@phdthesis{Pfeffer2017, author = {Pfeffer, Wolfgang}, title = {Qualitative Entwicklung der Begriffsbildung im Fach Mathematik in der Studieneingangsphase}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-4948}, school = {Universit{\"a}t Passau}, pages = {x, 331 Seiten}, year = {2017}, abstract = {Der {\"U}bergang vom sekund{\"a}ren zum terti{\"a}ren Bildungsbereich im Fach Mathematik stellt in den vergangenen Jahren ein zentrales Forschungsinteresse der mathematikdidaktischen Forschung dar. Das Forschungsfeld macht hierbei drei grundlegende Bedingungsfaktoren aus: epistemologische und kognitive, soziologische und kulturelle sowie didaktische Schwierigkeiten. Der Schwerpunkt der vorliegenden Arbeit liegt auf dem ersten Bereich, der den sich wandelnden Charakter der Mathematik inkludiert. F{\"u}r viele Studienanf{\"a}ngerinnen und -anf{\"a}nger ist der Wechsel von der Schule zur Universit{\"a}t "traumatisch", da sich das Anforderungsprofil an der Hochschule doch deutlich von dem bisher Gewohnten abhebt. Gerade die Studieneingangsphase ist f{\"u}r den sp{\"a}teren Studienerfolg als wichtigster Abschnitt anzusehen; werden die vielfach vorhandenen fachlichen Defizite sowie falschen Vorstellungen nicht beseitigt, f{\"u}hrt dies oft zu einem vorzeitigen Studienabbruch. Untersuchungen zur mathematischen Begriffsbildung {\"u}ber einen l{\"a}ngeren Zeitraum im Verlauf eines Hochschulstudiums fehlen bislang. Hieraus ergibt sich die Motivation f{\"u}r das in dieser Arbeit beschriebene Forschungsprojekt, das sich in drei Teile gliedert. Der erste Teil fasst den Stand der Forschung zum {\"U}bergang Schule -- Hochschule im Fach Mathematik zusammen. Der Fokus richtet sich dabei auf den sich wandelnden Charakter der Mathematik sowie die unterschiedlichen Anforderungen in Bezug auf die Begriffsbildung. Der zweite Teil stellt mathematisches Vorwissen als m{\"o}glichen Bedingungsfaktor f{\"u}r den Studienerfolg dar und beschreibt anschließend Konzeption, Auswertung sowie die Diskussion der Ergebnisse eines Fragebogens zum Hintergrundwissen Mathematik. Der dritte Teil der Arbeit beschreibt eine qualitative und l{\"a}ngsschnittlich durchgef{\"u}hrte, leitfadengest{\"u}tzte Interviewstudie, in deren Rahmen die gleiche Stichprobe im Laufe der ersten beiden Semester zu drei unterschiedlichen Zeitpunkten befragt wurde. Im Hinblick auf die praktische Relevanz f{\"u}r die Lehre wurden anhand der {\"A}ußerungen der Studierenden Fehlvorstellungen hinsichtlich der mathematischen Konzepte identifiziert und kategorisiert. Weiter wird eine Klassifikation der studentischen Vorstellungen hinsichtlich deren wissenschaftlichen Qualit{\"a}t vorgenommen sowie die Entwicklung dieser {\"u}ber die drei Interviewzeitpunkte hinweg beschrieben. Die Ergebnisse der Arbeit zeigen einerseits massive Defizite der Studienanf{\"a}ngerinnen und -anf{\"a}nger in Kenntnissen und Fertigkeiten mathematischen Grundwissens. Dar{\"u}ber hinaus erfolgt auch die Begriffsbildung resp. -entwicklung hinsichtlich zentraler mathematischer Konzepte der Hochschulmathematik auf sehr niedrigem Niveau. Als Indiz hierf{\"u}r zeugen die vielf{\"a}ltig identifizierten Fehlvorstellungen, die sich {\"u}ber alle Interviewzeitpunkte hinweg zeigten. Hieraus resultiert eine verst{\"a}rkte Notwendigkeit der F{\"o}rderung der Begriffsbildung sowie -entwicklung in der Studieneingangsphase.}, subject = {Begriffsbildung}, language = {de} } @phdthesis{Ruppert2017, author = {Ruppert, Julia}, title = {Asymptotic Expansion for the Time Evolution of the Probability Distribution Given by the Brownian Motion on Semialgebraic Sets}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-5069}, school = {Universit{\"a}t Passau}, pages = {vi, 174 S.}, year = {2017}, abstract = {In this thesis, we examine whether the probability distribution given by the Brownian Motion on a semialgebraic set is definable in an o-minimal structure and we establish asymptotic expansions for the time evolution. We study the probability distribution as an example for the occurrence of special parameterized integrals of a globally subanalytic function and the exponential function of a globally subanalytic function. This work is motivated by the work of Comte, Lion and Rolin, which considered parameterized integrals of globally subanalytic functions, of Cluckers and Miller, which examined parameterized integrals of constructible functions, and by the work of Cluckers, Comte, Miller, Rolin and Servi, which treated oscillatory integrals of globally subanalytic functions. In the one dimensional case we show that the probability distribution on a family of sets, which are definable in an o-minimal structure, are definable in the Pfaffian closure. In the two-dimensional case we investigate asymptotic expansions for the time evolution. As time t approaches zero, we show that the integrals behave like a Puiseux series, which is not necessarily convergent. As t tends towards infinity, we show that the probability distribution is definable in the expansion of the real ordered field by all restricted analytic functions if the semialgebraic set is bounded. For this purpose, we apply results for parameterized integrals of globally subanalytic functions of Lion and Rolin. By establishing the asymptotic expansion of the integrals over an unbounded set, we demonstrate that this expansion has the form of convergent Puiseux series with negative exponents and their logarithm. Subsequently, we get that the asymptotic expansion is definable in an o-minimal structure. Finally, we study the three-dimensional case and give the proof that the probability distribution given by the Brownian Motion behaves like a Puiseux series as time t tends towards zero. As t approaches infinity and the semialgebraic set is bounded, it can be ascertained that the probability distribution has the form of a constructible function by results of Cluckers and Miller and therefore it is definable in an o-minimal structure. If the semialgebraic set is unbounded, we establish the asymptotic expansions and prove that the probability distribution given by the Brownian Motion on unbounded sets has an asymptotic expansion of the form of a constructible function. In consequence of that, the asymptotic expansion is definable in an o-minimal structure.}, subject = {Brownsche Bewegung}, language = {en} } @phdthesis{Guetschow2018, author = {G{\"u}tschow, Silja}, title = {Roulets - Eine Integraltransformation zur Bestimmung gerichteter Singularit{\"a}ten}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-5929}, school = {Universit{\"a}t Passau}, pages = {vii, 104 Seiten}, year = {2018}, abstract = {In dieser Arbeit wird eine neue Integraltransformation, die Roulettransformation, eingef{\"u}hrt. Diese arbeitet mit anisotropen Skalierungen und Rotationen. Es wird gezeigt, dass die Roulettransformation allgemeine gerichtete Singularit{\"a}ten im Sinne von temperierten Distributionen aufl{\"o}st. Die Abklingraten an Punkt- sowie Liniensingularit{\"a}ten werden explizit angegeben.}, subject = {Integraltransformation}, language = {de} } @phdthesis{Kell2019, author = {Kell, Christian}, title = {A Structure-based Attack on the Linearized Braid Group-based Diffie-Hellman Conjugacy Problem in Combination with an Attack using Polynomial Interpolation and the Chinese Remainder Theorem}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-6476}, school = {Universit{\"a}t Passau}, pages = {xi, 209 Seiten}, year = {2019}, abstract = {This doctoral thesis is dedicated to improve a linear algebra attack on the so-called braid group-based Diffie-Hellman conjugacy problem (BDHCP). The general procedure of the attack is to transform a BDHCP to the problem of solving several simultaneous matrix equations. A first improvement is achieved by reducing the solution space of the matrix equations to matrices that have a specific structure, which we call here the left braid structure. Using the left braid structure the number of matrix equations to be solved reduces to one. Based on the left braid structure we are further able to formulate a structure-based attack on the BDHCP. That is to transform the matrix equation to a system of linear equations and exploiting the structure of the corresponding extended coefficient matrix, which is induced by the left braid structure of the solution space. The structure-based attack then has an empirically high probability to solve the BDHCP with significantly less arithmetic operations than the original attack. A third improvement of the original linear algebra attack is to use an algorithm that combines Gaussian elimination with integer polynomial interpolation and the Chinese remainder theorem (CRT), instead of fast matrix multiplication as suggested by others. The major idea here is to distribute the task of solving a system of linear equations over a giant finite field to several much smaller finite fields. Based on our empirically measured bounds for the degree of the polynomials to be interpolated and the bit size of the coefficients and integers to be recovered via the CRT, we conclude an improvement of the run time complexity of the original algorithm by a factor of n^8 bit operations in the best case, and still n^6 in the worst case.}, subject = {Kryptologie}, language = {en} } @phdthesis{Tueno2020, author = {Tueno, Anselme}, title = {Multiparty Protocols for Tree Classifiers}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-8251}, school = {Universit{\"a}t Passau}, pages = {xvii, 171 Seiten}, year = {2020}, abstract = {Cryptography is the scientific study of techniques for securing information and communication against adversaries. It is about designing and analyzing encryption schemes and protocols that protect data from unauthorized reading. However, in our modern information-driven society with highly complex and interconnected information systems, encryption alone is no longer enough as it makes the data unintelligible, preventing any meaningful computation without decryption. On the one hand, data owners want to maintain control over their sensitive data. On the other hand, there is a high business incentive for collaborating with an untrusted external party. Modern cryptography encompasses different techniques, such as secure multiparty computation, homomorphic encryption or order-preserving encryption, that enable cloud users to encrypt their data before outsourcing it to the cloud while still being able to process and search on the outsourced and encrypted data without decrypting it. In this thesis, we rely on these cryptographic techniques for computing on encrypted data to propose efficient multiparty protocols for order-preserving encryption, decision tree evaluation and kth-ranked element computation. We start with Order-preserving encryption (OPE) which allows encrypting data, while still enabling efficient range queries on the encrypted data. However, OPE is symmetric limiting, the use case to one client and one server. Imagine a scenario where a Data Owner (DO) outsources encrypted data to the Cloud Service Provider (CSP) and a Data Analyst (DA) wants to execute private range queries on this data. Then either the DO must reveal its encryption key or the DA must reveal the private queries. We overcome this limitation by allowing the equivalent of a public-key OPE. Decision trees are common and very popular classifiers because they are explainable. The problem of evaluating a private decision tree on private data consists of a server holding a private decision tree and a client holding a private attribute vector. The goal is to classify the client's input using the server's model such that the client learns only the result of the classification, and the server learns nothing. In a first approach, we represent the tree as an array and execute only d interactive comparisons (instead of 2 d as in existing solutions), where d denotes the depth of the tree. In a second approach, we delegate the complete tree evaluation to the server using somewhat or fully homomorphic encryption where the ciphertexts are encrypted under the client's public key. A generalization of a decision tree is a random forest that consists of many decision trees. A classification with a random forest evaluates each decision tree in the forest and outputs the classification label which occurs most often. Hence, the classification labels are ranked by their number of occurrences and the final result is the best ranked one. The best ranked element is a special case of the kth-ranked element. In this thesis, we consider the secure computation of the kth-ranked element in a distributed setting with applications in benchmarking and auctions. We propose different approaches for privately computing the kth-ranked element in a star network, using either garbled circuits or threshold homomorphic encryption.}, subject = {Mathematik}, language = {en} } @phdthesis{Tran2015, author = {Tran, Nguyen Khanh Linh}, title = {Kaehler Differential Algebras for 0-Dimensional Schemes and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-3290}, school = {Universit{\"a}t Passau}, pages = {158}, year = {2015}, abstract = {The aim of this dissertation is to investigate Kaehler differential algebras and their Hilbert functions for 0-dimensional schemes in P^n. First we give relations between Kaehler differential 1-forms of fat point schemes and another fat point schemes. Then we determine the Hilbert polynomial and give a sharp bound for the regularity index of the module of Kaehler differential m-forms, for 0