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Smith, Michael M. "PRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS." University of Akron / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=akron1164048974.
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Moore, Todd. "What calculus do students learn after calculus?" Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14090.
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Doctor of PhilosophyDepartment of Mathematics
Andrew Bennett
Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.
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Norris, J. R. "Malliavin calculus." Thesis, University of Oxford, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355791.
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Tall, Aliou. "From mathematics in logic to logic in mathematics : Boole and Frege." Thesis, University of York, 2002. http://etheses.whiterose.ac.uk/14163/.
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This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme.
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Stockham, Ty. "CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/898.
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This study investigates the effects of implementing a remediation program in a high school Advanced Placement Calculus AB course on student class grades and success in passing the AP Calculus AB exam.
A voluntary remediation program was designed to help students understand the key concepts and big ideas in beginning Calculus. Over a period of eight years the program was put into practice and data on student participation and achievement was collected. Students who participated in this program were given individualized recitation activities targeting their specific misunderstandings, and then given an opportunity to retest on chapter exams that they had taken prior to remediation. Students were able to improve their scores on the original chapter exams and their grade in the class by demonstrating a greater understanding of the material after participating in the remediation sessions. This process was repeated for all chapter exams given during the academic year.
In this study, a data analysis comparing the percent gain, after remediation, in each student’s overall class grade to their AP Calculus AB exam scores was conducted. Additionally, AP Calculus AB exam scores of students enrolled in these classes were compared to AP Calculus AB exam scores globally both pre and post implementation of the remediation program.
The results of this study demonstrate that there is a substantial positive correlation between student participation in the remediation program and greater success on the AP Calculus AB exam. The average AP Calculus AB score for the students enrolled in AP Calculus AB during the eight-year period of implementing the remediation program increased by over 9%.
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Stelljes, Scott. "Applications of Stochastic Calculus to Finance." UNF Digital Commons, 2004. http://digitalcommons.unf.edu/etd/267.
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Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The Pricing of Options and Corporate Liabilities" in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process.
As part of this paper, we present proofs of Ito's Formula and Girsanov's Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call option. There are two methods that result in the same price: the risk neutral valuation and the Black-Scholes partial differential equation.
A new model of the stock price process is presented in Section 4. This model was inspired by the model of Cox and Ross published in 1976. We develop the model such that a martingale measure will exist for the present value of the stock price. We fit data to the traditional geometric Brownian motion model and the new model and compare the resulting prices. The data fit some stocks well, but in some cases the new model provided a better fit. The price of a European call is calculated for both models for several different stocks.
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Bressler, Paul. "Schubert calculus in generalized cohomology." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/67103.
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Lindsay, J. M. "A quantum stochastic calculus." Thesis, University of Nottingham, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.356028.
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Stiles, Nancy L. Hathway Robert G. "Graphing calculators and calculus." Normal, Ill. Illinois State University, 1994. http://wwwlib.umi.com/cr/ilstu/fullcit?p9510432.
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Thesis (D.A.)--Illinois State University, 1994.Title from title page screen, viewed March 31, 2006. Dissertation Committee: Robert G. Hathway (chair), Lynn H. Brown, John A. Dossey, Arnold J. Insel, Patricia H. Klass. Includes bibliographical references (leaves 33-34) and abstract. Also available in print.
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Diepenveen, Emily. "Relational models of the lambda calculus." Thesis, University of Ottawa (Canada), 2008. http://hdl.handle.net/10393/27679.
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In [7], Ehrhard et al. present a model of the untyped lambda calculus built from an object without enough points in a cartesian closed category MRel. This thesis presents the background needed to construct and understand this model. In particular we describe what it means for models to have enough points and exhibit connections between MRel with various categorical models of lambda calculus in the literature. In particular, we are able to relate the graph model to MRel. We also describe connections with various kinds of Kleisli categories arising from comonads and their associated theory.
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Helfgott, Michel. "Calculus of One Variable: An Eclectic Approach." Digital Commons @ East Tennessee State University, 2012. http://amzn.com/1477633871.
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This textbook is intended for a two-semester course on calculus of one variable. The target audience is comprised of first-year students in biology, chemistry, physics and other related disciplines. The title of the book reflects the fact that it is not limited to one single approach to calculus. Rather, we use graphing calculators or applications whenever they are necessary to introduce certain topics. Nonetheless, as expected, a conceptual framework permeates the whole book. A distinctive characteristic of the book is the early introduction of sequences and geometric series, and a gradual development of simple differential equations, as well as the use of linear regression to analyze data. The core of the book is to be found in the first three chapters, in which examples from biology, chemistry and physics are analyzed with care, emphasizing the close links between calculus and the natural sciences. The last two chapters, or sections thereof, can be used as a sort of capstone in order to show how mathematics helps in the understanding of enzyme kinetics and transport across cell membranes.https://dc.etsu.edu/etsu_books/1064/thumbnail.jpg
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Scalfano, Denise. "Pricing Financial Derivatives Using Stochastic Calculus." Ohio University Honors Tutorial College / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ouhonors1492772147858348.
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Helfgott, Michel, and Darrell Moore. "Introductory Calculus for the Natural Sciences." Digital Commons @ East Tennessee State University, 2013. http://amzn.com/1453880836.
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This textbook is intended for first-year college students in biology, chemistry, or physics. Its most distinctive feature is the central role played by applications to the natural sciences. Considering that nowadays students have access to graphing calculators that can solve complicated integrals, little or no space has been devoted in the book to integrals that require subtle changes of variables. Rather, we choose to concentrate on the basic techniques of integration and stress the solution of applied problems, especially those that use real data. We envision a calculus course where students not only learn to calculate derivatives or solve integrals, but are also able to discuss the validity of a model and estimate parameters.https://dc.etsu.edu/etsu_books/1059/thumbnail.jpg
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Joshi, Mark S. (Mark Suresh). "A precision calculus of paired Lagrangian distributions." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/33512.
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Gibson, Kathleen Renae. "Nonstandard analysis based calculus." CSUSB ScholarWorks, 1994. https://scholarworks.lib.csusb.edu/etd-project/915.
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In the first part of the project the elementary development of an extended number system called Hyperreals is discussed. The second half of this project develops the basics of Nonstandard Analysis, including the theory of ultrafilters, and the formal construction of the Hyperreals.
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Ho, Theang. "Analysis of an online placement exam for calculus." Kansas State University, 2010. http://hdl.handle.net/2097/4650.
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Master of ScienceDepartment of Mathematics
Andrew G. Bennett
An online mathematics placement exam was administered to new freshmen enrolled at Kansas State University for the Fall of 2009. The purpose of this exam is to help determine which students are prepared for a college Calculus I or Calculus II course. Problems on the exam were analyzed and grouped together using different techniques including expert analysis and item response theory to determine which problems were similar or even relevant to placement. Student scores on the exam were compared to their performance on the final exam at the end of the course as well as ACT data. This showed how well the placement exam indicated which students were prepared. A model was created using ACT information and the new information from the placement exam that improved prediction of success in a college calculus course. The new model offers a significant improvement upon what the ACT data provides to advisers. Suggestions for improvements to the test and methodology are made based upon the analysis.
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Permpoontanalarp, Yongyuth. "Embedding dynamic logic in situation calculus." Thesis, Imperial College London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299487.
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Teuscher, Dawn. "Two paths to advanced placement calculus an examination of secondary students' mathematical understanding emerging from integrated and single-subject curricula /." Diss., Columbia, Mo. : University of Missouri-Columbia, 2008. http://hdl.handle.net/10355/5530.
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Thesis (Ph. D.)--University of Missouri-Columbia, 2008.The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 17, 2009) Vita. Includes bibliographical references.
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Schrohe, Elmar. "A short introduction to Boutet de Monvel's calculus." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2569/.
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Maniccia, L., and M. Mughetti. "Weyl calculus for a class of subelliptic operators." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2603/.
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Schulze, Bert-Wolfgang. "Pseudo-differential calculus on manifolds with geometric singularities." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2009/3020/.
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Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order
of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.
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Beaulieu, Guy. "Finding presheaf models for the finite pi-calculus." Thesis, University of Ottawa (Canada), 2002. http://hdl.handle.net/10393/6206.
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This thesis provides a fully-abstract (set theoretical) model for the finite pi-calculus with respect to late-bisimulation and late-equivalence relations. This is achieved by amalgamating the works by M. P. Fiore, E. Moggi and D. Sangiorgi, and I. Stark. In their respective works the authors construct categorical models, and define a meta-language in which the finite pi-calculus can be interpreted. We discuss the general properties a categorical model should satisfy to be considered an appropriate model for the finite pi-calculus. In particular, I show that the categorical model based on the syntax provides a fully abstract model for the finite pi-calculus. Finally, I include all the details of the model which were often omitted by the above authors. We extend the discussion by examining alternative categorical constructs for the model of the finite pi-calculus, for example we use doubly closed categories which are a main focus of Bunched Logic by P. W. O'Hearn and D. J. Pym.
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Kalisz, Piotr. "On partial semigroup models for the Lambek calculus." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq29728.pdf.
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Deserti, Francesca. "Aspects of learning and understanding in multivariable calculus." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/37140.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (leaf 55).
In this thesis we study the processes by which university students solve problems in multivariable calculus. Our data consists of a series of questionnaires and interviews with students enrolled in a vector calculus class at MIT. We interpret our observations in the light of previous research into the acquisition of mathematical knowledge and understanding.
by Francesca Deserti.
Ph.D.
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Pereira, Luis Alexandre Meira Fernandes Alves. "Goodwillie calculus and algebras over a spectral operad." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/82440.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2013.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 131-133).
The overall goal of this thesis is to apply the theory of Goodwillie calculus to the category Algo of algebras over a spectral operad. Its first part generalizes many of the original results of Goodwillie in [14] so that they apply to a larger class of model categories and hence be applicable to Algo. The second part then applies that generalized theory to the Algo categories. The main results here are: an understanding of finitary homogeneous between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary n-excisive functors can not distinguish between Algo and Algo,, the category of algebras over the truncated operad O<; and a weak form of the chain rule between the algebra categories, analogous to the one found in [1].
by Luis Alexandre Meira Fernandes Alves Pereira.
Ph.D.
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Tynan, Philip Douglas. "Equivariant Weiss Calculus and Loops of Stiefel Manifolds." Thesis, Harvard University, 2016. http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493281.
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In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds O(V ;W) and U(V ;W) of orthogonal and unitary, respectively, maps V -> V ⊕W stably split as a wedge sum of Thom spaces defined over Grassmanians. Additionally, they produced a similar filtration for loops on SU(V), with a similar splitting. A few years later, Michael Crabb made explicit the equivariance of the Stiefel manifold splittings and conjectured that the splitting of the loop space was equivariant as well. However, it has long been unknown whether the loop space of the real Steifel manifold (or even the special case of ΩSO_n) has a similar splitting.
In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds O(V ;W) and U(V ;W) of orthogonal and unitary, respectively, maps V → V ⊕W stably split as a wedge sum of Thom spaces defined over Grassmanians. Additionally, they produced a similar filtration for loops on SU(V), with a similar splitting. A few years later, Michael Crabb made explicit the equivariance of the Stiefel manifold splittings and conjectured that the splitting of the loop space was equivariant as well. However, it has long been unknown whether the loop space of the real Steifel manifold (or even the special case of ΩSOn) has a similar splitting.
Here, inspired by the work of Greg Arone that made use of Weiss’ orthogonal calculus to generalize the results of Mitchell and Richter, we obtain an Z~2Z-equivariant splitting theorem using an equivariant version of Weiss calculus. In particular, we show that ΩU(V ;W) has an equivariant stable splitting when dim W > 0. By considering the (geometric) fixed points of this loop space, we also obtain, as a corollary, a stable splitting of the space Ω(U(V ;W),O(V_R;W_R)) of paths in U(V ;W) from I to a point of O(V_R;W_R) as well. In particular, by setting W = C, this gives us a stable splitting of Ω(SUn / SOn). In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds O(V ;W) and U(V ;W) of orthogonal and unitary, respectively, maps V → V ⊕W stably split as a wedge sum of Thom spaces defined over Grassmanians. Additionally, they produced a similar filtration for loops on SU(V), with a similar splitting. A few years later, Michael Crabb made explicit the equivariance of the Stiefel manifold splittings and conjectured that the splitting of the loop space was equivariant as well. However, it has long been unknown whether the loop space of the real Steifel manifold (or even the special case of ΩSOn) has a similar splitting.
Here, inspired by the work of Greg Arone that made use of Weiss’ orthogonal calculus to generalize the results of Mitchell and Richter, we obtain an Z~2Z-equivariant splitting theorem using an equivariant version of Weiss calculus. In particular, we show that ΩU(V ;W) has an equivariant stable splitting when dim W > 0. By considering the (geometric) fixed points of this loop space, we also obtain, as a corollary, a stable splitting of the space Ω(U(V ;W),O(V_R;W_R)) of paths in U(V ;W) from I to a point of O(V_R;W_R) as well. In particular, by setting W = C, this gives us a stable splitting of Ω(SUn / SOn).Mathematics
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Charoenphon, Sutthirut. "Green's Functions of Discrete Fractional Calculus Boundary Value Problems and an Application of Discrete Fractional Calculus to a Pharmacokinetic Model." TopSCHOLAR®, 2014. http://digitalcommons.wku.edu/theses/1327.
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Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. In Chapter 2, we use these basic results in the analysis of a linear fractional difference equation. Existence of solutions to this difference equation is then established for both initial conditions (IVP) and two-point boundary conditions (BVP). In Chapter 3, Green's functions are introduced and discussed, along with examples. Instead of using Cauchy functions, the technique of finding Green's functions by a traditional method is demonstrated and used throughout this chapter. The solutions of the BVP play an important role in analysis and construction of the Green's functions. Then, Green's functions for the discrete calculus case are calculated using particular problems, such as boundary value problems, discrete boundary value problems (DBVP) and fractional boundary value problems (FBVP). Finally, we demonstrate how the Green's functions of the FBVP generalize the existence results of the Green's functions of DVBP. In Chapter 4, different compartmental pharmacokinetic models are discussed. This thesis limits discussion to the one-compartmental model. The Mathematica FindFit command and the statistical computational techniques of mean square error (MSE) and cross-validation are discussed. Each of the four models (continuous, continuous fractional, discrete and discrete fractional) is used to compute the MSE numerically with the given data of drug concentration. Then, the best fit and the best model are obtained by inspection of the resulting MSE. In the last Chapter, the results are summarized, conclusions are drawn, and directions for future work are stated.
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Poloczek, Felix. "Stochastic network calculus with martingales." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/89853/.
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The practicality of the stochastic network calculus (SNC) is often questioned on grounds of looseness of its performance bounds. The reason for its inaccuracy lies in the usage of too elementary tools from probability theory, such as Boole’s inequality, which is unable to account for correlations and thus inappropriate to properly model arrival flows. In this thesis, we propose an extension of stochastic network calculus that characterizes its main objects, namely arrival and service processes, in terms of martingales. This characterization allows to overcome the shortcomings of the classical SNC by leveraging Doob’s inequality to provide more accurate performance bounds. Additionally, the emerging stochastic network calculus with martingales is quite versatile in the sense that queueing related operations like multiplexing and scheduling directly translate into operations of the corresponding martingales. Concretely, the framework is applied to analyze the per-flow delay of various scheduling policies, the performance of random access protocols, and queueing scenarios with a random number of parallel flows. Moreover, we show our methodology is not only relevant within SNC but can be useful also in related queueing systems. E.g., in the context of multi-server systems, we provide a martingale-based analysis of fork-join queueing systems and systems with replications. Throughout, numerical comparisons against simulations show that the Martingale bounds obtained with Doob’s inequality are not only remarkably accurate, but they also improve the Standard SNC bounds by several orders of magnitude.
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Marko, Benjamin David. "Teaching Concepts Foundational to Calculus Using Inquiry and Technology." University of Akron / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=akron1144777991.
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Ling, Paul David. "The Ito calculus : a vector integral approach." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.238210.
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Gratwick, Richard. "Singular minimizers in the calculus of variations." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/47653/.
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This thesis examines the possible failure of regularity for minimizers of onedimensional variational problems. The direct method of the calculus of variations gives rigorous assurance that minimizers exist, but necessarily admits the possibility that minimizers might not be smooth. Regularity theory seeks to assert some extra smoothness of minimizers. Tonelli's partial regularity theorem states that any absolutely continuous minimizer has a (possibly infinite) classical derivative everywhere, and this derivative is continuous as a function into the extended real line. We examine the limits of this theorem. We find an example of a reasonable problem where partial regularity fails, and examples where partial regularity holds, but the infinite derivatives of minimizers permitted by the theorem occur very often, in precise senses. We construct continuous Lagrangians, strictly convex and superlinear in the third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem. Davie showed that any compact null set can occur as the singular set of a minimizer to a problem given via a smooth Lagrangian with quadratic growth. The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We give a new proof of the result, but constructing also a Lagrangian with arbitrary superlinear growth, and in which the Lavrentiev phenomenon does not occur in the problem. Universal singular sets record how often a given Lagrangian can have minimizers with infinite derivative. Despite being negligible in terms of both topology and category, they can have dimension two: any compact purely unrectifiable set can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity. We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible characterization of universal singular sets.
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Taheri, Ali. "Local minimizers in the calculus of variations." Thesis, Heriot-Watt University, 1997. http://hdl.handle.net/10399/656.
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Davidson, Timothy A. S. "Formal verification techniques using quantum process calculus." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/51368/.
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Quantum communication is a rapidly growing area of research and development. While the successful construction of a large-scale quantum computer may be some years away, there are already commercial implementations of secure communication using quantum cryptography. The application of formal methods to classical communication and cryptographic systems has been very successful, and is now widely used in industry by organisations such as Intel, Microsoft and NASA. There is reason to believe that similar benefits can be expected for the verification of quantum systems. In this thesis, we focus on the use of process calculus, specifically Communicating Quantum Processes (CQP), for the analysis of quantum protocols. Congruence relations are an important aspect of process calculus, since they provide the foundation for equational reasoning. Previous work on congruence relations for quantum processes excluded the classical information arising from measurements, and was therefore unable to analyse many of the interesting known quantum communication protocols. Developing a congruence relation for general quantum processes is difficult because of the interaction between measurement, entanglement and parallel composition. We define a labelled transition relation for CQP in order to describe external interactions. Based on this semantics, we define a notion of observational equivalence for CQP processes, namely probabilistic branching bisimilarity. We find that this relation is not preserved by parallel composition, however we are able to gain a deeper understanding of the link between probabilistic branching and measurement. Based on this newfound understanding, we present a novel semantics for quantum processes, combining mixed quantum states with probabilistic branching. With respect to this new semantic model, we define full probabilistic branching bisimilarity and prove that it is a congruence. We use this congruence relation to discuss an axiomatic approach to the verification of quantum processes. The quantum teleportation protocol is used as a primary example throughout, and we prove that it is congruent to a quantum channel. We define a translation from CQP to the Quantum Model Checker (QMC) in order to provide automated verification techniques using CQP specifications. We prove that this translation preserves the semantics of CQP processes, thereby enabling a multifaceted approach to formal verification by enhancing the manual techniques of process calculus with the benefits of model checking.
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Arora, Raman. "Analysis of Economic Models Through Calculus of Variations." TopSCHOLAR®, 2005. http://digitalcommons.wku.edu/theses/453.
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This thesis is a combination of two science fields: Mathematics and Economics. Mathematics is often used to formulate a clear and concise solution to economic problems. In my observation calculus of variation has often been used in various macroeconomic problems. This mathematical method deals with maximizing or minimizing of various objective functions given a set of constraints. This topic brings out one of the best ways to show the relationship between mathematics and economics. My thesis consists of three parts: The first chapter contains a review of the calculus of variations. Basic definitions and important conditions have been stated. The aim of this chapter was to set the groundwork for understanding calculus of variations so that it can be used in solving various economics models. In the second chapter we study an economic model from which calculus of variations has been used to solve it. The macroeconomic model deals with optimizing the social welfare function. The entire working of the model has been discussed and documented in the thesis report. The third chapter deals with the analysis of the Lucas model which concentrated on how the accumulation of human capital impacts the growth rate of the economy. Lucas assumes that the growth rate of the human capital is linearly related to its level. If we abandon this assumption, will the optimal value of the time devoted to education in the steady state exist? If it exists, will it be same or different? So we introduced a new model in which the only modification we made to the Lucas model was in the equation that describes the process of human accumulation by introducing a nonlinear component. On investigation of this new model we have shown that it is possible that optimal behavior for an individual can be not to educate himself.
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Krimpogiannis, Michail. "The Double Layer Potential Operator Through Functional Calculus." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-81924.
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Layer potential operators associated to elliptic partial differential equations have been an object of investigation for more than a century, due to their contribution in the solution of boundary value problems through integral equations. In this Licentiate thesis we prove the boundedness of the double layer potential operator on the Hilbert space of square integrable functions on the boundary, associated to second order uniformly elliptic equations in divergence form in the upper half-space, with real, possibly non-symmetric, bounded measurable coefficients, that do not depend on the variable transversal to the boundary. This uses functional calculus of bisectorial operators and is done through a series of four steps. The first step consists of reformulating the second order partial differential equation as an equivalent first order vector-valued ordinary differential equation in the upper halfspace. This ordinary differential equation has a particularly simple form and it is here that the bisectorial operator corresponding to the original divergence form equation appears as an infinitesimal generator. Solving this ordinary differential through functional calculus comprises the second step. This is done with the help of the holomorphic semigroup associated to the restriction of the bisectorial operator to an appropriate spectral subspace; the restriction of the operator is a sectorial operator and the holomorphic semigroup is well-defined on the spectral subspace. The third step is the construction of the fundamental solution to the original divergence form equation. The behaviour of this fundamental solution is analogous to the behaviour of the fundamental solution to the classical Laplace equation and its conormal gradient of the adjoint fundamental solution is used as the kernel of the double layer potential operator. This third step is of a different nature than the others, insofar as it does not involve tools from functional calculus. In the last step Green’s formula for solutions of the divergence form partial differential equation is used to give a concrete integral representation of the solutions to the divergence form equation. Identifying this Green’s formula with the abstract formula derived by functional calculus yields the sought-after boundedness of the double layer potential operator, for coefficients of the particular form mentioned above.
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Loya, Paul 1971. "On the b-pseudodifferential calculus on manifolds with corners." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/47472.
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Bowie, Lynn Heather. "A learning theory approach to students' misconceptions in calculus." Master's thesis, University of Cape Town, 1998. http://hdl.handle.net/11427/9556.
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Bibliography: leaves 129-138.This study analyses students' errors in calculus through the lens of learning theories. The subjects in this study were 117 students enrolled in a calculus course for students from disadvantaged educational backgrounds at the University of Cape Town. A coding scheme to categorise the errors that these students made in the final examination was developed. This categorisation was supported by error data generated through the administration of a conceptual test and follow-up interviews. The pattern of errors in the coding scheme suggests that the students' perception of algebra is largely that of a "game of letters". As a result of this their construction of calculus knowledge is based on the rehearsal of algorithmic procedures. Their errors indicate that they develop linking and extending mechanisms to deal with the multiplicity of rules that are generated from this process of rehearsal.
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Hamm, D. Michael (Don Michael). "The Association Between Computer- Oriented and Non-Computer-Oriented Mathematics Instruction, Student Achievement, and Attitude Towards Mathematics in Introductory Calculus." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc332306/.
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The purposes of this study were (a) to develop, implement, and evaluate a computer-oriented instructional program for introductory calculus students, and (b) to explore the association between a computer-oriented calculus instructional program, a non-computer-oriented calculus instructional program, student achievement on three selected calculus topics, and student attitude toward mathematics.
An experimental study was conducted with two groups of introductory calculus students during the Spring Semester, 1989. The computer-oriented group consisted of 32 students who were taught using microcomputer calculus software for in-class presentations and homework assignments. The noncomputer-oriented group consisted of 40 students who were taught in a traditional setting with no microcomputer intervention.
Each of three experimenter-developed achievement examinations was administered in a pretest/posttest format with the pretest scores being used both as a covariate and in determining the two levels of student prior knowledge of the topic.
For attitude toward mathematics, the Aiken-Dreger Revised Math Attitude Scale was administered in a pretest/ posttest format with the pretest scores being used as a covariate. Students were also administered the MAA Calculus Readiness Test to determine two levels of calculus prerequisite skill mastery.
An ANCOVA for achievement and attitude toward mathematics was performed by treatment, level, and interaction of treatment and level. Using a .05 level of significance, there was no significant difference in treatments, levels of prior knowledge of topic, nor interaction when achievement was measured by each of the three achievement examination posttests. Furthermore, there was no significant difference between treatments, levels of student prerequisite skill mastery, and interaction when attitude toward mathematics was measured, at the .05 level of significance.
It was concluded that the use of the microcomputer in introductory calculus instruction does not significantly effect either student achievement in calculus or student attitude toward mathematics.
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Kimeu, Joseph M. "Fractional Calculus: Definitions and Applications." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/115.
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Rabb-Liu, Amy Felice 1968. "Teaching methods and student understanding in calculus." Diss., The University of Arizona, 1997. http://hdl.handle.net/10150/288725.
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This study is a comparative case study of what three college calculus teachers did in their classrooms and what their students understood about the concept of derivatives. The teachers were solicited on the basis of peer, supervisor and student recommendations as being good teachers; several volunteer student subjects were selected from each class. Using a naturalistic participant-observer paradigm, the data were collected primarily via extensive classroom observations and in-depth interviews with the teachers and students. Examination of written work, such as student exams, was employed for additional confirmation of hypotheses generated in the field. This study contributes to the bodies of knowledge on pedagogy, effective teaching, classroom dynamics, student understanding and teacher beliefs. The results should be of interest to teachers, teacher educators, mathematics text authors and people interested in how students learn and think about mathematics at the collegiate level. The study of these three classrooms reveals that there is a variety of effective teaching models for undergraduate calculus classrooms. There were, however, important commonalties among these models, the examination of which leads to some characterization of effective teaching practices. These teachers kept the focus on what their students were learning, rather than on covering material. In three different ways, these teachers each gave their students the opportunity to interact with the mathematics before the lesson ended. All three teachers displayed a willingness to grow and learn as teachers. Calculus students do not always learn what their teachers think they have taught. The students in this study displayed a variety of mistaken ideas about the concept of derivative and about other mathematical topics. For example, many students had trouble distinguishing between properties of the function and properties of the derivative. Some students believed that the derivative at a point was a line, rather than the numerical value associated with the slope of a line. Students and teachers disagreed about the correct definition of the derivative, with students attributing little importance to the idea of limits.
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Blum, William. "The safe lambda calculus." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:537d45e0-01ac-4645-8aba-ce284ca02673.
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We consider a syntactic restriction for higher-order grammars called safety that constrains occurrences of variables in the production rules according to their type-theoretic order. We transpose and generalize this restriction to the setting of the simply-typed lambda calculus, giving rise to what we call the safe lambda calculus. We analyze its expressivity and obtain a result in the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus: the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a similar characterization for representable word functions. We then examine the complexity of deciding beta-eta equality of two safe simply-typed terms and show that this problem is PSPACE-hard. The safety restriction is then extended to other applied lambda calculi featuring recursion and references such as PCF and Idealized Algol (IA for short). The next contribution concerns game semantics. We introduce a new concrete presentation of this semantics using the theory of traversals. It is shown that the revealed game denotation of a term can be computed by traversing some souped-up version of the term's abstract syntax tree using adequately defined traversal rules. Based on this presentation and via syntactic reasoning we obtain a game-semantic interpretation of safety: the strategy denotations of safe lambda-terms satisfy a property called P-incremental justification which says that the player's moves are always justified by the last pending opponent's move of greater order occurring in the player's view. Next we look at models of the safe lambda calculus. We show that these are precisely captured by Incremental Closed Categories. A game model is constructed and is shown to be fully abstract for safe IA. Further, it is effectively presentable: two terms are equivalent just if they have the same set of complete O-incrementally justified plays---where O-incremental justification is defined as the dual of P-incremental justification. Finally we study safety from the point of view of algorithmic game semantics. We observe that in the third-order fragment of IA, the addition of unsafe contexts is conservative for observational equivalence. This implies that all the upper complexity bounds known for the lower-order fragments of IA also hold for the safe fragment; we show that the lower-bounds remain the same as well. At order 4, observational equivalence is known to be undecidable for IA. We conjecture that for the order-4 safe fragment of IA, the problem is reducible to the DPDA-equivalence problem and is thus decidable.
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Witt, Ingo. "A calculus for a class of finitely degenerate pseudodifferential operators." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2624/.
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For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.
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Hillyard, Cinnamon. "A Maple Package for the Variational Calculus." DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7124.
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The HELMHOLTZ package, written in Maple V, is a collection of commands to support research in the variational calculus. These commands include the standard operators on differential forms, Euler-Lagrange operators, homotopy operators, Lie bracket, Lie derivatives, and the prolongation of a vector field. We give a brief introduction to the variational calculus. We describe each of the commands in the HELMHOLTZ package completely along with numerous examples of each. Applications of the package include verification of symmetry groups for differential equations, solving the inverse problem of the calculus of variations, computing generalized symmetries, and finding variational integrating factors. A complete listing of the Maple code for HELMHOLTZ is found in an appendix.
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Jensen, Taylor Austin. "A study of the relationship between introductory calculus students' understanding of function and their understanding of limit." Thesis, Montana State University, 2009. http://etd.lib.montana.edu/etd/2009/jensen/JensenT.pdf.
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Introductory calculus students are often successful in doing procedural tasks in calculus even when their understanding of the underlying concepts is lacking, and these conceptual difficulties extend to the limit concept. Since the concept of limit in introductory calculus usually concerns a process applied to a single function, it seems reasonable to believe that a robust understanding of function is beneficial to and perhaps necessary for a meaningful understanding of limit. Therefore, the main goal of this dissertation is to quantitatively correlate students' understanding of function and their understanding of limit. In particular, the correlation between the two is examined in the context of an introductory calculus course for future scientists and engineers at a public, land grant research university in the west. In order to measure the strength of the correlation between understanding of function and understanding of limit, two tests-the Precalculus Concept Assessment (PCA) to measure function understanding and the Limit Understanding Assessment (LUA) to measure limit understanding-were administered to students in all sections of the aforementioned introductory calculus course in the fall of 2008. A linear regression which included appropriate covariates was utilized in which students' scores on the PCA were correlated with their scores on the LUA. Nonparametric bivariate correlations between students' PCA scores and students' scores on particular subcategories of limit understanding measured by the LUA were also calculated. Moreover, a descriptive profile of students' understanding of limit was created which included possible explanations as to why students responded to LUA items the way they did. There was a strong positive linear correlation between PCA and LUA scores, and this correlation was highly significant (p<0.001). Furthermore, the nonparametric correlations between PCA scores and LUA subcategory scores were all statistically significant (p<0.001). The descriptive profile of what the typical student understands about limit in each LUA subcategory supplied valuable context in which to interpret the quantitative results. Based on these results, it is concluded that understanding of function is a significant predictor of future understanding of limit. Recommendations for practicing mathematics educators and indications for future research are provided.
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Calzi, Mattia. "Functional Calculus on Homogeneous Groups." Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85740.
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In the first part of the thesis, we consider the following problem.
Let G be a homogeneous group, and let (L_1,...,L_n) be a jointly hypoelliptic commutative finite family of formally self-adjoint, homogeneous, left-invariant differential operators without constant terms.
Then, the operators L_j are essentially self-adjoint as operators on L^2(G) with domain C^infty_c(G), and their closures commute emph{as self-adjoint operators}.
Therefore, one may consider the joint functional calculus associated with the family (L_1,...,L_n).
More precisely, for every bounded Borel measurable function $m$ on $R^n$, the corresponding operator m(L_1,...,L_n) commutes with left translations, so that it admits a unique right convolution kernel K(m).
The so-defined kernel transform K then maps S(R^n) continuously into S(G), and L^2(eta) isometrically into L^2(G) for some uniquely determined positive Radon measure eta on R^n; this latter property can be considered as an analogue of the Plancherel isomorphism.
In addition, K maps L^1(eta) continuously into C_0(G), and this property can be considered as an analogue of the Riemann--Lebesgue lemma.
We focus on the following properties of K:
(RL) if K(m)in L^1(G), then m can be taken in C_0(R^n): this is again an analogue of the Riemann--Lebesgue lemma;
(S) if K(m)in S(G), then m can be taken in S(R^n).
We prove that properties (RL) and (S) are compatible with products, and we characterize the Rockland operators which satisfy property (S) when the underlying group G is abelian.
We then consider the case of 2-step stratified groups, and families whose elements are either sub-Laplacians or vector fields of homogeneous degree 2. In this setting, we prove several sufficient conditions, as well as some necessary ones, for properties (RL) and (S); we even characterize them in some more specific settings.
In addition, we study the case of general (that is, not necessarily homogeneous) sub-Laplacians on 2-step stratified groups, and prove that they always satisfy properties (RL) and (S).
We also prove that, under some mild assumptions, a multiplier m can be taken so as to satisfy Mihlin--Hormander conditions of order infinity if and only if the corresponding kernel K(m) satisfies Calderon--Zygmund conditions of order infinity.
In the second part of the thesis, we present some results which are joint work with T. Bruno.
We fix the standard sub-Laplacian on an H-type group, and consider its heat kernel (p_s)_{s>0}. We provide sharp asymptotic estimates at $infty$ for basically all the derivatives of p_1. Because of the homogeneity of the family (p_s), these estimates can also be considered as short-time asymptotics.
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Rector, R. Blake. "Generalized Differential Calculus and Applications to Optimization." PDXScholar, 2017. https://pdxscholar.library.pdx.edu/open_access_etds/3627.
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This thesis contains contributions in three areas: the theory of generalized calculus, numerical algorithms for operations research, and applications of optimization to problems in modern electric power systems. A geometric approach is used to advance the theory and tools used for studying generalized notions of derivatives for nonsmooth functions. These advances specifically pertain to methods for calculating subdifferentials and to expanding our understanding of a certain notion of derivative of set-valued maps, called the coderivative, in infinite dimensions. A strong understanding of the subdifferential is essential for numerical optimization algorithms, which are developed and applied to nonsmooth problems in operations research, including non-convex problems. Finally, an optimization framework is applied to solve a problem in electric power systems involving a smart solar inverter and battery storage system providing energy and ancillary services to the grid.
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Cockburn, Andre. "Martingale representation in a distribution-theoretic quantum stochastic calculus." Thesis, University of Nottingham, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319603.
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Krainer, Thomas. "The calculus of Volterra Mellin pseudodifferential operators with operator-valued symbols." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2618/.
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We introduce the calculus of Mellin pseudodifferential operators parameters based on "twisted" operator-valued Volterra symbols as well aas the abstract Mellin calclus with holomorphic symbols. We establish the properties of the symblic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e. g., for the Leibniz-product, kernel cut-off, and Mellin quantization.
Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi.
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Krainer, Thomas. "On the calculus of pseudodifferential operators with an anisotropic analytic parameter." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2620/.
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We introduce the Volterra calculus of pseudodifferential operators with an anisotropic analytic parameter based on "twisted" operator-valued Volterra symbols. We establish the properties of the symbolic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side. In particular, we investigate the kernel cut-off operator via direct oscillatory integral techniques purely on symbolic level.
We discuss the notion of parabolic for the calculus of Volterra operators, and construct Volterra parametrices for parabolic operators within the calculus.
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Gosson, Maurice A. de. "Extended Weyl calculus and application to the phase-space Schrödinger equation." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2987/.
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We show that the Schr¨odinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extendedWeyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.