교육용 소프트웨어를 이용한 수학 문제 해결력 신장

金光福 경기대학교 교육대학원 2002 국내석사

According to development and popularization of computer, the various teaching-learning methods are used in the scene of education. Under this situation, many students think mathematics is difficult and uninteresting. It is necessary to introduce computer to change the degree of interest and attitude affirmatively and to teach students according to the individual difference. Especially, if the teacher apply software with graphic-animation function to the students, the students can take the greater effect by arousing interest about the new learning method and the problem solving abilities in mathematics. In this thesis, through the teaching-learning activities which apply the educational software about the diagram-unit of the high school mathematics directly developed by the program of the visual basic, the teacher compared changes of the students' interest and attitude and also after comparing the difference of the problem solving abilities in upper and lower group composed by the level of scholastic achievement, the teacher concluded as follows. 1. The teaching-learning method applying the educational software rather than the traditional teaching method enhanced the students' inte rest and changed the students' attitude affirmatively in mathematics. 2. The teaching-learning method applying the educational software rather than the traditional teaching method improved the students' problem-solving abilities. In particular, the lower group losing the inter est in mathematics was more improved in the problem-solving abilities rather than the upper group. 3. Through the level-step learning which the students can search and learn spontaneously, the teacher can help the students increase the expansion of the 'mathematical power' aiming in the 7-th curriculum. 4. The students easily can understand the total-learning activities by the software which the teacher directly produced and the students can improve concentration applying graphs and simulations. In particular, through the related internet site, the students can improve the greater interest in mathematics and the greater changes in the learningactivities.

On the mathematics of self-assembly

Reishus, Dustin University of Southern California 2009 해외공개박사

Self-assembly is the ubiquitous process by which simple objects come together under simple rules to form more complex objects. Self-assembly occurs in nature to produce structures of extraordinary complexity. In the future, it may be possible to harness the power of self-assembly to manufacture useful devices in enormous quantities at little cost. In order to do so, it would be valuable to have a deep understanding of self-assembly, at both theoretical and practical levels. I first describe experimental work with DNA self-assembly. DNA is an ideal substance to use in experimental self-assembly: It has well-understood structure; it has readily-available tools to synthesize, manipulate, and visualize it; and it has "programmable" interactions with other molecules of DNA. I describe two self-assembling DNA complexes that can further self-assemble into regular lattices. Mathematical models of self-assembly have been created to aid in the analysis of the power and limits of self-assembly. I explore decidability questions in a mathematical model of self-assembly known as the tile assembly model. I prove the undecidability of distinguishing self-assembling systems in which infinite structures can be assembled from systems in which only finite structures can be assembled. Many self-assembly processes are rooted in chemistry. The event-systems model generalizes the classical theory of chemical thermodynamics and places the kinetic theory of chemical reactions on a firm mathematical foundation. I prove that many of the expectations acquired through empirical study are warranted. Finally, I use the event-systems model to explore questions in pure mathematics. The atomic hypothesis in chemistry (the theory that every substance is composed of a unique set of atoms) is analogous to the fundamental theorem of arithmetic in mathematics (the theory that every natural number is the product of a unique set of primes). I exploit this analogy by creating event-systems in which the basic components are natural numbers that can "react" through multiplication. Important thermodynamic properties such as temperature and pressure have purely mathematical implications in these systems. In particular, the pressure at equilibrium is the Riemann zeta function's value of the temperature of the system.

Dynamical Systems in Pure Mathematics

Lipton, Max Cornell University ProQuest Dissertations & Theses 2023 해외박사(DDOD)

The author's research program involves several topics which differ at first glance. However, they all share the common theme of exploring how geometry and topology influences dynamical equilibria. The dissertation is broken into three parts: the hyperbolic geometry of higher-dimensional Kuramoto oscillators, electrostatic knot theory, and minimal surfaces with Mobius energy on the boundary. Each part is further divided into chapters adapting the author's preprints and published papers, which have appeared in journals in physics, applied mathematics, and pure mathematics.

The Cognitive and Demographic Variables that Underlie Notetaking and Review in Mathematics: Does Quality of Notes Predict Test Performance in Mathematics?

Belanfante, Elizabeth Columbia University 2013 해외박사(DDOD)

Taking and reviewing lecture notes is an effective and prevalent method of studying employed by students at the post-secondary level (Armbruster, 2000; Armbruster, 2009; Dunkel & Davy, 1989; Peverly et al., 2009). However, few studies have examined the cognitive variables that underlie this skill. In addition, these studies have focused on more verbally based domains, such as history and psychology. The current study examined the practical utility of notes in actual class settings. It is the first study that has attempted to examine the outcomes and cognitive skills associated with note-taking and review in any area of mathematics. It also set out to establish the importance of quality of notes and quality of review sheets to test performance in graduate level probability and statistics courses. Finally, this dissertation sought to explore the extent to which variables besides notes also contribute to test performance in this domain. Participants included 74 graduate students enrolled in introductory probability and statistics courses at a private graduate teacher education college in a large city in the Northeast United States. Participants took notes during class and provided the researcher with a copy of their notes for several lectures. Participants were also required to write down additional information on the back of two formula sheets that were used as an aid on the midterm exam. The independent variables included handwriting speed, gender, spatial visualization ability, background knowledge, verbal ability, and working memory. The dependent variables were quality of lecture notes, quality of supplemental review sheets, and midterm performance. All measures were group administered. Results revealed that gender was the only predictor of quality of lecture notes. Quality of lecture notes was the only significant predictor of quality of supplemental review sheets. Neither quality of lecture notes nor quality of supplemental review sheets predicted overall test performance. Instead, background knowledge and instructor significantly predicted overall test performance. Handwriting speed was a marginally significant predictor of overall test performance. Future research aimed at replicating these findings and expanding the results to include other mathematical domains and educational levels is recommended.

Spectral Expansions and Excursion Theory for Non-Self-Adjoint Markov Semigroups with Applications in Mathematical Finance

Zhao, Yi Xuan ProQuest Dissertations & Theses Cornell University 2017 해외박사(DDOD)

This dissertation consists of three parts. In the first part, we establish a spectral theory in the Hilbert space L2( R+) of the C0-semigroup P and its adjoint Pˆ having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index 1 < alpha < 2. These linear operators, which are nonlocal and non-self-adjoint, appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative alpha-stable process. We establish an intertwining relationship between these semigroups and the semigroup of a Bessel type process which is self-adjoint. Relying on this commutation identity, we characterize the spectrum and the (weak) eigenfunctions and provide the spectral expansions of these semigroups on (at least) a dense subset of L2(R+). We also obtain an integral representation of their transition kernels that enables to derive regularity properties. Inspired by this development, we further exploit, in the second part of this dissertation, the concept of intertwining between general Markov semigroups. More specifically, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of It ˆ o, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (nonself- adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein's theory of strings by showing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium. The third part of this dissertation is devoted to the applications of some of these theoretical results to some substantial problems arising in financial mathematics. Keeping in mind the fundamental theorem of asset pricing, we suggest several transformations on a tractable and flexible Markov process (or equivalently, its respective semigroup) in order that the discounted transformed process becomes a (local) martingale while still keeping its tractability. In particular, we suggest using an intertwining approach and/or Bochner's subordination (random time-change via a subordinator) to achieve this goal. Moreover, in order to illustrate our approach, we discuss in details several examples that include the class of L´evy, self-similar and generalized CIR processes that reveal the usefulness of our result. Furthermore, we provide for the non-self-adjoint pricing semigroups associated to the latter family of processes a spectral expansions on which we carry out some numerical analysis.

Strong unique continuation for hyperbolic operators

Luca, Mihnea-Paul Purdue University 2013 해외박사(DDOD)

The topic of unique continuation is a topic of interest in many areas of mathematics. Perhaps the most well-known case is the one encountered in complex analysis. In general, it can be stated as follows: if Q is a differential operator in R n, u is a function verifying Qu = 0, and u vanishes to infinite order at a point x0, then u is identically 0 near x. There is a long history of strong unique continuation problems in the elliptic case, starting with the work of Aronszajn and Cordes. In this thesis, we study the hyperbolic case, namely the wave operator, with the Laplacian defined with respect to a certain metric in a Riemannian manifold. We write the Laplacian in the normal form, then we conjugate Q with the aid of a Gaussian, using a method of Tataru. Then we take advantage of an a priori estimate for the conjugate operator (which is the same as a Carleman estimate for Q) to get u = 0 near x, by using a technique of Hormander.

Topics in optimal stopping and fundamental theorem of asset pricing

Zhou, Zhou University of Michigan 2015 해외박사(DDOD)

In this thesis, we investigate several problems in optimal stopping and fundamental theorem of asset pricing (FTAP). In Chapter II, we study the controller-stopper problems with jumps. By a backward induction, we decompose the original problem with jumps into controller- stopper problems without jumps. Then we apply the decomposition result to indifference pricing of American options under multiple default risk. In Chapters III and IV, we consider zero-sum stopping games, where each player can adjust her own stopping strategies according to the other's behavior. We show that the values of the games and optimal stopping strategies can be characterized by corresponding Dynkin games. We work in discrete time in Chapter III and continuous time in Chapter IV. In Chapter V, we analyze an optimal stopping problem, in which the investor can peek epsilon amount of time into the future before making her stopping decision. We characterize the solution of this problem by a path-dependent reflected backward stochastic differential equation. We also obtain the order of the value as epsilon goes to zero. In Chapters VI-VIII, we investigate arbitrage and hedging under non-dominated model uncertainty in discrete time, where stocks are traded dynamically and liquid European-style options are traded statically. In Chapter VI we obtain the FTAP and hedging dualities under some convex and closed portfolio constraints. In Chapter VII we study arbitrage and super-hedging in the case when the liquid options are quoted with bid-ask spreads. In Chapter VIII we investigate the dualities for sub and super-hedging prices of American options. Note that for these three chapters, since we work in the frameworks lacking dominating measures, many classical tools in probability theory cannot be applied. In Chapter IX, we consider arbitrage, hedging, and utility maximization in a given model, where stocks are available for dynamic trading, and both European and American options are available for static trading. Using a separating hyperplane argument, we get the result of FTAP, which implies the dualities of hedging prices. Then the hedging dualities lead to the duality for the utility maximization.

Topics in Metric Approximation

Leeb, William Edward Yale University 2015 해외박사(DDOD)

This thesis develops effective approximations of certain metrics that occur frequently in pure and applied mathematics. We show that distances that often arise in applications, such as the Earth Mover's Distance between two probability measures, can be approximated by easily computed formulas for a wide variety of ground distances. We develop simple and easily computed characterizations both of norms measuring a function's regularity -- such as the Lipschitz norm -- and of their duals. We are particularly concerned with the tensor product of metric spaces, where the natural notion of regularity is not the Lipschitz condition but the mixed Lipschitz condition. A theme that runs throughout this thesis is that snowflake metrics (metrics raised to a power less than 1) are often better-behaved than ordinary metrics. For example, we show that snowflake metrics on finite spaces can be approximated by the average of tree metrics with a distortion bounded by intrinsic geometric characteristics of the space and not the number of points. Many of the metrics for which we characterize the Lipschitz space and its dual are snowflake metrics. We also present applications of the characterization of certain regularity norms to the problem of recovering a matrix that has been corrupted by noise. We are able to achieve an optimal rate of recovery for certain families of matrices by exploiting the relationship between mixed-variable regularity conditions and the decay of a function's coefficients in a certain orthonormal basis.

Choosing a dissimilarity representation for classification

Cardinal-Stakenas, Adam James The Johns Hopkins University 2011 해외박사(DDOD)

The dissimilarity representation is a vital component of modern statistical analysis. By examining all pairs of dissimilarities between the elements of an observed data set, one can leverage the more than 50-year history of statistical pattern recognition on high- and infinite-dimensional spaces, spaces that exhibit non-Euclidean geometry, and spaces that defy analysis by traditional means. However, for most data sets observed in spaces like these, there are many dissimilarities that could be successfully applied. A critical question facing the researcher is: how should one choose which dissimilarity to use in these circumstances, and, if there are many that perform well, can they be combined to optimize inferential performance? We will begin to address these questions by applying a variety of methods from matrix analysis, factor analysis, optimization, combinatorics, and statistics.

Algebraic Operations via Solvable Lattice Models

Hardt, Andrew ProQuest Dissertations & Theses University of Minn 2022 해외박사(DDOD)

This thesis explores solvable lattice models in several contexts. Our overarching goal is to un- derstand and exploit the flexibility of lattice models in their ability to express functorial operations in algebra. In particular, we study lattice models whose partition functions are special functions in representation theory and Schubert calculus. These functions tend to have nice properties relating to their algebraic structure, and we try to connect these properties to combinatorial operations on lattice models.Chapter 2 studies the connection between solvable lattice models and discrete-time Hamiltonian operators. We give general conditions for the existence of a Hamiltonian operator whose discrete time evolution matches the partition function of certain solvable lattice models. In particular, we examine two classes of lattice models: the classical six-vertex model and a generalized family of (2n+4)-vertex models for each positive integer n. These models depend on a statistic called charge, and are associated to the quantum group U_q(gl(1|n)) [1]. Our results show a close and unexpected connection between Hamiltonian operators and solvability.The six-vertex model can be associated with Hamiltonians from classical Fock space, and we show that such a correspondence exists precisely when the Boltzmann weights are free fermionic. This allows us to prove that the free fermionic partition function is always a (skew) supersymmetric Schur function and then use the Berele-Regev formula to correct a result from [2]. Then, we prove a sharp solvability criterion for the six-vertex model with charge that provides the proper analogue of the free fermion condition. Building on results in [3], we show that this criterion exactly dictates when a charged model has a Hamiltonian operator acting on a Drinfeld twist of q-Fock space. The resulting partition function is then always a (skew) supersymmetric LLT polynomial.Chapter 3 considers the connections between lattice models and formal group laws. In particular, we exhibit a substitution corresponding to any formal group law into any solution to the Yang-Baxter equation. When applied to the R-matrix from the standard evaluation module for U_q(sl(n+1)), the resulting lattice models are related to those studied in [4], and their partition functions may have interpretations in higher cohomology of Schubert varieties. Then, Chapter 4 gives an exposition of lattice model proofs of some well-known identities for Schur polynomials. In addition, we introduce what we call a symmetrized version of the Yang-Baxter algebra and show how the Schur polynomial identities come from relations in this algebra.Running through this work is a thematic aspiration: that lattice models are “unreasonably effective” (in Ben Brubaker’s words) in expressing various phenomena throughout mathematics.