@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{Lorenz2018, author = {Lorenz, Florian}, title = {Analyse und Erzeugung von glatten Fl{\"a}chen{\"u}berg{\"a}ngen f{\"u}r das CNC-Fr{\"a}sen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-5166}, school = {Universit{\"a}t Passau}, pages = {i, 149 Seiten}, year = {2018}, abstract = {In dieser Arbeit werden numerisch stabile Methoden zur Pr{\"u}fung von Stetigkeiten an Fl{\"a}chen{\"u}berg{\"a}ngen vorgestellt und Algorithmen zur Erzeugung von G^2-stetigen Fl{\"a}chen{\"u}berg{\"a}ngen hergeleitet.}, subject = {Differentialgeometrie}, 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{Klaus2019, author = {Klaus, Tina}, title = {Complexity Analysis of Quantizations of Multidimensional Stochastic Differential Equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-7665}, school = {Universit{\"a}t Passau}, pages = {iv, 166 Seiten}, year = {2019}, abstract = {The dissertation is located in the field of quantizations of certain stochastic processes, namely a solution X of a multidimensional stochastic differential equation (SDE). The quantization problem for X consists in approximating X by a a random element which takes only finitely many values. Our main interest lies in the investigation of the asymptotic behavior of the Nth minimal quantization error of X as N tends to infinity, which incorporates the determination of both the sharp rate of convergence and explicit asymptotic constants. Especially explicit asymptotic constants have been so far unknown in the context of multidimensional SDEs. Furthermore, as part of our analysis, we provide a method which yields a strongly asymptotically optimal sequence of N-quantization of X. In certain special cases our method is fully constructive and the algorithm is easy to implement.}, subject = {Stochastische Differentialgleichung}, language = {en} } @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{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{Handeck2016, author = {Handeck, J{\"o}rg}, title = {Analyse und Korrektur von Teileprogrammen f{\"u}r numerisch gesteuerte Werkzeugmaschinen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-4546}, school = {Universit{\"a}t Passau}, pages = {102 S.}, year = {2016}, abstract = {Splinekurven sind oft das erste Mittel der Wahl, wenn Daten interpoliert oder approximiert werden sollen. Sie spielen in vielen praktischen Anwendungsbereichen eine wichtige Rolle und sind in Bereichen des CAD/CAM nicht mehr weg zu denken. In der vorliegenden Arbeit werden in diesem Kontext Bahnpunkte zur Steuerung von Werkzeugmaschinen untersucht. Die Analyse wird mit Hilfe eines Multiresolution (MRA) Ansatzes f{\"u}r Splinekurven mit adaptiven Knotenfolgen realisiert. Dieser MRA Ansatz basiert auf einer Least-Squares-Projektion zum Knotenentfernen und unterscheidet sich somit zu bekannten Ans{\"a}tzen, die auf orthogonalen Komplementen aufbauen. Des Weiteren wird ein Konzept zur Approximation von Orientierungsdaten mittels homogenen Quaternionensplines vorgestellt. Diese Splines leben auf der Sph{\"a}re und lassen sich mittels Knotenentfernen bzw. einf{\"u}gen verfeinern. Somit l{\"a}sst sich das vorgestellte MRA-Analyseverfahren ebenfalls auf diese Kurven anwenden. Weiter konnte f{\"u}r diese Kurven eine konvexe H{\"u}lle-Eigenschaft nachgewiesen werden.}, subject = {Spline}, language = {de} } @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 00, linear in the dimension d of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse N~(ε,d), our bounds are optimal with respect to the dependence on ε.}, language = {en} } @article{KreuzerWalsh2024, author = {Kreuzer, Martin and Walsh, Florian}, title = {Computing the binomial part of a polynomial ideal}, series = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, volume = {2024}, journal = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, number = {124}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jsc.2024.102298}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18708}, pages = {30 Seiten}, year = {2024}, abstract = {Given an ideal I in a polynomial ring K[x1,...,xn] over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal Bin(I) of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate Bin(I) when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of Bin(I) for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called (s,t)-binomial parts. All algorithms have been implemented in SageMath.}, language = {en} } @article{FruehwirthProchno2024, author = {Fr{\"u}hwirth, Lorenz and Prochno, Joscha}, title = {Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls}, series = {Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813)}, volume = {2024}, journal = {Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813)}, number = {536,1}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jmaa.2024.128169}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18723}, pages = {25 Seiten}, year = {2024}, abstract = {In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincar{\´e}-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. In more geometric parlance, the latter shows that if V1 and V2 are Orlicz functions, then random points in the V1-Orlicz ball, conditioned on having a small V2-Orlicz radius, look like an appropriately scaled V2-Orlicz ball. In fact, we show that the limiting distribution in our Poincar{\´e}-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V2-Orlicz radius.}, language = {en} } @article{PenaFerrandezSauer2024, author = {Pe{\~n}a Ferr{\´a}ndez, Juan Manuel and Sauer, Thomas}, title = {Stabilized recovery and model reduction for multivariate exponential polynomials}, series = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, volume = {2024}, journal = {Journal of Symbolic Computation (Online ISSN: 1095-855X)}, number = {125}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jsc.2024.102313}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18748}, pages = {20 Seiten}, year = {2024}, abstract = {Recovery of multivariate exponential polynomials, i.e., the multivariate version of Prony's problem, can be stabilized by using more than the minimally needed multiinteger samples of the function. We present an algorithm that takes into account this extra information and prove a backward error estimate for the algebraic recovery method SMILE. In addition, we give a method to approximate data by an exponential polynomial sequence of a given structure as a step in the direction of multivariate model reduction.}, language = {en} } @article{KabluchkoProchnoSonnleitner2024, author = {Kabluchko, Zakhar and Prochno, Joscha and Sonnleitner, Mathias}, title = {A probabilistic approach to Lorentz balls l(^n)(q,1)}, series = {Journal of Functional Analysis (Online ISSN: 1096-0783)}, volume = {2025}, journal = {Journal of Functional Analysis (Online ISSN: 1096-0783)}, number = {288, 1}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jfa.2024.110682}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18878}, pages = {32 Seiten}, year = {2024}, abstract = {We develop a probabilistic approach to study the volumetric and geometric properties of unit balls |B(^n)(q,1) of finite-dimensional Lorentz sequence spaces l(^n)(q,1). More precisely, we show that the empirical distribution of a random vector X^(n) uniformly distributed on its volume normalized unit ball converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincar{\´e}-Maxwell-Borel principle for any fixed number k in |N of coordinates of X^(n) as n grows infinitly. Moreover, we prove a central limit theorem for the largest coordinate of X^(n), demonstrating a quite different behavior than in the case of the l(^n)(q) balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschl{\"a}ger type result for the asymptotic volume of intersections of volume normalized l(^n)(q,1) and l(^n)(p) balls.}, language = {en} } @article{MuellerGronbachYaroslavtseva2024, author = {M{\"u}ller-Gronbach, Thomas and Yaroslavtseva, Larisa}, title = {On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient}, series = {Journal of Complexity (Online ISSN: 1090-2708)}, volume = {2024}, journal = {Journal of Complexity (Online ISSN: 1090-2708)}, number = {85}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jco.2024.101870}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18819}, pages = {17 Seiten}, year = {2024}, abstract = {We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.}, language = {en} } @article{EpperleinWirth2024, author = {Epperlein, Jeremias and Wirth, Fabian}, title = {The joint spectral radius is pointwise H{\"o}lder continuous}, series = {Linear Algebra and its Applications}, volume = {2025}, journal = {Linear Algebra and its Applications}, number = {704}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1873-1856}, doi = {10.1016/j.laa.2024.09.016}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19140}, pages = {92 -- 122}, year = {2024}, abstract = {We show that the joint spectral radius is pointwise H{\"o}lder continuous. In addition, the joint spectral radius is locally H{\"o}lder continuous for ε-inflations. In the two-dimensional case, local H{\"o}lder continuity holds on the matrix sets with positive joint spectral radius.}, language = {en} } @article{GlockMunhaCorreiaSudakov2024, author = {Glock, Stefan and Munh{\´a} Correia, David and Sudakov, Benny}, title = {Hamilton cycles in pseudorandom graphs}, series = {Advances in Mathematics}, volume = {2024}, journal = {Advances in Mathematics}, number = {458 B}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.aim.2024.109984}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19151}, pages = {45 Seiten}, year = {2024}, abstract = {Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well-known conjecture in the area states that any d-regular n-vertex graph G whose second largest eigenvalue in absolute value λ(G) is at most d/C, for some universal constant C > 0, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree d is at least a small power of n. Secondly, in the general case we show that λ(G) ≤ d/C(log n)1/3 implies the existence of a Hamilton cycle, improving the 20-year old bound of d/ log1-o(1) n of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust P{\´o}sa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas. Our results have several interesting applications. In particular, they imply the currently best-known bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lov{\´a}sz. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.}, language = {en} } @article{KreuzerLinhLong2024, author = {Kreuzer, Martin and Linh, Tran N. K. and Long, Le N.}, title = {Differential theory of zero-dimensional schemes}, series = {Journal of Pure and Applied Algebra}, volume = {229 (2025)}, journal = {Journal of Pure and Applied Algebra}, number = {1}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1873-1376}, doi = {10.1016/j.jpaa.2024.107815}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19130}, pages = {31 Seiten}, year = {2024}, abstract = {To study a 0-dimensional scheme |X in |P^n over a perfect field K, we use the module of K{\"a}hler differentials (Omega)(^1)(_{R/K}) of its homogeneous coordinate ring R and its exterior powers, the higher modules of K{\"a}hler differentials (Omega)(^m)(_{R/K}). One of our main results is a characterization of weakly curvilinear schemes |X by the Hilbert polynomials of the modules (Omega)(^m)(_{R/K}) which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of |X. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules (Omega)(^m)(_{R/K}) for a fat point scheme |X which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring R into its truncated integral closure ~R. Then we use the corresponding map from the module of K{\"a}hler differentials (Omega)(^1)(_{R/K}) to (Omega)(^1)(_{~R/K}) to find a formula for the Hilbert polynomial HP((Omega)(^1)(_{R/K})) and a sharp bound for the regularity index ri((Omega)(^1)(_{R/K})). Next we extend this to formulas for the Hilbert polynomials HP((Omega)(^m)(_{R/K})) and bounds for the regularity indices of the higher modules of K{\"a}hler differentials. As a further application, we characterize uniformity conditions on |X using the Hilbert functions of the K{\"a}hler differential modules of |X and its subschemes.}, language = {en} } @article{KreuzerMiasnikovWalsh2024, author = {Kreuzer, Martin and Miasnikov, Alexei and Walsh, Florian}, title = {Decomposing finite Z-algebras}, series = {Journal of Algebra}, volume = {2025}, journal = {Journal of Algebra}, number = {664 B}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1090-266X}, doi = {10.1016/j.jalgebra.2024.10.027}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19184}, pages = {206 -- 246}, year = {2024}, abstract = {For a finite Z-algebra R, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of R/Ann(R) into directly indecomposable factors under weak hypotheses. The method is based on constructing and decomposing a ring of scalarsS, and then lifting the decomposition ofSto the bilinear map given by the multiplication of R, and finally to R/Ann(R). All steps of the construction are given as explicit algorithms and it is shown that the entire procedure has a probabilistic polynomial time complexity in the bit size of the input, except for the possible need to calculate the prime factorization of an integer. In particular, in the case when Ann(R)=0, these algorithms compute direct decompositions of R into directly indecomposable factors.}, language = {en} } @article{GishbolinerGlockSgueglia2025, author = {Gishboliner, Lior and Glock, Stefan and Sgueglia, Amedeo}, title = {Tight Hamilton cycles with high discrepancy}, series = {Combinatorics, Probability and Computing (1469-2163)}, volume = {34 (2025)}, journal = {Combinatorics, Probability and Computing (1469-2163)}, number = {4}, publisher = {Cambridge University Press}, address = {Cambridge}, issn = {1469-2163}, doi = {10.1017/S0963548325000057}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19166}, pages = {565 -- 584}, year = {2025}, abstract = {In this paper, we study discrepancy questions for spanning subgraphs of k-uniform hypergraphs. Our main result is that, for any integers k ≥ 3 and r ≥ 2, any r-colouring of the edges of a k-uniform n-vertex hypergraph G with minimum (k-1)-degree δ(G) ≥ (1/2+o(1))n contains a tight Hamilton cycle with high discrepancy, that is, with at least n/r +� (n) edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Tur{\´a}n-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.}, language = {en} } @article{ChenMuellerYokoyama2024, author = {Chen, Yijia and M{\"u}ller, Moritz and Yokoyama, Keita}, title = {A parameterized halting problem, Δ0 truth and the MRDP theorem}, series = {The Journal of Symbolic Logic (ISSN 1943-5886)}, volume = {90 (2025)}, journal = {The Journal of Symbolic Logic (ISSN 1943-5886)}, number = {2}, publisher = {Cambridge University Press}, address = {Cambridge}, issn = {1943-5886}, doi = {10.1017/jsl.2024.44}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19308}, pages = {483 -- 508}, year = {2024}, abstract = {We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given Δ0-formula ϕ(x); the parameter is the size of ϕ. This parameterization focusses attention on instances where n is large compared to the size of ϕ.We show unconditionally that this problem does not belong to the parameterized analogue of AC0. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that IΔ0 proves the MRDP theorem in a certain weak sense.}, language = {en} } @article{Hofstadler2025, author = {Hofstadler, Julian}, title = {Optimal convergence rates of MCMC integration for functions with unbounded second moment}, series = {Journal of Applied Probability}, volume = {62 (2025)}, journal = {Journal of Applied Probability}, number = {3}, publisher = {Cambridge University Press}, address = {Cambridge}, doi = {10.1017/jpr.2024.108}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19310}, pages = {1069 -- 1075}, year = {2025}, abstract = {We study the Markov chain Monte Carlo estimator for numerical integration for func- tions that do not need to be square integrable with respect to the invariant distribution. For chains with a spectral gap we show that the absolute mean error for L^p functions, with p ∈ (1, 2), decreases like n^(1/p)-1 , which is known to be the optimal rate. This improves currently known results where an additional parameter δ > 0 appears and the convergence is of order n^((1+δ)/p)-1 .}, language = {en} } @phdthesis{Henle2025, author = {Henle, Mona}, title = {Multi-Leader Congestion Games with an Adversary}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19683}, school = {Universit{\"a}t Passau}, pages = {112 Seiten}, year = {2025}, abstract = {In this thesis, we introduced a congestion game with multiple leaders and a single follower (adversary) which is motivated by security applications with congestion effects. Our objective was to understand the result and the impact of selfish acting individuals in these games. In this regard, we analyzed the existence, the computation and the quality of (approximate) pure Nash equilibria. First, we observed that an exact pure Nash equilibrium always exists in the resulting strategic game among the leaders if the resource cost coefficients are identical and the underlying congestion game is a matroid congestion game. If one of these two conditions is not fulfilled, the existence of PNE is not ensured anymore in general. Consequently, we focused on approximate equilibria. For the case of symmetric singleton strategies, one of our main result established that K ≈ 1.1974, the unique solution of a cubic polynomial equation, is the smallest possible factor such that the existence of a K-approximate equilibrium is guaranteed for all instances of the game. To this end, we presented an efficient algorithm which computes a K-approximate PNE. Furthermore, we showed that the factor K is tight by providing an instance where no α-approximate PNE with α < K exists. However, for a specific symmetric singleton instance there might be a better α-approximate PNE, i.e., with α < K. A given instance could even admit an exact PNE. We provided therefore a polynomial time procedure that computes a best approximate PNE of a given instance. In particular, this procedure can verify the existence of an exact PNE in a given instance efficiently and, if it exists, can also determine the corresponding load vector. Finally, for symmetric singleton instances with two resources, we compared the total cost of a best (cheapest) and worst (most expensive) PNE to the total cost of an optimal outcome, termed by the price of stability and the price of anarchy, respectively. In particular, we verified that the PoS and the PoA are 4/3.}, subject = {Spieltheorie}, language = {en} } @phdthesis{Schiermeier2025, author = {Schiermeier, Kathrin}, title = {Multidimensional Wavelets and Neural Networks}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19742}, school = {Universit{\"a}t Passau}, pages = {xx, 168 Seiten}, year = {2025}, abstract = {The construction of scaling functions and wavelets in multiple dimensions and for arbitrary scaling matrices is a challenging task entailing some complexities. Existing approaches mainly focus on the two-dimensional case using dyadic or quincunx sampling. This thesis aims to develop a method to construct multidimensional scaling and wavelet filters yielding orthogonal scaling functions and wavelets under the usage of convolutional neural networks. We start by recalling substantial fundamentals of ideals, modules, Fourier analysis, filterbanks and multiresolution analyses, where the mentioned concepts are already considered in an arbitrary dimensional setting to prepare the proof of the main result. There, we show the connection between multivariate scaling functions and multidimensional filters possessing certain properties. This enables us to construct scaling functions and corresponding wavelets by discrete filter design. Exploiting the link between the discrete wavelet decomposition, filterbanks and neural networks, we utilize the latter to do so. Being the main difficulty of this process, we especially focus on the Cohen criterion, which concerns the zeros of the Fourier transform of the scaling filter in modulus representing a multivariate trigonometric polynomial. After transferring the Bernstein inequality for univariate trigonomic polynomials to multiple dimensions, we present a method to derive a finite set of inequality constraints implying that the Cohen criterion holds true for a given multivariate cosine sum. Afterwards, we introduce neural networks and TensorFlow as the main tools to execute the described approach, formulate the described objective as an optimization problem and present some smaller numerical experiments and their results. A second objective of this thesis is the construction of filters possessing a unimodular modulation vector and therefore the ability to be completed to a perfect reconstruction filterbank. Both - the construction and the filterbank completion - can also be considered in a neural network framework as we will detail in the last section of this thesis alongside with the presentation of corresponding numerical experiments. In the context of filterbank completion, a further observation which allows to complete any given interpolatory filter to a perfect reconstruction filterbank in a very intuitive and simple way is presented. Furthermore, we explain that any given unimodular filter can be rendered interpolatory through prefiltering.}, language = {en} } @phdthesis{Neuwirth2026, author = {Neuwirth, Daniel}, title = {Einbettung und Charakterisierung von aligned bar 1-visibility Graphen und outer fan free Graphen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-19930}, school = {Universit{\"a}t Passau}, pages = {VII, 247 Seiten}, year = {2026}, abstract = {In dieser Arbeit werden drei verschiedene Klassen von Graphen untersucht. Die Klassen sind die bar (1;1)-visibilty Graphen, die aligned bar 1-visibility Graphen und die outer fan free Graphen. Die Klassen werden durch ihre m{\"o}glichen Einbettungen charakterisiert. Die Repr{\"a}sentation der bar (1; j)-visibility Graphen ist, dass jeder Knoten als horizontaler Strich und jede Kante als vertikaler Strich gezeichnet wird. Eine Kante kann einen Knoten genau einmal schneiden und ein Knoten kann j-mal geschnitten werden. Wir erweitern die Ergebnisse von Dean et. al. und geben Beispiele mit einer maximalen Dichte an f{\"u}r bar (1; 2)-visibility, bar (1; 3)-visibility und bar (1; 4)-visibility Graphen und geben einen maximal d{\"u}nnen Graphen f{\"u}r die Klasse der bar (1;1) visibility Graphen an. Wir zeigen, dass die Klassen der bar (1; j)-visibility Graphen f{\"u}r 1 < j < 1eine unendliche Hierarchie bilden. Abschließend beweisen wir, dass das Erkennungsproblem ob ein Graph eine bar (1;1)-visibility Repr{\"a}sentation hat, NP-vollst{\"a}ndig ist. Die Klasse der aligned bar 1-visibility Graphen (AB1V ) erh{\"a}lt man, indem man die bar (1;1)-visibility Repr{\"a}sentation um 90 Grad dreht und alle Knoten verl{\"a}ngert, so dass diese alle mit der y-Koordinate 0 starten. Die relative Position bzgl. der x-Koordinate wird mit der t-Ordnung beschrieben und mit der r-Ordnung die relative Position bzgl. der y-Koordinate. Wir erweitern die Erkenntnisse von Felsner und Massow f{\"u}r die Klasse der AB1V Graphen bzgl. ihrer maximalen Dichte, der minimale Grad eines Knotens. Wir f{\"u}hren die Methode Pfadaddition ein, um anhand deren Abschlusseigenschaften zu unterscheiden, ob ein Graph in einer Klasse liegt oder nicht. Diese Methode nutzen wir, um die Beziehung der Klasse der AB1V Graphen mit anderen Klassen zu untersuchen. F{\"u}r die Klasse der maximalen Graphen geben wir einen d{\"u}nnen Graphen und eine untere Schranke bzgl. der Dichte an. Wir geben einen Algorithmus an, welcher eine Bucheinbettung aus einer AB1V Einbettung berechnet. F{\"u}r die Klassen der optimalen AB1V Graphen geben wir einen Einbettungsalgorithmus an. Wir verbessern den Erkennungsalgorithmus von Felsner und Massow, ob ein Graph mit einer gegebenen t-Ordnung eine AB1V Einbettung besitzt. F{\"u}r die Klasse der distinkt strong AB1V Graphen, Graphen in der jeder Knoten ein unterschiedliche r-Ordnung hat und maximal f{\"u}r die r-Ordnung ist, geben wir einen Algorithmus an, der in O(n6) eine m{\"o}gliche Einbettung berechnet. Zum Schluss zeigen wir f{\"u}r diese Klasse, dass es exponentiell viele verschiedene Einbettungen gibt. Ein Graph hat eine outer fan free Einbettung, wenn alle Knoten inzident zu einer Fl{\"a}che sind und keine Kante von zwei Kanten geschnitten wird, die adjazent zu einem Knoten sind. Wir untersuchen diese Klasse zuerst auf die Dichte. Weiter erforschen wir die Beziehung zwischen den Klassen der AB1V , RAC und k-planaren Graphen. Abschließend geben wir eine Reduktion von NAE-3-SAT auf das Erkennungsproblem von outer fan free Graphen an.}, language = {de} }