Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures improved, references update

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

Posterior fossa meningiomas: perioperative predictors of extent of resection, overall survival and progression-free survival

BACKGROUND: Posterior fossa meningiomas (PFMs) often represent surgical challenges due to their proximity to neurovascular structures. Factors predicting the extent of resection (EOR), overall survival (OS), and progression-free survival (PFS) were identified and integrated in a prediction tool to offer evidence-based personalized therapeutic strategies. METHODS: All meningiomas managed surgically from 1990 to 2010 from a single-center were reviewed. A classification tree was created using the classification and regression tree recursive partitioning analysis that incorporated patient and tumor data available before surgery in order to predict the rates of gross total resection (GTR). RESULTS: A total of 198 patients were identified (female-to-male ratio, 2.7; mean age, 59.1 years) and compared with 1271 supratentorial meningiomas (STMs) operated in the same institution during the same time period. GTR was achieved less often (59.6% versus 81.9%; p < 0.01) in PFMs than STMs. Preoperative neurological symptoms were predictive of higher Simpson grades (OR, 2.19 [1.05; 4.58]; p = 0.04). Age was associated with reduced OS (OR, 1.08 [1.04;1.12]; p < 0.001). A KPS ≥ 70 was associated with higher survival rates (OR, 2.70 [2.19;2.92]; p = 0.02). Higher WHO grades were associated with reduced OS (OR, 3.56 [1.02;12.47]; p = 0.05). The GTR rate varies from 80% in patients without a preoperative deficit to 40% patients with a preoperative deficit, younger than 60 years old, and with adjacent bone invasion. CONCLUSIONS: This study provides a classification tree of the predictors of EOR in PFMs, based upon preoperative demographic, clinical, and radiological variables. An evidence-based management protocol with estimated EORs may guide the decision-making process in PFMs

On hydrogen bond correlations at high pressures

In situ high pressure neutron diffraction measured lengths of O H and H O pairs in hydrogen bonds in substances are shown to follow the correlation between them established from 0.1 MPa data on different chemical compounds. In particular, the conclusion by Nelmes et al that their high pressure data on ice VIII differ from it is not supported. For compounds in which the O H stretching frequencies red shift under pressure, it is shown that wherever structural data is available, they follow the stretching frequency versus H O (or O O) distance correlation. For compounds displaying blue shifts with pressure an analogy appears to exist with improper hydrogen bonds.Comment: 12 pages,4 figure

WHO grade I meningiomas: classification-tree for prognostic factors of survival

World Health Organization (WHO) grade I meningiomas are intracranial extracerebral tumors, in which microsurgery as a stand-alone therapy provides high rates of disease control and low recurrence rates. Our aim was to identify prognostic factors of overall survival and time-to-retreat (OS; TTR) in a cohort of patients with surgically managed WHO grade I meningioma. Patients with WHO grade I meningiomas from a retrospectively (1990 to 2002) and prospectively managed (2003 to 2010) databank of Oslo University Hospital, Norway, were included. The mean follow-up was 9.2 ± 5.7 years, with a total of 11,414 patient-years. One thousand three hundred fifty-five patients were included. The mean age was 58 ± 13.2, mean Karnofsky Performance Status (KPS) 92.6 ± 26.1 and female-to-male ratio 2.5:1. The 1-year, 5-year, 10-year, 15-year, and 20-year probabilities were 0.98, 0.91, 0.87, 0.84, and 0.8 for TTR. Patient age (OR 0.92 [0.91, 0.94]), male sex (OR 0.59 [0.45, 0.76]), preoperative KPS ≥ 70 (OR 2.22 [1.59, 3.13]), skull base location (OR 0.77 [0.60, 1]), and the occurrence of a postoperative hematoma (OR 0.44 [0.26, 0.76]) were identified as independent prognostic factors of OS. Patient age (OR 1.02 [1.01, 1.03]) and skull base location (OR 0.30 [0.21, 0.45]) were independent predictors of decreased PFS. Using a recursive partitioning analysis, we suggest a classification tree for the prediction of 5-year PFS based on patient and tumor characteristics. The findings from this cohort of meningioma WHO I patients helps to identify patients at risk of recurrence and tailor the therapeutic management

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

MPS Editor

Previously, it was time-consuming to hand-edit data and then set up simulation runs to find the effect and impact of the input data on a spacecraft. MPS Editor provides the user the capability to create/edit/update models and sequences, and immediately try them out using what appears to the user as one piece of software. MPS Editor provides an integrated sequencing environment for users. It provides them with software that can be utilized during development as well as actual operations. In addition, it provides them with a single, consistent, user friendly interface. MPS Editor uses the Eclipse Rich Client Platform to provide an environment that can be tailored to specific missions. It provides the capability to create and edit, and includes an Activity Dictionary to build the simulation spacecraft models, build and edit sequences of commands, and model the effects of those commands on the spacecraft. MPS Editor is written in Java using the Eclipse Rich Client Platform. It is currently built with four perspectives: the Activity Dictionary Perspective, the Project Adaptation Perspective, the Sequence Building Perspective, and the Sequence Modeling Perspective. Each perspective performs a given task. If a mission doesn't require that task, the unneeded perspective is not added to that project's delivery. In the Activity Dictionary Perspective, the user builds the project-specific activities, observations, calibrations, etc. Typically, this is used during the development phases of the mission, although it can be used later to make changes and updates to the Project Activity Dictionary. In the Adaptation Perspective, the user creates the spacecraft models such as power, data store, etc. Again, this is typically used during development, but will be used to update or add models of the spacecraft. The Sequence Building Perspective allows the user to create a sequence of activities or commands that go to the spacecraft. It provides a simulation of the activities and commands that have been created