An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Combined Molecular Phylogenetic Analysis of the Orthoptera (Arthropoda, Insecta) and Implications for Their Higher Systematics

A phylogenetic analysis of mitochondrial and nuclear rDNA sequences from species of all the superfamilies of the insect order Orthoptera (grasshoppers, crickets, and relatives) confirmed that although mitochondrial sequences provided good resolution of the youngest superfamilies, nuclear rDNA sequences were necessary to separate the basal groups. To try to reconcile these data sets into a single, fully resolved orthopteran phylogeny, we adopted consensus and combined data strategies. The consensus analysis produced a partially resolved tree that lacked several well-supported features of the individual analyses. However, this lack of resolution was explained by an examination of resampled data sets, which identified the likely source of error as the relatively short length of the individual mitochondrial data partitions. In a subsequent comparison in which the mitochondrial sequences were initially combined, we observed less conflict. We then used two approaches to examine the validity of combining all of the data in a single analysis: comparative analysis of trees recovered from resampled data sets, and the application of a randomization test. Because the results did not point to significant levels of heterogeneity in phylogenetic signal between the mitochondrial and nuclear data sets, we therefore proceeded with a combined analysis. Reconstructing phylogenies under the minimum evolution and maximum likelihood optimality criteria, we examined monophyly of the major orthopteran groups, using nonparametric and parametric bootstrap analysis and Kishino-Hasegawa tests. Our analysis suggests that phylogeny reconstruction under the maximum likelihood criteria is the most discriminating approach for the combined sequences. The results indicate, moreover, that the caeliferan Pneumoroidea and Pamphagoidea, as previously suggested, are polyphyletic. The Acridoidea is redefined to include all pamphagoid families other than the Pyrgomorphidae, which we propose should be accorded superfamily statu

Model of Low-pass Filtering of Local Field Potentials in Brain Tissue

Local field potentials (LFPs) are routinely measured experimentally in brain tissue, and exhibit strong low-pass frequency filtering properties, with high frequencies (such as action potentials) being visible only at very short distances ($\approx$10~$\mu m$) from the recording electrode. Understanding this filtering is crucial to relate LFP signals with neuronal activity, but not much is known about the exact mechanisms underlying this low-pass filtering. In this paper, we investigate a possible biophysical mechanism for the low-pass filtering properties of LFPs. We investigate the propagation of electric fields and its frequency dependence close to the current source, i.e. at length scales in the order of average interneuronal distance. We take into account the presence of a high density of cellular membranes around current sources, such as glial cells. By considering them as passive cells, we show that under the influence of the electric source field, they respond by polarisation, i.e., creation of an induced field. Because of the finite velocity of ionic charge movement, this polarization will not be instantaneous. Consequently, the induced electric field will be frequency-dependent, and much reduced for high frequencies. Our model establishes that with respect to frequency attenuation properties, this situation is analogous to an equivalent RC-circuit, or better a system of coupled RC-circuits. We present a number of numerical simulations of induced electric field for biologically realistic values of parameters, and show this frequency filtering effect as well as the attenuation of extracellular potentials with distance. We suggest that induced electric fields in passive cells surrounding neurons is the physical origin of frequency filtering properties of LFPs.Comment: 10 figs, revised tex file and revised fig

Complementary vertices and adjacency testing in polytopes

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we improve adjacency testing for vertices in both simple and non-simple polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq 0} and a list of its V vertices, we describe an O(n) test to identify whether any two given vertices are adjacent. For simple polytopes this test is perfect; for non-simple polytopes it may be indeterminate, and instead acts as a filter to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2) precomputation, which is acceptable in settings such as all-pairs adjacency testing. These results improve upon the more general O(nV) combinatorial and O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal version, which strengthens and extends the results in Section 2 (see p1 of the paper for details

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.Comment: 20 pages, 5 figure

Cassini RTG acceptance test results and RTG performance on Galileo and Ulysses

Flight acceptance testing has been completed for the RTGs to be used on the Cassini spacecraft which is scheduled for an October 6, 1997 launch to Saturn. The acceptance test program includes vibration tests, magnetic field measurements, properties (weight and c.g.) and thermal vacuum test. This paper presents The thermal vacuum test results. Three RTGs are to be used, F-2, F-6, and F-7. F-5 is tile back-up RTG, as it was for the Galileo and Ulysses missions launched in 1989 and 1990, respectively. RTG performance measured during the thermal vacuum tests carried out at die Mound Laboratory facility met all specification requirements. Beginning of mission (BOM) and end of mission (EOM) power predictions have been made based on than tests results. BOM power is predicted to be 888 watts compared to the minimum requirement of 826 watts. Degradation models predict the EOM power after 16 years is to be 640 watts compared to a minimum requirement of 596 watts. Results of small scale module tests are also showing. The modules contain couples from the qualification and flight production runs. The tests have exceeded 28,000 hours (3.2 years) and are continuing to provide increased confidence in the predicted long term performance of the Cassini RTGs. All test results indicate that the power requirements of the Cassini spacecraft will be met. BOM and EOM power margins of over five percent are predicted. Power output from telemetry for the two Galileo RTGs are shown from the 1989 launch to the recent Jupiter encounter. Comparisons of predicted, measured and required performance are shown. Telemetry data are also shown for the RTG on the Ulysses spacecraft which completed its planned mission in 1995 and is now in the extended mission

Measurement of bronchial hyperreactivity : comparison of three Nordic dosimetric methods

Clinical testing of bronchial hyperreactivity (BHR) provides valuable information in asthma diagnostics. Nevertheless, the test results depend to a great extent on the testing procedure: test substance, apparatus and protocol. In Nordic countries, three protocols predominate in the testing field: Per Malmberg, Nieminen and Sovijarvi methods. However, knowledge of their equivalence is limited. We aimed to find equivalent provocative doses (PD) to obtain similar bronchoconstrictive responses for the three protocols. We recruited 31 patients with suspected asthma and health care workers and performed BHR testing with methacholine according to Malmberg and Nieminen methods, and with histamine according to Sovijarvi. We obtained the individual response-dose slopes for each method and predicted equivalent PD values. Applying a mixed-model, we found significant differences in the mean (standard error of mean) response-dose (forced expiratory volume in one second (FEV1)%/mg): Sovijarvi 7.2 (1.5), Nieminen 13.8 (4.2) and Malmberg 26 (7.3). We found that the earlier reported cut-point values for moderate BHR and marked BHR between the Sovijarvi (PD15) and Nieminen (PD20) methods were similar, but with the Malmberg method a significant bronchoconstrictive reaction was measured with lower PD20 values. We obtained a relationship between slope values and PD (mg) between different methods, useful in epidemiological research and clinical practice.Peer reviewe