The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

The last few years witnessed a dramatic increase in the demand on high-rate reliable wireless communications. In order to meet these new requirements, resorting to Multiple-Input Multiple-Output (MIMO) techniques was inevitable as they may offer high-rate reliable wireless communications without any additional bandwidth. In the case where the transmitter does not have any prior knowledge about the channel state information, space-time coding techniques have proved to efficiently exploit the MIMO channel degrees of freedom while taking advantage of the maximum diversity gain. On the other hand, the ML decoding complexity of Space-Time Codes (STCs) generally increases exponentially with the rate which imposes an important challenge to their incorporation in recent communications standards. Recognizing the importance of the low-complexity criterion in the STC design for practical considerations, this thesis focuses on the design of new low-complexity Space-Time Block Codes (STBCs) where the transmitted code matrix can be expressed as a weighted linear combination of information symbols and we propose new codes that are decoded with a lower complexity than that of their rivals in the literature while providing better or slightly lower performance.

A small group of profit seeking publishers dominates the American textbook market and guides the learning of the majority of our nation’s calculus students. The College Board’s AP Calculus curriculum is a de facto national standard for this gateway course that is critically important to 21st century STEM careers. A multi-representational understanding of calculus is a central pillar of the AP curriculum. This dissertation asks whether this multi-representational vision is manifest in popular calculus textbooks. This dissertation began with a survey of all AP Calculus AB Examination free response items, 2002-2011, and found that students score worse on items characterized by numerical anchors or verbal targets. Based on previously elucidated models, a new cognitive model of five levels and six principles is developed for the purpose of calculus textbook task analysis. This model explicates complexity as a function of representational input and output. Eight popular secondary calculus textbooks were selected for study based on Amazon sales rank data. All verbally anchored mathematical tasks (n=555) from sections of those books concerning the mean value theorem and all AP Calculus AB prompts (n=226) were analyzed for cognitive complexity and representational diversity using the model. The textbook study found that calculus textbooks underrepresented the numerical anchor and verbal target. It found that the textbooks were both explicitly and implicitly less cognitively complex than the AP test. The article suggested that textbook tasks should be less dense, avoid cognitive attenuation, move away from the stand-alone item, juxtapose anchor representations, scaffold student solutions, incorporate previously considered overarching concepts and include more profound follow-up questions. To date there have been no studies of calculus textbook content based on established research on cognitive learning. Given the critical role that their calculus course plays in the lives of hundreds of thousands of students annually, it is incumbent upon the College Board to establish a textbook review process at the very least in the same vain as the teacher syllabus auditing process established in recent years.

The potential of a broad enactive approach in education has yet to be realized. This thesis contributes to the development of a well-rounded enactive educational theory and practice. This thesis argues that a broad enactive perspective has the potential to challenge, reframe and reconfigure problems, issues and practices in education in ways that improve teaching, learning and research communities. It establishes that a broad enactive approach as a theory of embodied mind, a dynamic co-emergence theory, and a method of examining human experience helps to realize the meaning, scope, and potential of enactive education. It takes as its point of departure Dewey’s broad enactive philosophy of mind, cognition, embodiment, experience, and dynamic co-emergence. It shows, through an examination of an actual public classroom encounter, that a broad enactive approach has the potential to reconfigure responsibility, ethics and justice in education. It demonstrates using a case study of the enactment of impostor feelings in higher education how a broad enactive approach to education as the potential to reconfigure teaching, learning and research practices.

Storage is an important part of the background in many archaeological studies of the origins of early complex societies. Yet, a problem with many of these studies of formation and change in complex societies is that the social significance of storage is assumed rather than demonstrated. In this dissertation, I examine the practice of storage in three regions of prehistoric Korea and its relationship with socio-political structural changes. I analyze the distribution of storage artifacts and features such as pits, large-capacity pottery, and raised-floor structures in the context of their spatial relationships with other archaeological features and elite precincts at the household and settlement levels. The archaeological features used for storage in the Mumun Pottery Period (3390-2290 calibrated years B.P.) changed in form diachronically and show that underground pit storage remained constant during the period of study and that clandestine storage was not completely replaced by above-ground visible storage. Elite actors seem to have had some influence on the nature of storage in at least two central settlements, Daepyeong I and II and Songguk-ri, but appear to have been unable to completely control stored agricultural surplus in the two settlements and had little control over the surrounding areas.

The purpose of this study was to examine the relative effectiveness of three-dimensional visualization techniques for learning about protein conformation and molecular motion in association with a ligand and receptor binding event. Increasingly complex versions of the same binding event were depicted in each of four animated treatments. Students (n = 131) were tested at three time points, and over both the short and longer term, the most complex of the four animated treatments was the most successful at fostering students’ understanding of the events depicted. A follow-up study including eight biology students was conducted to gain greater insight into the students’ underlying thought processes and better characterize their understanding of the animated representations. Analysis of verbal reports and eye tracking data suggest that students are able to attend to the same narrative elements regardless of the level of complexity depicted in each animation. Analysis of verbal protocol data revealed a positive correlation between the number of explanatory statements expressed by participants and the complexity of the animation viewed. As well, prior knowledge was positively correlated with the number of explanatory statements contained in each protocol. Overall, students demonstrated an understanding of protein conformation and molecular crowding. However results suggest that students have difficulty understanding and associating randomness with molecular events. The verbal reports contained several instances of students’ attaching agency to protein and ligand, anthropomorphizing their movements and subsequent binding. Ordinarily cellular events, owing to their sheer complexity, are depicted in a highly schematized, simplified form. The results of this study would suggest that under select circumstances this may not be the most appropriate approach to depicting dynamic events. However additional attention must be given to exploring techniques that can satisfactorily balance the random nature of molecular events with narrative explanations of these processes.

The growing number of people living below the poverty line has made homelessness a topic of interest, once again. This paper focuses on the homeless adolescent population that is often overlooked, and explores the complexity of the homeless situation, and how there is no definite solution to overcome homelessness. At-risk and homeless adolescents are affected by many negative factors that cause them to seek early independence, such as parenting style, finical instability, lack of an education, drugs and alcohol, physical and sexual abuse, all of which are discussed in this paper. Along with the negative factors, there are protective barriers that can potentially help an at-risk adolescent but are ineffective once the adolescent is homeless. This paper also addresses how we as a society can be more proactive in helping this population, and be aware if the warning signs that can lead a youth to decide to run away and eventually end up homeless.

Communication complexity is an area of complexity theory that studies an abstract model of computation called a communication protocol. In a $k$-player communication protocol, an input to a known function is partitioned into $k$ pieces of $n$ bits each, and each piece is assigned to one of the players in the protocol. The goal of the players is to evaluate the function on the distributed input by using as little communication as possible. In a Number-On-Forehead (NOF) protocol, the input piece assigned to each player is metaphorically placed on that player's forehead, so that each player sees everyone else's input but its own. In a Number-In-Hand (NIH) protocol, the piece assigned to each player is seen only by that player. Overall, the study of communication protocols has been used to obtain lower bounds and impossibility results for a wide variety of other models of computation. Two of the main contributions presented in this thesis are negative results on the NOF model of communication, identifying limitations of NOF protocols. Together, these results consitute stepping stones towards a better fundamental understanding of this model. As the first contribution, we show that randomized NOF protocols are exponentially more powerful than deterministic NOF protocols, as long as $k \le n^c$ for some constant $c$. As the second contribution, we show that nondeterministic NOF protocols are exponentially more powerful than randomized NOF protocols, as long as $k \le \delta \cdot \log n$ for some constant $\delta < 1$. For the third major contribution, we turn to the NIH model and we present a positive result. Informally, we show that a NIH communication protocol for a function $f$ can simulate a Stack Machine (a Turing Machine augmented with a stack) for a related function $F$, consisting of several instances of $f$ bundled together. Using this simulation and known communication complexity lower bounds, we obtain the first known (space vs. number of passes) trade-off lower bounds for Stack Machines.

The potential of a broad enactive approach in education has yet to be realized. This thesis contributes to the development of a well-rounded enactive educational theory and practice. This thesis argues that a broad enactive perspective has the potential to challenge, reframe and reconfigure problems, issues and practices in education in ways that improve teaching, learning and research communities. It establishes that a broad enactive approach as a theory of embodied mind, a dynamic co-emergence theory, and a method of examining human experience helps to realize the meaning, scope, and potential of enactive education. It takes as its point of departure Dewey’s broad enactive philosophy of mind, cognition, embodiment, experience, and dynamic co-emergence. It shows, through an examination of an actual public classroom encounter, that a broad enactive approach has the potential to reconfigure responsibility, ethics and justice in education. It demonstrates using a case study of the enactment of impostor feelings in higher education how a broad enactive approach to education as the potential to reconfigure teaching, learning and research practices.

Storage is an important part of the background in many archaeological studies of the origins of early complex societies. Yet, a problem with many of these studies of formation and change in complex societies is that the social significance of storage is assumed rather than demonstrated. In this dissertation, I examine the practice of storage in three regions of prehistoric Korea and its relationship with socio-political structural changes. I analyze the distribution of storage artifacts and features such as pits, large-capacity pottery, and raised-floor structures in the context of their spatial relationships with other archaeological features and elite precincts at the household and settlement levels. The archaeological features used for storage in the Mumun Pottery Period (3390-2290 calibrated years B.P.) changed in form diachronically and show that underground pit storage remained constant during the period of study and that clandestine storage was not completely replaced by above-ground visible storage. Elite actors seem to have had some influence on the nature of storage in at least two central settlements, Daepyeong I and II and Songguk-ri, but appear to have been unable to completely control stored agricultural surplus in the two settlements and had little control over the surrounding areas.