@phdthesis{Stadler2015, author = {Stadler, Thomas}, title = {Eine Anwendung der Invariantentheorie auf das Korrespondenzproblem lokaler Bildmerkmale}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-4026}, school = {Universit{\"a}t Passau}, pages = {XII, 324 S.}, year = {2015}, abstract = {Als sich in der ersten H{\"a}lfte des 19. Jahrhunderts zunehmend mehr bedeutende Mathematiker mit der Suche nach Invarianten besch{\"a}ftigten, konnte nat{\"u}rlich noch niemand vorhersehen, dass die Invariantentheorie mit Beginn des Computerzeitalters in der Bildverarbeitung bzw. dem Rechnersehen ein {\"a}ußerst fruchtbares Anwendungsgebiet finden wird. In dieser Arbeit wird eine neue Anwendungsm{\"o}glichkeit der Invariantentheorie in der Bildverarbeitung vorgestellt. Dazu werden lokale Bildmerkmale betrachtet. Dabei handelt es sich um die Koordinaten einer Polynomfunktion bzgl. einer geeigneten Orthonormalbasis von P_n(R^2,R), die die zeitintegrierte Sensorinputfunktion auf lokalen Pixelfenstern bestm{\"o}glich approximiert. Diese Bildmerkmale werden in vielen Anwendungen eingesetzt, um Objekte in Bildern zu erkennen und zu lokalisieren. Beispiele hierf{\"u}r sind die Detektion von Werkst{\"u}cken an einem Fließband oder die Verfolgung von Fahrbahnmarkierungen in Fahrerassistenzsystemen. Modellieren l{\"a}sst sich die Suche nach einem Muster in einem Suchbild als Paar von Stereobildern, auf denen lokal die affin-lineare Gruppe AGL(R) operiert. Will man also feststellen, ob zwei lokale Pixelfenster in etwa Bilder eines bestimmten dreidimensionalen Oberfl{\"a}chenausschnitts sind, ist zu kl{\"a}ren, ob die Bildausschnitte durch eine Operation der Gruppe AGL(R) n{\"a}herungsweise ineinander {\"u}bergef{\"u}hrt werden k{\"o}nnen. Je nach Anwendung gen{\"u}gt es bereits, passende Untergruppen G von AGL(R) zu betrachten. Dank der lokalen Approximation durch Polynomfunktionen induziert die Operation einer Untergruppe G eine Operation auf dem reellen Vektorraum P_n(R^2,R). Damit l{\"a}sst sich das Korrespondenzproblem auf die Frage reduzieren, ob es eine Transformation T in G gibt so, dass p ungef{\"a}hr mit der Komposition von q und T f{\"u}r die zugeh{\"o}rigen Approximationspolynome p,q in P_n(R^2,R) gilt. Mit anderen Worten, es ist zu kl{\"a}ren, ob sich p und q n{\"a}herungsweise in einer G-Bahn befinden, eine typische Fragestellung der Invariantentheorie. Da nur lokale Bildausschnitte betrachtet werden, gen{\"u}gt es weiter, Untergruppen G von GL_2(R) zu betrachten. Dann erh{\"a}lt man sofort auch die Antwort f{\"u}r das semidirekte Produkt von R^2 mit G. Besonders interessant f{\"u}r Anwendungen ist hierbei die spezielle orthogonale Gruppe G=SO_2(R) und damit insgesamt die eigentliche Euklidische Gruppe. F{\"u}r diese Gruppe und spezielle Pixelfenster ist das Korrespondenzproblem bereits gel{\"o}st. In dieser Arbeit wird das Problem in eben dieser Konstellation ebenfalls gel{\"o}st, allerdings auf elegante Weise mit Methoden der Invariantentheorie. Der Ansatz, der hier vorgestellt wird, ist aber nicht auf diese Gruppe und spezielle Pixelfenster begrenzt, sondern leicht auf weitere F{\"a}lle erweiterbar. Dazu ist insbesondere zu kl{\"a}ren, wie sich sogenannte fundamentale Invarianten von lokalen Bildmerkmalen, also letztendlich Invarianten von Polynomfunktionen, berechnen lassen, d.h. Erzeugendensysteme der entsprechenden Invariantenringe. Mit deren Hilfe l{\"a}sst sich die Zugeh{\"o}rigkeit einer Polynomfunktion zur Bahn einer anderen Funktion auf einfache Weise untersuchen. Neben der Vorstellung des Verfahrens zur Korrespondenzfindung und der daf{\"u}r notwendigen Theorie werden in dieser Arbeit Erzeugendensysteme von Invariantenringen untersucht, die besonders "sch{\"o}ne" Eigenschaften besitzen. Diese sch{\"o}nen Erzeugendensysteme von Unteralgebren werden, analog zu Gr{\"o}bner-Basen als Erzeugendensysteme von Idealen, SAGBI-Basen genannt ("Subalgebra Analogs to Gr{\"o}bner Bases for Ideals"). SAGBI-Basen werden hier insbesondere aus algorithmischer Sicht behandelt, d.h. die Berechnung von SAGBI-Basen steht im Vordergrund. Dazu werden verschiedene Algorithmen erarbeitet, deren Korrektheit bewiesen und implementiert. Daraus resultiert ein Software-Paket zu SAGBI-Basen f{\"u}r das Computeralgebrasystem ApCoCoA, dessen Funktionalit{\"a}t in diesem Umfang in keinem Computeralgebrasystem zu finden sein wird. Im Zuge der Umsetzung der einzelnen Algorithmen konnte außerdem die Theorie der SAGBI-Basen an zahlreichen Stellen erweitert werden.}, language = {de} } @phdthesis{Jovanovic2015, author = {Jovanovic, Philipp}, title = {Analysis and Design of Symmetric Cryptographic Algorithms}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-3319}, school = {Universit{\"a}t Passau}, pages = {216}, year = {2015}, abstract = {This doctoral thesis is dedicated to the analysis and the design of symmetric cryptographic algorithms. In the first part of the dissertation, we deal with fault-based attacks on cryptographic circuits which belong to the field of active implementation attacks and aim to retrieve secret keys stored on such chips. Our main focus lies on the cryptanalytic aspects of those attacks. In particular, we target block ciphers with a lightweight and (often) non-bijective key schedule where the derived subkeys are (almost) independent from each other. An attacker who is able to reconstruct one of the subkeys is thus not necessarily able to directly retrieve other subkeys or even the secret master key by simply reversing the key schedule. We introduce a framework based on differential fault analysis that allows to attack block ciphers with an arbitrary number of independent subkeys and which rely on a substitution-permutation network. These methods are then applied to the lightweight block ciphers LED and PRINCE and we show in both cases how to recover the secret master key requiring only a small number of fault injections. Moreover, we investigate approaches that utilize algebraic instead of differential techniques for the fault analysis and discuss advantages and drawbacks. At the end of the first part of the dissertation, we explore fault-based attacks on the block cipher Bel-T which also has a lightweight key schedule but is not based on a substitution-permutation network but instead on the so-called Lai-Massey scheme. The framework mentioned above is thus not usable against Bel-T. Nevertheless, we also present techniques for the case of Bel-T that enable full recovery of the secret key in a very efficient way using differential fault analysis. In the second part of the thesis, we focus on authenticated encryption schemes. While regular ciphers only protect privacy of processed data, authenticated encryption schemes also secure its authenticity and integrity. Many of these ciphers are additionally able to protect authenticity and integrity of so-called associated data. This type of data is transmitted unencrypted but nevertheless must be protected from being tampered with during transmission. Authenticated encryption is nowadays the standard technique to protect in-transit data. However, most of the currently deployed schemes have deficits and there are many leverage points for improvements. With NORX we introduce a novel authenticated encryption scheme supporting associated data. This algorithm was designed with high security, efficiency in both hardware and software, simplicity, and robustness against side-channel attacks in mind. Next to its specification, we present special features, security goals, implementation details, extensive performance measurements and discuss advantages over currently deployed standards. Finally, we describe our preliminary security analysis where we investigate differential and rotational properties of NORX. Noteworthy are in particular the newly developed techniques for differential cryptanalysis of NORX which exploit the power of SAT- and SMT-solvers and have the potential to be easily adaptable to other encryption schemes as well.}, subject = {Kryptologie}, language = {en} } @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{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{Lehner2016, author = {Lehner, Sabrina}, title = {The Asymptotic Behaviour of the Riemann Mapping Function at Analytic Cusps}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-3587}, school = {Universit{\"a}t Passau}, pages = {97}, year = {2016}, abstract = {The well-known Riemann Mapping Theorem states the existence of a conformal map of a simply connected proper domain of the complex plane onto the upper half plane. One of the main topics in geometric function theory is to investigate the behaviour of the mapping functions at the boundary of such domains. In this work, we always assume that a piecewise analytic boundary is given. Hereby, we have to distinguish regular and singular boundary points. While the asymptotic behaviour for regular boundary points can be investigated by using the Schwarz Reflection at analytic arcs, the situation for singular boundary points is far more complicated. In the latter scenario two cases have to be differentiated: analytic corners and analytic cusps. The first part of the thesis deals with the asymptotic behaviour at analytic corners where the opening angle is greater than 0. The results of Lichtenstein and Warschawski on the asymptotic behaviour of the Riemann map and its derivatives at an analytic corner are presented as well as the much stronger result of Lehman that the mapping function can be developed in a certain generalised power series which in turn enables to examine the o-minimal content of the Riemann Mapping Theorem. To obtain a similar statement for domains with analytic cusps, it is necessary to investigate the asymptotic behaviour of a Riemann map at the cusp and based on this result to determine the asymptotic power series expansion. Therefore, the aim of the second part of this work is to investigate the asymptotic behaviour of a Riemann map at an analytic cusp. A simply connected domain has an analytic cusp if the boundary is locally given by two analytic arcs such that the interior angle vanishes. Besides the asymptotic behaviour of the mapping function, the behaviour of its derivatives, its inverse, and the derivatives of the inverse are analysed. Finally, we present a conjecture on the asymptotic power series expansion of the mapping function at an analytic cusp.}, subject = {Geometrische Funktionentheorie}, language = {en} } @phdthesis{Kriegl2015, author = {Kriegl, Markus}, title = {Generalizations and Applications of Border Bases}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-3628}, school = {Universit{\"a}t Passau}, year = {2015}, abstract = {This doctoral thesis is devoted to generalize border bases to the module setting and to apply them in various ways. First, we generalize the theory of border bases to finitely generated modules over a polynomial ring. We characterize these generalized border bases and show that we can compute them. As an application, we are able to characterize subideal border bases in various new ways and give a new algorithm for their computation. Moreover, we prove Schreyer's Theorem for border bases of submodules of free modules of finite rank over a polynomial ring. In the second part of this thesis, we study the effect of homogenization to border bases of zero-dimensional ideals. This yields the new concept of projective border bases of homogeneous one-dimensional ideals. We show that there is a one-to-one correspondence between projective border bases and zero-dimensional closed subschemes of weighted projective spaces that have no point on the hyperplane at infinity. Applying that correspondence, we can characterize uniform zero-dimensional closed subschemes of weighted projective spaces that have a rational support over the base field in various ways. Finally, we introduce projective border basis schemes as specific subschemes of border basis schemes. We show that these projective border basis schemes parametrize all zero-dimensional closed subschemes of a weighted projective space whose defining ideals possess a projective border basis. Assuming that the base field is algebraically closed, we are able to prove that the set of all closed points of a projective border basis scheme that correspond to a uniform subscheme is a constructive set with respect to the Zariski topology.}, language = {en} } @phdthesis{Sipal2017, author = {Sipal, Bilge}, title = {Border Basis Schemes}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-4702}, school = {Universit{\"a}t Passau}, year = {2017}, abstract = {The basic idea of border basis theory is to describe a zero-dimensional ring P/I by an order ideal of terms whose residue classes form a K-vector space basis of P/I. The O-border basis scheme is a scheme that parametrizes all zero-dimensional ideals that have an O-border basis. In general, the O-border basis scheme is not an affine space. Subsequently, in [Huib09] it is proved that if an order ideal with "d" elements is defined in a two-dimensional polynomial ring and it is of some special shapes, then the O-border basis scheme is isomorphic to the affine space of dimension 2d. This thesis is dedicated to find a more general condition for an O-border basis scheme to be isomorphic to an affine space of dimension "nd" that is independent of the shape of the order ideal with "d" elements and "n" is the dimension of the polynomial ring that the order ideal is defined in. We accomplish this in 6 Chapters. In Chapters 2 and 3 we develop the concepts and properties of border basis schemes. In Chapter 4 we transfer the smoothness criterion (see [Huib05]) for the point (0,...,0) in a Hilbert scheme of points to the monomial point of the border basis scheme by employing the tools from border basis theory. In Chapter 5 we explain trace and Jacobi identity syzygies of the defining equations of a O-border basis scheme and characterize them by the arrow grading. In Chapter 6 we give a criterion for the isomorphism between 2d dimensional affine space and O-border basis scheme by using the results from Chapters 3 and Chapter 4. The techniques from other chapters are applied in Chapter 6.1 to segment border basis schemes and in Chapter 6.2 to O-border basis schemes for which O is of the sawtooth form.}, subject = {Polynomring}, language = {en} } @phdthesis{Fink2019, author = {Fink, Thomas}, title = {Curvature Detection by Integral Transforms}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-7684}, school = {Universit{\"a}t Passau}, pages = {viii, 194 Seiten}, year = {2019}, abstract = {In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practical relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature. We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions and the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform. Furthermore, we introduce an extension of the continuous shearlet transform which additionally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order geometric information of edges. We introduce novel vanishing moment conditions which enable a more robust detection of the geometric edge features and examine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R^2 into R^3 and thereby allow for the analysis of the geometry of object surfaces.}, subject = {Kr{\"u}mmung}, language = {en} } @phdthesis{Hatzesberger2020, author = {Hatzesberger, Simon}, title = {Strongly Asymptotically Optimal Methods for the Pathwise Global Approximation of Stochastic Differential Equations with Coefficients of Super-linear Growth}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-8100}, school = {Universit{\"a}t Passau}, pages = {ii, 116 Seiten}, year = {2020}, abstract = {Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum and the L_p error criteria, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. For the supremum error, we prove strong asymptotic optimality for specific tamed Euler schemes relating to certain adaptive and to equidistant time discretizations. For the L_p error, we prove strong asymptotic optimality for specific tamed Milstein schemes relating to certain adaptive and to equidistant time discretizations. To illustrate our findings, we numerically analyze the SDE associated with the Heston-3/2-model originating from mathematical finance.}, subject = {Stochastische Differentialgleichung}, language = {en} } @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{Apfler2018, author = {Apfler, Sabine}, title = {Erwerb mathematischer Kompetenzen in der Regelvolksschule}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-5593}, school = {Universit{\"a}t Passau}, pages = {447 Seiten}, year = {2018}, abstract = {Durch die Einf{\"u}hrung der Bildungsstandards{\"u}berpr{\"u}fungen erh{\"a}lt die Frage nach der Kompetenzentwicklung von Kindern einen zentralen Stellenwert in Bildungsangelegenheiten. Lehrkr{\"a}fte stehen dadurch vor der Aufgabe, ihren Unterricht diesbez{\"u}glich zu ver{\"a}ndern und L{\"o}sungen zu finden. Die Frage, ob der Einsatz der Montessorip{\"a}dagogik die Entwicklung mathematischer Kompetenzen positiv beeinflusst, wurde im Rahmen einer Dissertation diskutiert und empirisch erforscht werden. Der Arbeit liegt folgende Fragestellung zugrunde: Kann die Montessorip{\"a}dagogik dazu beitragen, dass Sch{\"u}lerinnen und Sch{\"u}ler in {\"o}sterreichischen Regelvolksschulen ein hohes Kompetenzniveau in Mathematik erreichen? Wie unterscheidet sich die Kompetenzentwicklung in Klassen, die von Lehrpersonen mit beziehungsweise ohne Montessoriausbildung unterrichtet werden? Bei der vorliegenden Studie handelt es sich um eine L{\"a}ngsschnittuntersuchung mit zwei Testzeitpunkten. Im Sinne einer summativen Evaluation wurden zu Beginn der Untersuchung die Basiskompetenzen mit dem „Entwicklungsorientierten Test zur Erfassung mathematischer Basiskompetenzen ab Schuleintritt (MBK 1+)" analysiert. Am Ende der dritten Schulstufe wurde eine abschließende Bewertung im Rahmen der IKM-Testung des BIFIE vorgenommen, um die Wirksamkeit des Einsatzes der Montessorimethode zu ermitteln. Untersucht wurden 14 Volksschulklassen, die entweder einer Untersuchungsgruppe oder einer Kontrollgruppe zugeordnet wurden. Die Klassen der Untersuchungsgruppe wurden von Lehrpersonen mit Montessori-Diplomausbildung unterrichtet, die Klassen der Kontrollgruppe von Lehrpersonen, die keine Montessoriausbildung absolviert haben. An der Studie nahmen 248 Sch{\"u}lerinnen und Sch{\"u}ler teil. Der Untersuchungszeitraum erstreckte sich {\"u}ber insgesamt drei Schuljahre. Die Ergebnisse der Studie zeigen Tendenzen, dass sich der Einsatz der Montessorip{\"a}dagogik positiv auf die Kompetenzentwicklung der Kinder im Bereich der Mathematik auswirkt.}, subject = {{\"O}sterreich}, language = {de} } @phdthesis{Schlenker2022, author = {Schlenker, Florian}, title = {Delaunay Configuration B-Splines}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-11225}, school = {Universit{\"a}t Passau}, pages = {xxiii, 223 Seiten}, year = {2022}, abstract = {The generalization of univariate splines to higher dimensions is not straightforward. There are different approaches, each with its own advantages and drawbacks. A promising approach using Delaunay configurations and simplex splines is due to Neamtu. After recalling fundamentals of univariate splines, simplex splines, and the wellknown, multivariate DMS-splines, we address Neamtu's DCB-splines. He defined two variants that we refer to as the nonpooled and the pooled approach, respectively. Regarding these spline spaces, we contribute the following results. We prove that, under suitable assumptions on the knot set, both variants exhibit the local finiteness property, i.e., these spline spaces are locally finite-dimensional and at each point only a finite number of basis candidate functions have a nonzero value. Additionally, we establish a criterion guaranteeing these properties within a compact region under mitigated assumptions. Moreover, we show that the knot insertion process known from univariate splines does not work for DCB-splines and reason why this behavior is inherent to these spline spaces. Furthermore, we provide a necessary criterion for the knot insertion property to hold true for a specific inserted knot. This criterion is also sufficient for bivariate, nonpooled DCB-splines of degrees zero and one. Numerical experiments suggest that the sufficiency also holds true for arbitrary spline degrees. Univariate functions can be approximated in terms of splines using the Schoenberg operator, where the approximation error decreases quadratically as the maximum distance between consecutive knots is reduced. We show that the Schoenberg operator can be defined analogously for both variants of DCB-splines with a similar error bound. Additionally, we provide a counterexample showing that the basis candidate functions of nonpooled DCB-splines are not necessarily linearly independent, contrary to earlier statements in the literature. In particular, this implies that the corresponding functions are not a basis for the space of nonpooled DCB-splines.}, subject = {Spline}, language = {en} } @phdthesis{Sonnleitner2022, author = {Sonnleitner, Mathias}, title = {The power of random information for numerical approximation and integration}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-11305}, school = {Universit{\"a}t Passau}, pages = {x, 165 Seiten}, year = {2022}, abstract = {This thesis investigates the quality of randomly collected data by employing a framework built on information-based complexity, a field related to the numerical analysis of abstract problems. The quality or power of gathered information is measured by its radius which is the uniform error obtainable by the best possible algorithm using it. The main aim is to present progress towards understanding the power of random information for approximation and integration problems. In the first problem considered, information given by linear functionals is used to recover vectors, in particular from generalized ellipsoids. This is related to the approximation of diagonal operators which are important objects of study in the theory of function spaces. We obtain upper bounds on the radius of random information both in a convex and a quasi-normed setting, which extend and, in some cases, improve existing results. We conjecture and partially establish that the power of random information is subject to a dichotomy determined by the decay of the length of the semiaxes of the generalized ellipsoid. Second, we study multivariate approximation and integration using information given by function values at sampling point sets. We obtain an asymptotic characterization of the radius of information in terms of a geometric measure of equidistribution, the distortion, which is well known in the theory of quantization of measures. This holds for isotropic Sobolev as well as H{\"o}lder and Triebel-Lizorkin spaces on bounded convex domains. We obtain that for these spaces, depending on the parameters involved, typical point sets are either asymptotically optimal or worse by a logarithmic factor, again extending and improving existing results. Further, we study isotropic discrepancy which is related to numerical integration using linear algorithms with equal weights. In particular, we analyze the quality of lattice point sets with respect to this criterion and obtain that they are suboptimal compared to uniform random points. This is in contrast to the approximation of Sobolev functions and resolves an open question raised in the context of a possible low discrepancy construction on the two-dimensional sphere.}, subject = {Komplexit{\"a}t / Algorithmus}, language = {en} } @phdthesis{Ullah2012, author = {Ullah, Ehsan}, title = {New Techniques for Polynomial System Solving}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-26815}, school = {Universit{\"a}t Passau}, year = {2012}, abstract = {Since any encryption map may be viewed as a polynomial map between finite dimensional vector spaces over finite fields, the security of a cryptosystem can be examined by studying the difficulty of solving large systems of multivariate polynomial equations. Therefore, algebraic attacks lead to the task of solving polynomial systems over finite fields. In this thesis, we study several new algebraic techniques for polynomial system solving over finite fields, especially over the finite field with two elements. Instead of using traditional Gr{\"o}bner basis techniques we focus on highly developed methods from several other areas like linear algebra, discrete optimization, numerical analysis and number theory. We study some techniques from combinatorial optimization to transform a polynomial system solving problem into a (sparse) linear algebra problem. We highlight two new kinds of hybrid techniques. The first kind combines the concept of transforming combinatorial infeasibility proofs to large systems of linear equations and the concept of mutants (finding special lower degree polynomials). The second kind uses the concept of mutants to optimize the Border Basis Algorithm. We study recent suggestions of transferring a system of polynomial equations over the finite field with two elements into a system of polynomial equalities and inequalities over the set of integers (respectively over the set of reals). In particular, we develop several techniques and strategies for converting the polynomial system of equations over the field with two elements to a polynomial system of equalities and inequalities over the reals (respectively over the set of integers). This enables us to make use of several algorithms in the field of discrete optimization and number theory. Furthermore, this also enables us to investigate the use of numerical analysis techniques such as the homotopy continuation methods and Newton's method. In each case several conversion techniques have been developed, optimized and implemented. Finally, the efficiency of the developed techniques and strategies is examined using standard cryptographic examples such as CTC and HFE. Our experimental results show that most of the techniques developed are highly competitive to state-of-the-art algebraic techniques.}, subject = {Polynoml{\"o}sung}, language = {en} } @phdthesis{Xiu2012, author = {Xiu, Xingqiang}, title = {Non-commutative Gr{\"o}bner Bases and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-26827}, school = {Universit{\"a}t Passau}, year = {2012}, abstract = {Commutative Gr{\"o}bner bases have a lot of applications in theory and practice, because they have many nice properties, they are computable, and there exist many efficient improvements of their computations. Non-commutative Gr{\"o}bner bases also have many useful properties. However, applications of non-commutative Gr{\"o}bner bases are rarely considered due to high complexity of computations. The purpose of this study was to improve the computation of non-commutative Gr{\"o}bner bases and investigate the applications of non-commutative Gr{\"o}bner bases. Gr{\"o}bner basis theory in free monoid rings was carefully revised and Gr{\"o}bner bases were precisely characterized in great detail. For the computations of Gr{\"o}bner bases, the Buchberger Procedure was formulated. Three methods, say interreduction on obstructions, Gebauer-M{\"o}ller criteria, and detecting redundant generators, were developed for efficiently improving the Buchberger Procedure. Further, the same approach was applied to study Gr{\"o}bner basis theory in free bimodules over free monoid rings. The Buchberger Procedure was also formulated and improved in this setting. Moreover, J.-C. Faugere's F4 algorithm was generalized to this setting. Finally, many meaningful applications of non-commutative Gr{\"o}bner bases were developed. Enumerating procedures were proposed to semi-decide some interesting undecidable problems. All the examples in the thesis were computed using the package gbmr of the computer algebra system ApCoCoA. The package was developed by the author. It contains dozens of functions for Gr{\"o}bner basis computations and many concrete applications. The package gbmr and a collection of interesting examples are available at http://www.apcocoa.org/.}, subject = {Gr{\"o}bner-Basis}, language = {en} } @phdthesis{Capco2010, author = {Capco, Jose}, title = {Real Closed * Rings}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-25915}, school = {Universit{\"a}t Passau}, year = {2010}, abstract = {In this dissertation I examine a definition of real closure of commutative unitary reduced rings. I also give a characterization of rings that are real closed in this context and how one is able to arrive to such a real closure. There are sufficient examples to help the reader get a feel for real closed * rings and the real closure * of commutative unitary rings.}, language = {en} } @unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {Entropy Density and Mismatch in High-Rate Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-26132}, year = {2011}, abstract = {Properties of scalar quantization with \$r\$th power distortion and constrained R\'enyi entropy of order \$\alpha\in (0,1)\$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate (\$\alpha=0\$) and Gray \emph{et al.}'s variable-rate (\$\alpha=1\$)mismatch results to general values of the entropy order parameter \$\alpha\$}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2008, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization for uniform distributions on Cantor-like sets}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-12449}, year = {2008}, abstract = {In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar \$N-\$adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first \$N\$ quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2005, author = {Kreitmeier, Wolfgang}, title = {Optimal Quantization for Dyadic Homogeneous Cantor Distributions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-3845}, year = {2005}, abstract = {For a large class of dyadic homogeneous Cantor distributions in \mathbb{R}, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to \frac{1}{3}. For these distributions we calculate the quantization errors explicitly.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2007, author = {Kreitmeier, Wolfgang}, title = {Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-7374}, year = {2007}, abstract = {For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2007, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization of probabilities concentrated on small balls}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-12010}, year = {2007}, abstract = {We consider probability distributions which are uniformly distributed on a disjoint union of balls with equal radius. For small enough radius the optimal quantization error is calculated explicitly in terms of the ball centroids. We apply the results to special self-similar measures.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2011, author = {Kreitmeier, Wolfgang}, title = {Optimal vector quantization in terms of Wasserstein distance}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-22502}, year = {2011}, abstract = {The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the R\'enyi-\$\alpha\$-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained \$(\alpha = 1)\$ and memory-size constrained \$(\alpha = 0)\$ quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gy\"{o}rgy and Linder.}, subject = {Maßtheorie}, language = {en} } @phdthesis{Ali2011, author = {Ali, Rashid}, title = {Weyl Gr{\"o}bner Basis Cryptosystems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-23195}, school = {Universit{\"a}t Passau}, year = {2011}, abstract = {In this thesis, we shall consider a certain class of algebraic cryptosystems called Gr{\"o}bner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gr{\"o}bner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gr{\"o}bner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks that do not require the computation of Gr{\"o}bner basis for breaking the system, for example, the attacks based on linear algebra. To secure these (commutative) Gr{\"o}bner basis cryptosystems against various attacks, among others, Ackermann and Kreuzer introduced a general class of Gr{\"o}bner Basis Cryptosystems that are based on the difficulty of computing module Gr{\"o}bner bases over general non-commutative rings. The objective of this research is to describe a special class of such cryptosystems by introducing the Weyl Gr{\"o}bner Basis Cryptosystems. We divide this class of cryptosystems in two parts namely the (left) Weyl Gr{\"o}bner Basis Cryptosystems and Two-Sided Weyl Gr{\"o}bner Basis Cryptosystems. We suggest to use Gr{\"o}bner bases for left and two-sided ideals in Weyl algebras to construct specific instances of such cryptosystems. We analyse the resistance of these cryptosystems to the standard attacks and provide computational evidence that secure Weyl Gr{\"o}bner Basis Cryptosystems can be built using left (resp. two-sided) Gr{\"o}bner bases in Weyl algebras.}, subject = {Gr{\"o}bner-Basis}, language = {en} } @unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {High-Resolution Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-23787}, year = {2011}, abstract = {We consider optimal scalar quantization with \$r\$th power distortion and constrained R\'enyi entropy of order \$\alpha\$. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for \$\alpha=0\$ (fixed-rate quantization) and \$\alpha=1\$ (entropy-constrained quantization). These results have recently been extended to quantization with R\'enyi entropy constraint of order \$\alpha \ge r+1\$. Here we consider the more challenging case \$\alpha\in [-\infty,0)\cup (0,1)\$ and for a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive.}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Hausdorff measure of uniform self-similar fractals}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-17948}, year = {2009}, abstract = {Let d \&\#8805; 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on Rd. Denote by D the Hausdorff dimension, by HD(E) the Hausdorff measure and by diam (E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set \ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on \ω, such that the Hausdorff dimension is smaller than one and HD = (E) D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where \ω depends on c and HD < diam(E)D, if c is small enough.}, subject = {Maßtheorie}, language = {en} } @phdthesis{Graf2007, author = {Graf, Simone}, title = {Kamerakalibrierung mit radialer Verzeichnung - die radiale essentielle Matrix}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-12711}, school = {Universit{\"a}t Passau}, year = {2007}, abstract = {In der Bildverarbeitung wird die beobachtende Kamera meist als Lochkamera modelliert: ein Modell, das zahlreiche theoretische Vorteile bietet. So kann etwa das Abbildungsverhalten als projektive Abbildung aufgefasst werden. In einem Stereokamerasystem dieses Modells stehen korrespondierende Punkte - das sind Bildpunkte desselben 3D-Punktes - in einem linearen Zusammenhang, der auch ohne Kenntnis der Kameraparameter aus beobachteten Korrespondenzen gesch{\"a}tzt werden kann. F{\"u}r die meisten Kameras, insbesondere f{\"u}r solche mit Weitwinkelobjektiven, ist die Modellannahme einer Lochkamera allerdings sichtbar unzureichend. Deshalb m{\"u}ssen zus{\"a}tzlich zur Lochkamera noch Verzeichnungsabbildungen ins Modell integriert werden. In dieser Arbeit wird gezeigt, dass bei polynomialer radialer Verzeichnung die Parameter der Projektionsabbildung die Verzeichnungsparameter bestimmen. Dieses theoretische Ergebnis fließt in Algorithmen zur Kamerakalibrierung, d.h. zur Bestimmung der Parameter eines Kameramodells, ein. Diese wurden experimentell getestet und mit bestehenden Verfahren verglichen. Weiterhin wird die radiale essentielle Matrix eingef{\"u}hrt, die die Beziehung von korrespondierenden Punkten im Stereokamerafall bei radialer Verzeichnung beschreibt. Es werden vier Algorithmen vorgestellt, die diese theoretische Beziehung verwerten. Sie geben an, wie aus korrespondierenden Punkten die radiale essentielle Matrix gesch{\"a}tzt werden kann und welche Kameraparameter daraus gewonnen werden k{\"o}nnen. Damit ist beispielsweise eine Nachkalibrierung m{\"o}glich. Auch diese Verfahren wurden implementiert und evaluiert. Umgekehrt ist bei bekannter radialer essentieller Matrix eine Einschr{\"a}nkung des Suchraums f{\"u}r korrespondierende Punkte m{\"o}glich, die f{\"u}r die Rekonstruktion ben{\"o}tigt werden.}, subject = {Optische Messtechnik}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization for the one-dimensional uniform distribution with R{\´e}nyi -α-entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16983}, year = {2009}, abstract = {We establish the optimal quantization problem for probabilities under constrained R{\´e}nyi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases α = 0 (restricted codebook size) and α = 1 (restricted Shannon entropy).}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Error bounds for high-resolution quantization with R{\´e}nyi - \&\#945; - entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16647}, year = {2009}, abstract = {We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on Rd under constrained R{\´e}nyi-\&\#945;-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases \&\#945; = 0 (memory-constrained quantization) and \&\#945; = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter \&\#945; \&\#8712; [1+r/d, \&\#8734;]. For \&\#945; \&\#8712; [0,1+r/d[ we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error decays exponentially fast with the entropy bound and the exact decay rate is determined for all \&\#945; \&\#8712; [0, \&\#8734;].}, subject = {Maßtheorie}, language = {en} } @phdthesis{Bachl2003, author = {Bachl, Walter}, title = {Interaktives orthogonales Zeichnen von planaren Graphen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-392}, school = {Universit{\"a}t Passau}, year = {2003}, abstract = {Die Arbeit besch{\"a}ftigt sich mit dem automatischen Zeichnen von Graphen. Hier wird ein interaktiver Ansatz untersucht, bei dem der Graph mit einer Menge von Operationen Schritt f{\"u}r Schritt aufgebaut wird. Der Zielgraph und die Einf{\"u}gereihenfolge sind dabei nicht fest vorgegeben, sondern werden vom Benutzer bestimmt. In der Arbeit wird vor allem ein Szenario f{\"u}r zweifach zusammenh{\"a}ngende Graphen untersucht und ein f{\"u}r diese Zwecke passendes Zeichenmodell entwickelt. Dieser Ansatz wird dann um verschiedene Varianten erweitert. Außerdem wird gezeigt, dass das fl{\"a}chenminimale Zeichnen in dem neu entwickelten Zeichenmodell NP-vollst{\"a}ndig ist.}, subject = {Graphenzeichnen}, language = {de} } @phdthesis{Kreitmeier2006, author = {Kreitmeier, Wolfgang}, title = {Optimale Quantisierung verallgemeinerter Cantor-Verteilungen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-913}, school = {Universit{\"a}t Passau}, year = {2006}, abstract = {F{\"u}r verallgemeinerte Cantor-Verteilungen, die im Eindimensionalen mittels klassischer Wischkonstruktion bzw. in h{\"o}heren Dimensionen {\"u}ber iterierte Funktionensysteme definiert werden, wird das Problem der optimalen Quantisierung unter bestimmten Voraussetzungen vollst{\"a}ndig gel{\"o}st. Es werden die optimalen Codeb{\"u}cher bestimmt und Formeln f{\"u}r den optimalen Quantisierungsfehler bewiesen. Im eindimensionalen Fall wird eine Existenzcharakterisierung der Quantisierungsdimension gegeben und unter bestimmten Voraussetzungen die Nichtexistenz des Quantisierungskoeffizienten gezeigt. Auch in h{\"o}heren Dimensionen wird f{\"u}r die betrachteten Verteilungen bewiesen, dass der Quantisierungskoeffizient, bei existenter Quantisierungsdimension, nicht existiert. Die gewonnenen Resultate werden auf die Gleichverteilungen von modifizierten klassischen fraktalen Mengen, wie das Sierpinski-Dreieck, die Cantormenge und den Cantor-Staub angewandt.}, subject = {Maßtheorie}, language = {de} } @phdthesis{Lauren2005, author = {Lauren, Verena}, title = {Semilineare Approximation in der Bildrekonstruktion}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-730}, school = {Universit{\"a}t Passau}, year = {2005}, abstract = {In der Bildverarbeitung spielen Segmentierungsalgorithmen, bei denen die Art der zu segmentierenden Menge schon im Voraus festgelegt werden kann und nur noch ihre Gr{\"o}ße und Lage angepasst werden muss, eine eher untergeordnete Rolle. Gr{\"u}nde hierf{\"u}r sind vor allem komplizierte Zielfunktionen und daraus resultierende lange Rechenzeiten, die zudem meist kein optimales Ergebnis liefern. Dabei kann eine mengenbasierte Segmentierung durchaus sinnvoll eingesetzt werden, wenn gewisse Rahmenbedingungen eingehalten werden. In dieser Arbeit wird eine Theorie zur allgemeinen mengenbasierten Segmentierung vorgestellt und untersucht, unter welchen Bedingungen optimale Segmentierungsergebnisse erreicht werden k{\"o}nnen. Die anschließenden Anwendungen best{\"a}tigen die N{\"u}tzlichkeit dieser Theorie.}, subject = {Optimale Rekonstruktion}, language = {de} } @phdthesis{Fink2023, author = {Fink, Simon Dominik}, title = {Constrained Planarity Algorithms in Theory and Practice}, doi = {10.15475/cpatp.2024}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-13817}, school = {Universit{\"a}t Passau}, pages = {216 Seiten}, year = {2023}, abstract = {In the constrained planarity setting, we ask whether a graph admits a crossing-free drawing that additionally satisfies a given set of constraints. These constraints are often derived from very natural problems; prominent examples are Level Planarity, where vertices have to lie on given horizontal lines indicating a hierarchy, Partially Embedded Planarity, where we extend a given drawing without modifying already-drawn parts, and Clustered Planarity, where we additionally draw the boundaries of clusters which recursively group the vertices in a crossing-free manner. In the last years, the family of constrained planarity problems received a lot of attention in the field of graph drawing. Efficient algorithms were discovered for many of them, while a few others turned out to be NP-complete. In contrast to the extensive theoretical considerations and the direct motivation by applications, only very few of the found algorithms have been implemented and evaluated in practice. The goal of this thesis is to advance the research on both theoretical as well as practical aspects of constrained planarity. On the theoretical side, we consider two types of constrained planarity problems. The first type are problems that individually constrain the rotations of vertices, that is they restrict the counter-clockwise cyclic orders of the edges incident to vertices. We give a simple linear-time algorithm for the problem Partially Embedded Planarity, which also generalizes to further constrained planarity variants of this type. The second type of constrained planarity problem concerns more involved planarity variants that come down to the question whether there are embeddings of one or multiple graphs such that the rotations of certain vertices are in sync in a certain way. Clustered Planarity and a variant of the Simultaneous Embedding with Fixed Edges Problem (Connected SEFE-2) are well-known problems of this type. Both are generalized by our Synchronized Planarity problem, for which we give a quadratic algorithm. Through reductions from various other problems, we provide a unified modelling framework for almost all known efficiently solvable constrained planarity variants that also directly provides a quadratic-time solution to all of them. For both our algorithms, a key ingredient for reaching an efficient solution is the usage of the right data structure for the problem at hand. In this case, these data structures are the SPQR-tree and the PC-tree, which describe planar embedding possibilities from a global and a local perspective, respectively. More specifically, PC-trees can be used to locally describe the possible cyclic orders of edges around vertices in all planar embeddings of a graph. This makes it a key component for our algorithms, as it allows us to test planarity while also respecting further constraints, and to communicate constraints arising from the surrounding graph structure between vertices with synchronized rotation. Bridging over to the practical side, we present the first correct implementation of PC-trees. We also describe further improvements, which allow us to outperform all implementations of alternative data structures (out of which we only found very few to be fully correct) by at least a factor of 4. We show that this yields a simple and competitive planarity test that can also yield an embedding to certify planarity. We also use our PC-tree implementation to implement our quadratic algorithm for solving Synchronized Planarity. Here, we show that our algorithm greatly outperforms previous attempts at solving related problems like Clustered Planarity in practice. We also engineer its running time and show how degrees of freedom in the theoretical algorithm can be leveraged to yield an up to tenfold speed-up in practice.}, language = {en} } @phdthesis{Seidl2006, author = {Seidl, Andreas}, title = {Cylindrical Decomposition Under Application-Oriented Paradigms}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-816}, school = {Universit{\"a}t Passau}, year = {2006}, abstract = {Quantifier elimination (QE) is a powerful tool for problem solving. Once a problem is expressed as a formula, such a method converts it to a simpler, quantifier-free equivalent, thus solving the problem. Particularly many problems live in the domain of real numbers, which makes real QE very interesting. Among the so far implemented methods, QE by cylindrical algebraic decomposition (CAD) is the most important complete method. The aim of this thesis is to develop CAD-based algorithms, which can solve more problems in practice and/or provide more interesting information as output. An algorithm that satisfies these standards would concentrate on generic cases and postpone special and degenerated ones to be treated separately or to be abandoned completely. It would give a solution, which is locally correct for a region the user is interested in. It would give answers, which can provide much valuable information in particular for decision problems. It would combine these methods with more specialized ones, for subcases that allow for. It would exploit degrees of freedom in the algorithms by deciding to proceed in a way that promises to be efficient. It is the focus of this dissertation to treat these challenges. Algorithms described here are implemented in the computer logic system REDLOG and ship with the computer algebra system REDUCE.}, subject = {Quantorenelimination}, language = {en} } @phdthesis{Fischer2005, author = {Fischer, Andreas}, title = {Peano-differentiable functions in O-Minimal structures}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-673}, school = {Universit{\"a}t Passau}, year = {2005}, abstract = {We discuss several aspects of Peano-differentiable functions which are definable in an o-minimal structure expanding a real closed field. After recalling some already known results about o-minimal structures we develop techniques for the intrinsic study of differentiable functions in these structures. After this we study (ordinary) differentiable functions definable in an o-minimal structure and their continuiuty properties along curves of different differentiability classes. Then we generalise (ordinary) differentiability to Peano-differentiability. We study differentiability of certain Peano-derivatives of definable functions and characterise the sets of non-continuity of these derivatives. In the end we study extendability of these functions defined on closed sets and give sufficient conditions by which we can extend functions as Peano-differentiable functions.}, subject = {Semialgebraische Menge}, language = {en} } @phdthesis{Le2015, author = {Le, Ngoc Long}, title = {Various Differents for 0-Dimensional Schemes and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-3386}, school = {Universit{\"a}t Passau}, pages = {205}, year = {2015}, abstract = {This thesis attempts to investigate the Noether, Dedekind, and K{\"a}hler differents for a 0-dimensional scheme X in the projective n-space P^n_K over an arbitrary field K. In particular, we focus on studying the relations between the algebraic structure of these differents and geometric properties of the scheme X. In Chapter 1 we give an outline to the problems this thesis is concerned with, a brief literature review for each problem, and the main results regarding these problems. Chapter 2 contains background results that we will need in the subsequent chapters. We introduce the concept of maximal p_j-subschemes of a 0-dimensional scheme X and give some descriptions of them and their Hilbert functions. Furthermore, we generalize the notion of a separator of a subscheme of X of degree deg(X)-1 to a set of separators of a maximal p_j-subscheme of X. In Chapter 3 we explore the Noether, Dedekind, and K{\"a}hler differents for 0-dimensional schemes X. First we define these differents for X, and take a look at how to compute these differents and examine their relations. Then we give an answer to the question "What are the Hilbert functions of these differents?" in some cases. In Chapter 4 we use the differents to investigate the Cayley-Bacharach property of 0-dimensional schemes over an arbitrary field K. The principal results of this chapter are characterizations of CB-schemes and of arithmetically Gorenstein schemes in terms of their Dedekind differents and a criterion for a 0-dimensional smooth scheme to be a complete intersection. We also generalize some results such as Dedekind's formula and the characterization of the Cayley-Bacharach property by using Liaison theory. In addition, several propositions on the uniformities are proven. In Chapter 5 we are interested in studying the Noether, Dedekind, and K{\"a}hler differents for finite special classes of schemes and finding out some applications of these differents. First, we investigate these differents for reduced 0-dimensional almost complete intersections X in P^n_K over a perfect field K. Then we investigate the relationships between these differents and the i-th Fitting ideals of the module of K{\"a}hler differentials of the homogeneous coordinate ring of X. Finally, we look more closely at the Hilbert functions and the regularity indices of these differents for fat point schemes.}, subject = {Kommutativer Ring}, 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