The Groebner basis of the ideal of vanishing polynomials

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.Comment: 15 pages, 1 table, 2 algorithms (corrected version with new Prop. 3.8 and proof); Journal of Symbolic Computation 46 (2011

Geometry of rank tests

We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear extensions of partially ordered sets specified by the data. Our methods refine rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for the exploratory analysis of ordinal data. Convex rank tests correspond to probabilistic conditional independence structures known as semi-graphoids. Submodular rank tests are classified by the faces of the cone of submodular functions, or by Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Graphical tests correspond to both graphical models and to graph associahedra, and they have excellent statistical and algorithmic properties.Comment: 8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/. v2: Expanded proofs, revised after reviewer comment

New developments in the theory of Groebner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Groebner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of Pure and Applied Algebr

Convex Rank Tests and Semigraphoids

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The subclass of submodular rank tests is derived from faces of the cone of submodular functions, or from Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Of particular interest are graphical tests, which correspond to both graphical models and to graph associahedra

Klinische Performance eines neuen SARS-CoV-2-Antigen-Tests in der Notaufnahme eines Maximalversorgers

Ein Baustein zur Eindämmung der COVID-19-Pandemie ist die Verfügbarkeit von Tests mit hoher Sensitivität und Spezifität zur Detektion von SARS-CoV-2, insbesondere um Infizierte in vulnerablen Einrichtungen, z. B. Krankenhäusern und Pflegeeinrichtungen, zeitnah identifizieren und isolieren zu können. Dies betrifft alle Personengruppen dieser Einrichtungen, also Patient*innen/Bewohner*innen, Besucher*innen als auch Personal. Bisheriger Goldstandard für den Nachweis einer SARS-CoV-2-Infektion ist die RT-PCR. SARS-CoV-2-Antigen-Tests sind aufgrund ihres Point-of-Care-Ansatzes, der einfachen Handhabung und des günstigeren Preises eine wertvolle Ergänzung zur RT-PCR-Diagnostik. Sie erkennen mit ausreichender Sicherheit SARS-CoV-2-Infektionen bei symptomatischen Patient*innen und in Proben mit niedrigen Ct-Werten in der RT-PCR. Als Einzeltestung bei asymptomatischen Patient*innen ist ihre Wertigkeit dagegen deutlich eingeschränkt. Hier sollten repetitive Antigen-Testungen oder primär PCR-basierte Verfahren zur Anwendung kommen

Algorithms for Symbolic Computation and their Applications - Standard Bases over Rings and Rank Tests in Statistics

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges. The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.Im ersten Teil der Dissertation wird die Theorie der Standardbasen in freien Moduln über (lokalisierten) Polynomialringen entwickelt. Falls lineare Gleichungen in den Koeffizienten der Polynome lösbar sind, entwickeln wir Algorithmen um Standardbasen bezüglich beliebiger (Modul-) Monomordnungen zu berechnen. Weiterhin untersuchen wir den Fall von Hauptidealringen mit Nullteilern. Wir implementieren für diesen Fall modifizierte neue Algorithmen, welche deutlich performanter als die generischen sind. Die Implementierung wurde durch bestehende Restriktionen bei der Verifikation von System-on-Chip Layouts motiviert. Wir zeigen, dass die neue Herangehensweise mit Methoden der Computeralgebra bestehende Beschränkungen der industriellen Anwendung überwindet. Der zweite Teil basiert auf gemeinsamer Forschung mit Jason Morton, Bernd Sturmfels und Anne Shiu. Wir entwickeln eine allgemeine Methode um eine bestimmte Klasse von Ranktests zu beschreiben und zu berechnen. Diese Klasse kann vereinfacht durch Partialordnungen sowie die Berechnung ihrer linearen Erweiterungen beschrieben werden. Um die Tests in der Praxis anwenden zu können entwickeln wir zwei Algorithmen. Die Implementation wurde für die Bewertungen von Genexpressionsdaten genutzt, welche am Stowers Institute for Medical Research erhoben wurden. Der verwendete Datensatz dient der Untersuchung der Entwicklung der Wirbelsäule in Embryonen. Die Rangfolgen, welche von den Tests generiert wurden, waren hilfreich

Algorithms for Symbolic Computation and their Applications - Standard Bases over Rings and Rank Tests in Statistics

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges. The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.Im ersten Teil der Dissertation wird die Theorie der Standardbasen in freien Moduln über (lokalisierten) Polynomialringen entwickelt. Falls lineare Gleichungen in den Koeffizienten der Polynome lösbar sind, entwickeln wir Algorithmen um Standardbasen bezüglich beliebiger (Modul-) Monomordnungen zu berechnen. Weiterhin untersuchen wir den Fall von Hauptidealringen mit Nullteilern. Wir implementieren für diesen Fall modifizierte neue Algorithmen, welche deutlich performanter als die generischen sind. Die Implementierung wurde durch bestehende Restriktionen bei der Verifikation von System-on-Chip Layouts motiviert. Wir zeigen, dass die neue Herangehensweise mit Methoden der Computeralgebra bestehende Beschränkungen der industriellen Anwendung überwindet. Der zweite Teil basiert auf gemeinsamer Forschung mit Jason Morton, Bernd Sturmfels und Anne Shiu. Wir entwickeln eine allgemeine Methode um eine bestimmte Klasse von Ranktests zu beschreiben und zu berechnen. Diese Klasse kann vereinfacht durch Partialordnungen sowie die Berechnung ihrer linearen Erweiterungen beschrieben werden. Um die Tests in der Praxis anwenden zu können entwickeln wir zwei Algorithmen. Die Implementation wurde für die Bewertungen von Genexpressionsdaten genutzt, welche am Stowers Institute for Medical Research erhoben wurden. Der verwendete Datensatz dient der Untersuchung der Entwicklung der Wirbelsäule in Embryonen. Die Rangfolgen, welche von den Tests generiert wurden, waren hilfreich

Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring $R[[t_1,\ldots,t_m]][x_1,\ldots,x_n]^s$ respectively of their localization with respect to a $t$-local monomial ordering for a certain class of noetherian rings $R$. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert--Hironaka and Mora and to generalize the Buchberger criterion. Everything else then translates naturally. Setting either $m=0$ or $n=0$ we get standard bases for polynomial rings respectively for power series rings over $R$ as a special case.Comment: 44 page