The Gomory-Chvátal closure : polyhedrality, complexity, and extensions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Vita. Cataloged from PDF version of thesis.Includes bibliographical references (p. 163-166).In this thesis, we examine theoretical aspects of the Gomory-Chvátal closure of polyhedra. A Gomory-Chvátal cutting plane for a polyhedron P is derived from any rational inequality that is valid for P by shifting the boundary of the associated half-space towards the polyhedron until it intersects an integer point. The Gomory-ChvAital closure of P is the intersection of all half-spaces defined by its Gomory-Chvátal cuts. While it is was known that the separation problem for the Gomory-Chvátal closure of a rational polyhedron is NP-hard, we show that this remains true for the family of Gomory-Chvátal cuts for which all coefficients are either 0 or 1. Several combinatorially derived cutting planes belong to this class. Furthermore, as the hyperplanes associated with these cuts have very dense and symmetric lattices of integer points, these cutting planes are in some- sense the "simplest" cuts in the set of all Gomory-Chvátal cuts. In the second part of this thesis, we answer a question raised by Schrijver (1980) and show that the Gomory-Chvátal closure of any non-rational polytope is a polytope. Schrijver (1980) had established the polyhedrality of the Gomory-Chvdtal closure for rational polyhedra. In essence, his proof relies on the fact that the set of integer points in a rational polyhedral cone is generated by a finite subset of these points. This is not true for non-rational polyhedral cones. Hence, we develop a completely different proof technique to show that the Gomory-Chvátal closure of a non-rational polytope can be described by a finite set of Gomory-Chvátal cuts. Our proof is geometrically motivated and applies classic results from polyhedral theory and the geometry of numbers. Last, we introduce a natural modification of Gomory-Chvaital cutting planes for the important class of 0/1 integer programming problems. If the hyperplane associated with a Gomory-Chvátal cut for a polytope P C [0, 1]' does not contain any 0/1 point, shifting the hyperplane further towards P until it intersects a 0/1 point guarantees that the resulting half-space contains all feasible solutions. We formalize this observation and introduce the class of M-cuts that arises by strengthening the family of Gomory- Chvátal cuts in this way. We study the polyhedral properties of the resulting closure, its complexity, and the associated cutting plane procedure.by Juliane DunkelPh.D

Neurophysiologische Korrelate der Verarbeitung von kohärenter Bewegung bei Kindern und Jugendlichen mit Autismus-Spektrum-Störungen

Die Autismus-Spektrum-Störungen (ASD) gehören zu den Tiefgreifenden Entwicklungsstörungen und sind durch Beeinträchtigungen in der sozialen Kommunikation und Interaktion gekennzeichnet. Eine Grundlage für die Interpretation von sozialen Kontexten ist die Wahrnehmung von visuellen Informationen und insbesondere der Verarbeitung von Bewegung. Diese Verarbeitung von Bewegung erfolgt vor allem über den dorsalen Pfad des visuellen Systems (Area MT/V5). Bei Untersuchungen zur visuellen Informationsverarbeitung zeigten Menschen mit ASD ein Defizit in der Wahrnehmung kohärenter Bewegung, also zusammenhängender und gleichzeitiger Bewegung. Da Personen mit ASD eine höhere Schwelle bei der Erkennung von kohärenter Bewegung zeigten, entstand die Theorie eines Defizits im dorsalen Pfad des visuellen Systems. Die Untersuchung dieser Theorie auf neurophysiologischer Ebene, vor allem mittels fMRT, ergab bisher uneinheitliche Ergebnisse. Eine weitere Möglichkeit der Erfassung neurophysiologischer Prozesse, besonders der zeitlichen Abläufe der Verarbeitung von Bewegungswahrnehmung, stellt die Messung ereigniskorrelierter Potentiale (ERP) während entsprechender Aufgaben dar. ERP spiegeln generell höhere Verarbeitungsprozesse des Kortex wider und die N200 stellt das dominante Potential bei der Erfassung von kohärenter Bewegung dar. Ein Defizit im dorsalen Pfad, auf dem die Informationen über kohärente Bewegung zur Area MT/V5 gelangen, müsste daher eine Minderaktivität bedingen und sich somit in einer reduzierten Amplitude der N200 widerspiegeln. Zur Überprüfung der Hypothese eines Defizits im dorsalen Pfad bei ASD wurde eine Untersuchung der Bewegungswahrnehmung mit 16 Kindern und Jugendlichen mit ASD und 12 normal entwickelten Kontrollpersonen durchgeführt. In der vorliegenden Studie erfolgte die Bearbeitung eines „Random dot kinematogram“ (RDK) unter Ableitung eines 64-Kanal-EEGs. Die Aufgabe bestand aus einer Präsentation von Punkten mit zunächst zufälliger Bewegung einer Punktwolke, aus der heraus die kohärente Bewegung einer bestimmten Anzahl von Punkten nach rechts oder links erfolgte (abhängig vom Kohärenzniveau 20%, 40% oder 60% der Punkte). Die Probanden sollten die Bewegungsrichtung der Punkte angeben. Es wurden Verhaltensdaten und EEG aufgezeichnet. Nach Auswertung der EEG-Daten ergab sich neben der N200 (indikativ für basale Bewegungswahrnehmung) als Hauptpotential noch eine spätere Positivität, hier P400 (indikativ für kognitive Bewertungs- bzw. Entscheidungsprozesse) genannt. Im Gruppenvergleich fand sich hierbei ein signifikanter Unterschied für die Amplitude der N200: die Teilnehmer mit ASD zeigten eine kleinere Amplitude als die Kontrollgruppe. Die Latenz der N200 unterschied sich nicht signifikant zwischen beiden Gruppen. Für die Amplitude und Latenz der P400 zeigte sich kein Gruppenunterschied. Bei der Auswertung der Verhaltensdaten fanden sich keine Gruppenunterschiede. Die Arbeit konnte somit zeigen, dass bei Kindern und Jugendlichen mit ASD auf neurophysiologischer Ebene eine abweichende Verarbeitung von kohärenter Bewegung stattfindet. Eine reduzierte Amplitude der N200 lässt auf ein Defizit in der Verarbeitung von kohärenter Bewegung, insbesondere in den frühen Anteilen des dorsalen Pfades schließen, bei erhaltenen kognitiven, bewertenden Prozessen. Die vorliegende Studie leistet somit einen wichtigen Beitrag zum Verständnis neurophysiologischer Grundlagen von Bewegungswahrnehmung als potentielle Grundlage sozial-kommunikativer Auffälligkeit bei ASD.

Seronegative myasthenic crisis: a multicenter analysis

Myasthenic crisis (MC) is a life-threatening condition for patients with myasthenia gravis (MG). Seronegative patients represent around 10–15% of MG, but data on outcome of seronegative MCs are lacking. We performed a subgroup analysis of patients who presented with MC with either acetylcholine-receptor-antibody-positive MG (AChR-MG) or seronegative MG between 2006 and 2015 in a retrospective German multicenter study. We identified 15 seronegative MG patients with 17 MCs and 142 AChR-MG with 159 MCs. Seronegative MCs were younger (54.3 ± 14.5 vs 66.5 ± 16.3 years; p = 0.0037), had a higher rate of thymus hyperplasia (29.4% vs 3.1%; p = 0.0009), and were more likely to be female (58.8% vs 37.7%; p = 0.12) compared to AChR-MCs. Time between diagnosis of MG and MC was significantly longer in seronegative patients (8.2 ± 7.6 vs 3.1 ± 4.4 years; p < 0.0001). We found no differences in duration of mechanical ventilation (16.2 ± 15.8 vs 16.5 ± 15.9 days; p = 0.94) and length of stay at intensive care unit (17.6 ± 15.2 vs 17.8 ± 15.4 days; p = 0.96), or in-hospital mortality (11.8% vs. 10.1%; p = 0.69). We conclude that MC in seronegative MG affects younger patients after a longer period of disease, but that crisis treatment efficacy and outcome do not differ compared to AChR-MCs

Massive X-ray screening reveals two allosteric drug binding sites of SARS-CoV-2 main protease

The coronavirus disease (COVID-19) caused by SARS-CoV-2 is creating tremendous health problems and economical challenges for mankind. To date, no effective drug is available to directly treat the disease and prevent virus spreading. In a search for a drug against COVID-19, we have performed a massive X-ray crystallographic screen of repurposing drug libraries containing 5953 individual compounds against the SARS-CoV-2 main protease (Mpro), which is a potent drug target as it is essential for the virus replication. In contrast to commonly applied X-ray fragment screening experiments with molecules of low complexity, our screen tested already approved drugs and drugs in clinical trials. From the three-dimensional protein structures, we identified 37 compounds binding to Mpro. In subsequent cell-based viral reduction assays, one peptidomimetic and five non-peptidic compounds showed antiviral activity at non-toxic concentrations. Interestingly, two compounds bind outside the active site to the native dimer interface in close proximity to the S1 binding pocket. Another compound binds in a cleft between the catalytic and dimerization domain of Mpro. Neither binding site is related to the enzymatic active site and both represent attractive targets for drug development against SARS-CoV-2. This X-ray screening approach thus has the potential to help deliver an approved drug on an accelerated time-scale for this and future pandemics

X-ray screening identifies active site and allosteric inhibitors of SARS-CoV-2 main protease

The coronavirus disease (COVID-19) caused by SARS-CoV-2 is creating tremendous human suffering. To date, no effective drug is available to directly treat the disease. In a search for a drug against COVID-19, we have performed a high-throughput X-ray crystallographic screen of two repurposing drug libraries against the SARS-CoV-2 main protease (M^(pro)), which is essential for viral replication. In contrast to commonly applied X-ray fragment screening experiments with molecules of low complexity, our screen tested already approved drugs and drugs in clinical trials. From the three-dimensional protein structures, we identified 37 compounds that bind to M^(pro). In subsequent cell-based viral reduction assays, one peptidomimetic and six non-peptidic compounds showed antiviral activity at non-toxic concentrations. We identified two allosteric binding sites representing attractive targets for drug development against SARS-CoV-2

Lattice Triangulations How many triangulations exist for a 6 × 6 grid?

In this paper we describe two different methods for counting the number of triangulations for planar point sets. The first method introduced by Kaibel and Ziegler [1] is shown to be effective only in the special case of planar grids of size m × n, denoted by Pm,n: = {0, 1,..., m}×{0, 1,..., n}. The second one, the path of a triangulation method du

Clinical spectrum in three families with familial hemiplegic migraine type 2 including a novel mutation in the ATP1A2 gene

Introduction Familial hemiplegic migraine (FHM) is a rare subtype of migraine with transient hemiplegic aura. Patients and methods We describe three unrelated families with familial hemiplegic migraine type II (FHM2). Retrospectively, information on 47 family members could be obtained, 15 by personal examination and 32 by indirect anamnesis from relatives. Genetic analyses were performed in 13 patients. Results One family had a novel missense mutation in the ATP1A2 gene (c.659C>T, p.Ser220Leu) that segregated with the phenotype in three generations. Two further unrelated families with different ethnic backgrounds (one from Germany and one from Russia) had a missense mutation that has not been described as yet in FHM, but occurred in only a single patient with sporadic hemiplegic migraine (c.2723G>A, p.Arg908Gln). Clinically the patients had severe attacks lasting up to several weeks as well as epileptic seizures. Three patients with a proven mutation in the ATP1A2 gene clinically presented without hemiparesis. Furthermore, there was a possible relation of FHM2 to mental retardation in another two patients. Conclusion Clinical symptoms may last for several weeks in some patients. Patients with FHM2 may also present without hemiplegia. Therefore, the full family history has to be taken into account to establish the diagnosis of FHM