@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{Walsh2024,
author = {Walsh, Florian},
title = {Computing the Binomial Part of Polynomial Ideals},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-15096},
school = {Universit{\"a}t Passau},
pages = {vi, 131 Seiten},
year = {2024},
abstract = {Given an ideal in a polynomial ring over a field, we present a complete algorithm to compute its binomial part.},
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{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{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}
}
@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}
}
@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}
}
@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}
}