Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes.Comment: 54 pages, 33 figure

Enumerating Foldings and Unfoldings between Polygons and Polytopes

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in Proceedings of the Japan Conference on Discrete and Computational Geometry, Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701

Limits on the validity of the thin-layer model of the ionosphere for radio interferometric calibration

For a ground-based radio interferometer observing at low frequencies, the ionosphere causes propagation delays and refraction of cosmic radio waves which result in phase errors in the received signal. These phase errors can be corrected using a calibration method that assumes a two-dimensional phase screen at a fixed altitude above the surface of the Earth, known as the thin-layer model. Here we investigate the validity of the thin-layer model and provide a simple equation with which users can check when this approximation can be applied to observations for varying time of day, zenith angle, interferometer latitude, baseline length, ionospheric electron content and observing frequency.Comment: 8 pages, 10 figures, accepted MNRA

The Oxygen Abundance of HE 1327-2326

From a newly obtained VLT/UVES spectrum we have determined the oxygen abundance of HE 1327-2326, the most iron-poor star known to date. UV-OH lines yield a 1D LTE abundance of [O/Fe]_OH = 3.7 (subgiant case) and [O/Fe]_OH = 3.4 (dwarf case). Using a correction of -1.0 dex to account for 3D effects on OH line formation, the abundances are lowered to [O/Fe] = 2.8 and [O/Fe] = 2.5, respectively, which we adopt. Without 3D corrections, the UV-OH based abundance would be in disagreement with the upper limits derived from the OI triplet lines: [O/Fe]_trip < 2.8 (subgiant) and [O/Fe]_trip < 3.0 (dwarf). We also correct the previously determined carbon and nitrogen abundances for 3D effects. Knowledge of the O abundance of HE 1327-2326 has implications for the interpretation of its abundance pattern. A large O abundance is in accordance with HE 1327-2326 being an early Population II star which formed from material chemically enriched by a first generation supernova. Our derived abundances, however, do not exclude other possibilities such as a Population III scenario.Comment: 13 pages, accepted for publication in ApJ

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Parental Investment Decisions in Response to Ambient Nest-Predation Risk Versus Actual Predation on the Prior Nest

Theory predicts that parents should invest less in dependent offspring with lower reproductive value, such as those with a high risk of predation. Moreover, high predation risk can favor reduced parental activity when such activity attracts nest predators. Yet, the ability of parents to assess ambient nest-predation risk and respond adaptively remains unclear, especially where nest-predator assemblages are diverse and potentially difficult to assess. We tested whether variation in parental investment by a multi-brooded songbird (Brewer\u27s Sparrow, Spizella breweri) in an environment (sagebrush steppe) with diverse predators was predicted by ambient nest-predation risk or direct experience with nest predation. Variation among eight sites in ambient nest-predation risk, assayed by daily probabilities of nest predation, was largely uncorrelated across four years. In this system risk may therefore be unpredictable, and aspects of parental investment (clutch size, egg mass, incubation rhythms, nestling-feeding rates) were not related to ambient risk. Moreover, investment at first nests that were successful did not differ from that at nests that were depredated, suggesting parents could not assess and respond to territory-level nest-predation risk. However, parents whose nests were depredated reduced clutch sizes and activity at nests attempted later in the season by increasing the length of incubation shifts (on-bouts) and recesses (off-bouts) and decreasing trips to feed nestlings. In this unpredictable environment parent birds may therefore lack sufficient cues of ambient risk on which to base their investment decisions and instead rely on direct experience with nest predation to inform at least some of their decisions

Trauma ICU Prevalence Project: the diversity of surgical critical care.

Background:Surgical critical care is crucial to the care of trauma and surgical patients. This study was designed to provide a contemporary assessment of patient types, injuries, and conditions in intensive care units (ICU) caring for trauma patients. Methods:This was a multicenter prevalence study of the American Association for the Surgery of Trauma; data were collected on all patients present in participating centers' trauma ICU (TICU) on November 2, 2017 and April 10, 2018. Results:Forty-nine centers submitted data on 1416 patients. Median age was 58 years (IQR 41-70). Patient types included trauma (n=665, 46.9%), non-trauma surgical (n=536, 37.8%), medical (n=204, 14.4% overall), or unspecified (n=11). Surgical intensivists managed 73.1% of patients. Of ICU-specific diagnoses, 57% were pulmonary related. Multiple high-intensity diagnoses were represented (septic shock, 10.2%; multiple organ failure, 5.58%; adult respiratory distress syndrome, 4.38%). Hemorrhagic shock was seen in 11.6% of trauma patients and 6.55% of all patients. The most common traumatic injuries were rib fractures (41.6%), brain (38.8%), hemothorax/pneumothorax (30.8%), and facial fractures (23.7%). Forty-four percent were on mechanical ventilation, and 17.6% had a tracheostomy. One-third (33%) had an infection, and over half (54.3%) were on antibiotics. Operations were performed in 70.2%, with 23.7% having abdominal surgery. At 30 days, 5.4% were still in the ICU. Median ICU length of stay was 9 days (IQR 4-20). 30-day mortality was 11.2%. Conclusions:Patient acuity in TICUs in the USA is very high, as is the breadth of pathology and the interventions provided. Non-trauma patients constitute a significant proportion of TICU care. Further assessment of the global predictors of outcome is needed to inform the education, research, clinical practice, and staffing of surgical critical care providers. Level of evidence:IV, prospective observational study

Competitive and Information Effects of Cross‐Border Stock Listings

We examine the effect of 269 cross-border listings on rivals in the listing and domestic markets and find that U.S. rivals experience significant gains whereas domestic rivals do not. Both competitive and information effects are important in explaining the reaction of U.S. rivals. Regarding the competitive effects, the reaction of rivals is less favorable when listings originate in developed countries and more favorable when listing firms do not have prior operating presence in the United States. Regarding the information effects, the reaction is less favorable when listings are combined with equity offerings and more favorable when the listing is the first to occur within an industry

A three‐dimensional placoderm (stem‐group gnathostome) pharyngeal skeleton and its implications for primitive gnathostome pharyngeal architecture

The pharyngeal skeleton is a key vertebrate anatomical system in debates on the origin of jaws and gnathostome (jawed vertebrate) feeding. Furthermore, it offers considerable potential as a source of phylogenetic data. Well‐preserved examples of pharyngeal skeletons from stem‐group gnathostomes remain poorly known. Here, we describe an articulated, nearly complete pharyngeal skeleton in an Early Devonian placoderm fish, Paraplesiobatis heinrichsi Broili, from Hunsrück Slate of Germany. Using synchrotron light tomography, we resolve and reconstruct the three‐dimensional gill arch architecture of Paraplesiobatis and compare it with other gnathostomes. The preserved pharyngeal skeleton comprises elements of the hyoid arch (probable ceratohyal) and a series of branchial arches. Limited resolution in the tomography scan causes some uncertainty in interpreting the exact number of arches preserved. However, at least four branchial arches are present. The final and penultimate arches are connected as in osteichthyans. A single median basihyal is present as in chondrichthyans. No dorsal (epibranchial or pharyngobranchial) elements are observed. The structure of the pharyngeal skeleton of Paraplesiobatis agrees well with Pseudopetalichthys from the same deposit, allowing an alternative interpretation of the latter taxon. The phylogenetic significance of Paraplesiobatis is considered. A median basihyal is likely an ancestral gnathostome character, probably with some connection to both the hyoid and the first branchial arch pair. Unpaired basibranchial bones may be independently derived in chondrichthyans and osteichthyans.A three‐dimensional articulated gill arch skeleton of a 400‐million‐year‐old placoderm fish is described. This adds to the diversity of feeding and respiratory structures in jawed vertebrate animals and informing hypotheses of gill arch evolution in the first jawed vertebrates.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/138208/1/jmor20706_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/138208/2/jmor20706.pd