Kolmogorov complexity and the Recursion Theorem

Several classes of DNR functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0,1}-valued DNR function, iff A can compute a function F such that F(n) is a string of length n and maximal C-complexity among the strings of length n. A solves the halting problem iff A can compute a function F such that F(n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem.Comment: Full version of paper presented at STACS 2006, Lecture Notes in Computer Science 3884 (2006), 149--16

Der Euro: Handel ohne Grenzen?

Die Einführung des Euro bietet auch für den Händel neue Perspektiven. Werden durch den Wegfall der Wechselkursrisiken und die höhere Preistransparenz die grenzüberschreitende Nachfrage und das Angebot zunehmen? Werden sich die Preise auf dem jeweils niedrigsten nationalen Ausgangsniveau angleichen? --

Separations of Non-monotonic Randomness Notions

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-L"of randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of Martin-L"of randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr\u27s model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent ``non-monotonic\u27\u27 notion of randomness, now called Kolmogorov-Loveland-randomness, has been shown to be quite close to Martin-L"of randomness, but whether these two classes coincide remains a fundamental open question. In order to get a better understanding of non-monotonic randomness notions, Miller and Nies introduced some interesting intermediate concepts, where one only allows non-adaptive strategies, i.e., strategies that can still bet non-monotonically, but such that the sequence of betting positions is known in advance (and computable). Recently, these notions were shown by Kastermans and Lempp to differ from Martin-L"of randomness. We continue the study of the non-monotonic randomness notions introduced by Miller and Nies and obtain results about the Kolmogorov complexities of initial segments that may and may not occur for such sequences, where these results then imply a complete classification of these randomness notions by order of strength

Solovay functions and K-triviality

As part of his groundbreaking work on algorithmic randomness, Solovay demonstrated in the 1970s the remarkable fact that there are computable upper bounds of prefix-free Kolmogorov complexity $K$ that are tight on infinitely many values (up to an additive constant). Such computable upper bounds are called Solovay functions. Recent work of Bienvenu and Downey~[STACS 2009, LIPIcs 3, pp 147-158] indicates that Solovay functions are deeply connected with central concepts of algorithmic randomness such as $Omega$ numbers, K-triviality, and Martin-Loef randomness. In what follows, among other results we answer two open problems posed by Bienvenu and Downey about the definition of $K$-triviality and about the Gacs-Miller-Yu characterization of Martin-Loef randomness. The former defines a sequence A to be K-trivial if K(A|n) =^+ n-K(n). So both involve the noncomputable function K. As our main results we show that in both cases K(n) can be equivalently replaced by any Solovay function, and, what is more, that among all computable functions such a replacement is possible exactly for the Solovay functions. Moreover, similar statements hold for the larger class of all right-c.e. in place of the computable functions. These full characterizations, besides having significant theoretical interest on their own, will be useful as tools when working with K-trivial and Martin-Loef random sequences

Two-stage high pressure anaerobic digestion for biomethane production

The use of natural gas for power and heat generation in the EU has become particularly prominent since the 1990s. As a result, the whole natural gas infrastructure has been continuously expanded and today has a total length of 2.15 million km and a storage capacity of about 108.3 billion m³. The production of biomethane in the EU and its distribution by natural gas network offers an interesting alternative for the reconfiguration of EUs energy supply system. Up to now, biomethane is obtained by purifying and upgrading raw biogas in a complex process. In this study, a novel two-stage high pressure anaerobic digestion system was developed. This innovative concept aims to integrate biogas production, purification and pressure boosting within one system. The process is based on the enhanced water solubility of carbon dioxide compared to methane. By operating the methane reactor for biogas production at increased pressures, high amounts of dissolved carbon dioxide can be removed with the liquid effluent from the reactor, resulting in a high-calorific biogas. In batch experiments at pressures up to 30 bar, a significant influence of pressure on the pH-value in the reactor was observed, due to the augmented formation of carbon hydroxide. The study on the effect of a rapid pressure increase up to 100 bar showed no inhibition of the microorganisms in the batch-rigs too, although the microorganisms were not adapted to these environmental conditions. Furthermore, a continuously operated methane reactor was run at pressures up to 50 bar for the first time. The experiments showed that a stable anaerobic digestion process could be run at these pressures nearly without any problems and methane contents above 90% could be achieved. The promising results showed that this technology has great potential in producing on-site high calorific gas also in smaller units. In addition, the costs of post-production gas purification can be significantly reduced, due to the fact that the size of a subsequent gas purification unit can be decreased. Furthermore, the produced gas can be injected into the transnational gas grids without post pressurization or can be used in the transportation sector.In der EU hat die Nutzung von Erdgas zur Erzeugung von Strom und Wärme seit den 1990er Jahren zunehmend an Bedeutung gewonnen. Infolge dessen wurde die Erdgasinfrastruktur kontinuierlich ausgebaut und umfasst heute eine Leitungslänge von 2,15 Mio. km und ein Speichervolumen von 108,3 Mrd. m³. Eine interessante Alternative zur Neugestaltung des Energieversorgungssystems in der EU ist die Erzeugung von Biomethan und die anschließende Einspeisung in das bestehende Erdgasnetz. Biomethan wird bisher in einem aufwendigen Reinigungs- und Aufbereitungsverfahren aus Rohbiogas gewonnen. In der vorliegenden Studie wurde ein neuartiges zweistufiges Hochdruckfermentationsverfahren entwickelt. Ziel dieses innovativen Konzepts ist die Integration von Biogaserzeugung, -aufbereitung und -verdichtung in ein einziges Verfahren. Dieses Verfahren basiert auf der erhöhten Wasserlöslichkeit von Kohlenstoffdioxid im Vergleich zum Methan. Durch den Betrieb des Methanreaktors unter erhöhtem Druck können große Mengen an gelöstem Kohlenstoffdioxid mit dem Effluent aus dem Reaktor ausgetragen werden, was die Erzeugung eines hochkalorischen Biogases ermöglicht. Mithilfe von Batch Experimenten im Labormaßstab bei Betriebsdrücken bis 30 bar konnte ein signifikanter Einfluss des Drucks auf den pH-Wert im Reaktor ermittelt werden, aufgrund der ansteigenden Bildung von Hydrogencarbonat. Die schnelle Druckerhöhung in den Batch-Reaktoren in weiteren Untersuchungen bis zu 100 bar zeigte keinerlei Hemmung der Mikroorganismen, obwohl diese nicht an die neuen Umgebungsbedingungen angepasst waren. Außerdem wurde erstmalig ein kontinuierlicher Methanreaktor bei Drücken bis 50 bar betrieben. Dabei konnte gezeigt werden, dass ein kontinuierlicher Betrieb des Methanreaktors bei Drücken bis 50 bar nahezu problemlos möglich ist und Methangehalte von über 90% realisiert werden können. Diese vielversprechenden Ergebnisse zeigen, dass diese Technologie ein großes Potential für die Herstellung von hochkalorischem Gas auch in kleineren Maßstäben vor Ort bietet. Außerdem können die Kosten für eine nachgeschaltete Aufbereitungseinheit signifikant reduziert werden, da diese deutlich kleiner ausfallen könnte. Darüber hinaus kann das erzeugte Gas ohne weitere Verdichtung in das Erdgasnetz eingespeist oder im Transportsektor als Treibstoff eingesetzt werden

06051 Abstracts Collection -- Kolmogorov Complexity and Applications

From 29.01.06 to 03.02.06, the Dagstuhl Seminar 06051 ``Kolmogorov Complexity and Applications\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Prognostic significance of cardiac magnetic resonance imaging: Update 2010

Cardiac magnetic resonance imaging (CMR) has become an indispensible imaging technique for the diagnosis and treatment of patients with cardiovascular diseases. Technical advances in the past have rendered CMR unique in the evaluation of cardiovascular anatomy, physiology, and pathophysiology due to its unique ability to produce high resolution tomographic images of the human heart and vessels in any arbitrary orientation, with soft tissue contrast that is superior to competing imaging modalities without the use of ionizing radiation. CMR imaging is the gold standard for assessing left and right ventricular function and for detecting myocardial tissue abnormalities like edema, infarction, or scars. For prognostic reasons abnormal structure and dysfunction of the heart, and the detection of myocardial ischemia and/or myocardial scars are the main targets for CMR imaging. In this review we briefly describe the prognostic significance of several CMR imaging techniques and special CMR parameters in patients with coronary artery disease (CAD), with cardiomyopathies, and with chronic heart failure. Myocardial ischemia proved to be a strong predictor of an adverse outcome in patients with CAD. Microvascular obstruction in acute myocardial infarction is a new and independent parameter of negative left ventricular remodeling and a worse prognosis. Myocardial scars in patients with CAD and unrecognized myocardial infarction heralds a negative outcome. Scar in patients with dilated or hypertrophic cardiomyopathy are a strong predictor of both life-threatening ventricular tachyarrhythmias and prognosis. CMR imaging may improve the assessment of inter- and intraventricular dyssynchrony and provide prognostic information by detecting myocardial scars. (Cardiol J 2010; 17, 6: 549-557