focus on the specific aggressive behaviors being enacted.