Ordering the Reidemeister moves of a classical knot

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type.Comment: This is the version published by Algebraic & Geometric Topology on 18 May 200

Nonequilibrium ionization states and cooling rates of the photoionized enriched gas

Nonequilibrium (time-dependent) cooling rates and ionization state calculations are presented for low-density gas enriched with heavy elements (metals) and photoionized by external ultraviolet/X-ray radiation. We consider a wide range of gas densities and metallicities and also two types of external radiation field: a power-law and the extragalactic background spectra. We have found that both cooling efficiencies and ionic composition of enriched photoionized gas depend significantly on the gas metallicity and density, the flux amplitude, and the shape of ionizing radiation spectrum. The cooling rates and ionic composition of gas in nonequilibrium photoionization models differ strongly (by a factor of several) from those in photoequilibrium due to overionization of the ionic states in the nonequilibrium case. The difference is maximal at low values of the ionization parameter and similar in magnitude to that between the equlibrium and nonequilibrium cooling rates in the collisionally controlled gas. In general, the nonequilibrium effects are notable at T\simlt 10^6 K. In this temperature range, the mismatch of the ionic states and their ratios between the photoequilibrium and the photo-nonequilibrium models reach a factor of several. The net result is that the time-dependent energy losses due to each chemical element (i.e. the contributions to the total cooling rate) differ singificantly from the photoequilibrium ones. We advocate the use of nonequilibrium cooling rates and ionic states for gas with near-solar (and above) metallicity exposed to an arbitrary ionizing radiation flux. We provide a parameter space (in terms of temperature, density, metallicity and ionizing radiation flux), where the nonequilibrium cooling rates are to be used. (abridged)Comment: 14 pages, 11 figures, accepted to MNRA

Stabilization, amalgamation, and curves of intersection of Heegaard splittings

We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken 3-manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer $n$ there are 3-manifolds containing an essential torus and a Heegaard splitting such that the torus and splitting surface must intersect in at least $n$ simple closed curves. These give the first examples of lower bounds on the minimum number of curves of intersection between an essential surface and a Heegaard surface that are greater than one.Comment: Version for publication. To appear in Algebraic and Geometric Topolog

Operator algebras and conjugacy problem for the pseudo-Anosov automorphisms of a surface

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra). It is proved that the assignment is functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A(f). The new invariants of the conjugacy of the pseudo-Anosov automorphisms are obtained from the known invariants of the stable isomorphisms of the AF-algebras. Namely, the main invariant is a triple (L, [I], K), where L is an order in the ring of integers in a real algebraic number field K and [I] an equivalence class of the ideals in L. The numerical invariants include the determinant D and the signature S, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated.Comment: 23 pages, 1 fig;; to appear Pacific J. Math. arXiv admin note: text overlap with arXiv:math/011022

Flipping bridge surfaces and bounds on the stable bridge number

We show that if $K$ is a knot in $S^3$ and $\Sigma$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $\Sigma$ via an isotopy is $n$. We also construct a knot with two different bridge spheres with $2n$ and $2n-1$ bridges respectively for which any common perturbation has at least $3n-1$ bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.Comment: 20 pages, 7 figure

Architecture of selected full planetary systems

Unser Wissen über den Aufbau von Planetensystemen ist noch immer im Wandel. Unser Sonnensystem mit seinen acht Planeten, dem inneren Asteroidengürtel, sowie dem Kuipergürtel am äußeren Rand gilt als Prototyp eines Planetensystems. Durch die Entdeckungen von zum Beispiel heißen Jupitern oder sehr massereichen und ausgedehnten Trümmerscheiben wurde dies in Frage gestellt. Natürlich wurden durch die Einschränkungen in der Beobachtungstechnik bisher nur die Extremfälle entdeckt. In den meisten Systemen kennen wir nur Planeten oder nur Trümmerscheiben. In einigen Fällen sind jedoch sowohl Planeten als auch Trümmerscheiben bekannt. Für diese Systeme führen wir den Begriff „vollständige Planetensysteme“ ein. In dieser Arbeit wurde die Architektur von vollständigen Planetensystemen anhand der zwei Beispielsysteme HR 8799 und " Eridani untersucht. HR 8799 ist ein Mehrfachplanetensystem, in dem eine innere und eine äußere Staubscheibe bekannt sind, deren jeweils äußerer bzw. innerer Rand von vier massereichen Gasriesen geformt wird. Das System wurde zum einen auf die Stabilität und die Orientierung mit Hilfe von N-Körper-Simulationen untersucht. Zum anderen wurde versucht die Lage, Ausdehnung und Masse der Scheiben anhand der spektralen Energieverteilung zu bestimmen. Zusätzlich wurde auch das Alter des Sterns, seine Inklination sowie die Masse der Planeten berücksichtigt

Fault Diagnosis in Enterprise Software Systems Using Discrete Monitoring Data

Success for many businesses depends on their information software systems. Keeping these systems operational is critical, as failure in these systems is costly. Such systems are in many cases sophisticated, distributed and dynamically composed. To ensure high availability and correct operation, it is essential that failures be detected promptly, their causes diagnosed and remedial actions taken. Although automated recovery approaches exists for specific problem domains, the problem-resolution process is in many cases manual and painstaking. Computer support personnel put a great deal of effort into resolving the reported failures. The growing size and complexity of these systems creates the need to automate this process. The primary focus of our research is on automated fault diagnosis and recovery using discrete monitoring data such as log files and notifications. Our goal is to quickly pinpoint the root-cause of a failure. Our contributions are: Modelling discrete monitoring data for automated analysis, automatically leveraging common symptoms of failures from historic monitoring data using such models to pinpoint faults, and providing a model for decision-making under uncertainty such that appropriate recovery actions are chosen. Failures in such systems are caused by software defects, human error, hardware failures, environmental conditions and malicious behaviour. Our primary focus in this thesis is on software defects and misconfiguration

The universal functorial equivariant Lefschetz invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.Comment: 33 pages; shortened version of the author's PhD thesis, supervised by Wolfgang Lueck, Westfaelische Wilhelms-Universitaet Muenster, 200

Combinatorial Heegaard Floer homology and nice Heegaard diagrams

We consider a stabilized version of hat Heegaard Floer homology of a 3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance properties

Hamming distance kernelisation via topological quantum computation

We present a novel approach to computing Hamming distance and its kernelisation within Topological Quantum Computation. This approach is based on an encoding of two binary strings into a topological Hilbert space, whose inner product yields a natural Hamming distance kernel on the two strings. Kernelisation forges a link with the field of Machine Learning, particularly in relation to binary classifiers such as the Support Vector Machine (SVM). This makes our approach of potential interest to the quantum machine learning community