Asset prices are forward looking. Therefore, expectations play a central role in shaping asset prices. In this dissertation, I challenge the rational expectation assumption that has been influential in the field of asset pricing over the past few decades. Different from previous approaches, which typically build on behavioral theories originated from psychology literature, my approach takes data on subjective beliefs seriously and proposes empirically grounded models of subjective beliefs to evaluate the merits of the rational expectation assumption. Specifically, this dissertation research: 1). collects and analyzes data on investors' actual subjective return expectations; 2). builds models of subjective expectation formation; 3). derives and tests the models' implications for asset prices. I document the results of the research in two chapters. In summary, the dissertation shows that investors do not hold full-information rational expectations. On the other hand, their subjective expectations are not necessarily irrational. Rather, they are bounded by the information environment investors face and reflect investors' personal experiences and preferences. The deviation from fully-rational expectations can explain asset pricing anomalies such as cross-sectional anomalies in the U.S. stock market. In the first chapter, I provide a framework to rationalize the evidence of extrapolative return expectations, which is often interpreted as investors being irrational. I first document that subjective return expectations of Wall Street (sell-side, buy-side) analysts are contrarian and counter-cyclical. I then highlight the identification problem investors face when theyform return expectations using imperfect predictors through Kalman Filters. Investors differ in how they impose subjective priors, the same way rational agents differ in different macro-finance models. Estimating the priors using surveys, I find Wall Street and Main Street (CFOs, pension funds) both believe persistent cash flows drive asset prices but disagree on how fundamental news relates to future returns. These results support models featuring heterogeneous agents with persistent subjective growth expectations. In the second chapter, I propose and test a unifying hypothesis to explain both cross-sectional return anomalies and subjective return expectation errors: some investors falsely ignore the dynamics of discount rates when forming return expectations. Consistent with the hypothesis: 1) stocks' expected cash flow growth and idiosyncratic volatility explain significant cross-sectional variation of analysts' return forecast errors; 2). a measure of mispricing at the firm level strongly predicts stock returns, even among stocks in the S&P500 and at long horizon; 3). a tradable mispricing factor explains the CAPM alphas of 12 leading anomalies including investment, profitability, beta, idiosyncratic volatility and cash flow duration.

This study engaged fourth and fifth graders in solving a set of proportional tasks with focused discussion and concept development by the teacher. In order to understand the students’ ability to generalize the concept, they were asked to write problems that reflected the underlying concepts in the tasks and lessons. A qualitative analysis of the student generated problems show that the majority of the students were able to generalize the concepts. The analysis allowed for a discussion of problems solving approaches and a rich description of how students applied multiplicative reasoning in composing mathematics problems. These results are couched in a discussion of how the students solved the proportional reasoning tasks.

In this study the teaching of mathematics has been developed in relation to rational numbers and towards a learning activity. At the same time topic-specific mediated tools have been studied. The iterative model for learning study has been used as research approach. The purpose of the study was to explore what in an algebraic learning activity enables knowledge of rational numbers to develop. The specific questions answered by the study are how an algebraic learning activity can be formed in an otherwise arithmetic teaching tradition, what knowledge is mediated in relation to different mediated tools and what in these tools that enable this knowledge. The result of the study shows how an algebraic learning activity can be developed to support the students to understand rational numbers even in an arithmetic teaching tradition. The important details that developed the algebraic learning activity were to identify the problem to create learning tasks and the opportunity for the students to reflect that are characteristic of a learning activity. The result also shows that the mediating tools, the algebraic symbols and the general model for fractional numbers, have had significant importance for the students' possibilities to explore rational numbers. The conditions for the algebraic symbols seem to be the possibilities for these symbols to include clues to the meaning of the symbol and that the same symbol can be used in relation to several of other mediated tools. The conditions in the general model consisted of that the integer numbers and the rational numbers in the model could be distinguished and that the students could reflect on the meaning of the different parts. The general model consists of the algebraic symbols, developed in the learning activity. The algebraic symbols make the structure of the numbers visible and the general model mediates the structure of additive and multiplicative conditions that are contained in a rational number. The result of the study contributes in part to the field of mathematics education research by examining Elkonin's and Davydov's Mathematical Curriculum in a western teaching practice and in part to a development of the model of Learning study as a didactical research approach by using an activity-theoretical perspective on design and analysis.

This thesis presents parallel algorithms for rational cones and affine monoids which pursue two main computational goals: finding the Hilbert basis, a minimal generating system of the monoid of lattice points of a cone; and counting elements degree-wise in a generating function, the Hilbert series.

The solution to a set of simultaneous equations is of the form A-1 B = X where A-1 is the inverse of A in the equation AX= B. The purpose of this study is to obtain an exact A-1 through the use of rational arithmetic, and to study the behavior of rational numbers when used in arithmetic calculations. This study describes a matrix inversion program written in SPS II, utilizing the concept of rational arithmetic. This program, using the Gaussian elimination matrix inversion method, is compared to the same method written in Fortran. Gaussian elimination was used by this study because of its simplicity and speed of inversion. The Adjoint method was ruled out because of its complexity and relative lack of speed when compared with Gaussian elimination. The Fortran program gives only an approximate inverse due to the rounding error while the rational arithmetic program gives an exact inverse.

Understanding the relationship between fractions and decimals is an important step in developing an overall understanding of rational numbers. Research has demonstrated the feasibility of technology in the form of virtual manipulatives for facilitating students’ meaningful understanding of rational number concepts. This exploratory dissertation study was conducted for the two closely related purposes: first, to investigate a sample of fifth-grade students’ reasoning regarding the relationship between fractions and decimals for fractions with terminating decimal representations while using virtual manipulative incorporating parallel number lines; second, to investigate the affordances of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. The study employed qualitative methods in which the researcher collected and analyzed data from fifth-grade students’ verbal explanations, hand gestures, and mouse cursor motions. During the course of the study, four fifth-grade students participated in an initial clinical interview, five task-based clinical interviews while using the number line-based virtual manipulatives, and a final clinical interview. The researcher coded the data into categories that indicated the students’ synthetic models, their strategies for converting between fractions and decimals, and evidence of students’ accessing the affordances of the virtual manipulatives (e.g., students’ hand gestures, mouse cursor motions, and verbal explanations). The study yielded results regarding the students’ conceptions of the decimal-fraction relationship. The students’ synthetic models primarily showed their recognition of the relationship between the unit fraction 1/8 and its decimal 0.125. Additionally, the students used a diversity of strategies for converting between fractions and decimals. Moreover, results indicate that the pattern of strategies students used for conversions of decimals to fractions was different from the pattern of strategies students used for conversions of fractions to decimals. The study also yielded results for the affordances of the virtual manipulatives for supporting the students’ reasoning regarding the decimal-fraction relationship. The analysis of students’ hand gestures, mouse cursor motions, and verbal explanations revealed the affordances of alignment and partition of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. Additionally, the results indicate that the students drew on the affordances of alignment and partition more frequently during decimal to fraction conversions than during fraction to decimal conversions.

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Indiana University-Purdue University Indianapolis (IUPUI) / We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions. We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k). The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results. We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)). To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.

Programmed death-ligand 1 (PD-L1) is a type 1 transmembrane protein that has been reported to play a vital role in mediating suppressed immunity. The interaction between PD-L1 and PD-1 delivers a negative signal that reduces the proliferation of these T cells and induces apoptosis at the same time. Antibodies that can block the Programmed death-ligand 1 (PD-L1) on tumor cells have been shown to alleviate cancer-induced immunosuppression. While antibodies have a great potential in various therapeutic uses, many drawbacks such as the high cost of production, huge molecular size, and poor permeability impose restrictions on the extensive use of full-length antibodies. These limitations have necessitated research for finding alternatives to antibodies, such as peptides, that have lower molecular weight and similar properties as antibodies but do not have the lengthy and complicated approach of producing antibodies. In this study, a novel approach based on molecular interactions of the PD1-PD-L1 complex was developed to design peptides against PD-L1 using Knob-Socket model as basis. Three generations of peptides, α-helix, over-packed and salt bridge function peptides, were designed. All designed peptides were docked in the Molecular Operating Environment (MOE) and the AutoDock Vina software for the docking energy and the detail interaction information. Synthesis and characterization of selected peptides were performed after simulation studies. Surface Plasmon Resonance (SPR) studies showed that α-helix and over-packed peptides can’t bind to the PD-L1 protein with no response on sensorgrams, while peptides with salt bridge function had a higher binding response than those two generations of peptides. In confocal microscopic studies, PD-L1 positive breast cancer cell line MDA-MB-231 was used to determine the binding specificity of the salt bridge function peptides to PD-L1 in vitro, while another breast cancer cell line (MCF-7, without PD-L1) was used as a control. After incubation with peptides, significant fluorescence intensities were detected on the MDA-MB-231 cells, while only background fluorescence was observed on MCF-7 cells. In conclusion, this study demonstrated that peptides against PD-L1 designed using the Knob-Socket model and molecular interaction between PD-L1-PD1 complex showed feasibility to bind specifically with PD-L1 receptors.

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