(Generalized) Maximum Cumulative Direct, Residual, and Paired Φ Entropy Approach

A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative Φ entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents so-called contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey λ and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey λ distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily S&P500 returns and time durations between mine explosions.

Resolving the puzzle of type IIP SN 2016X

ABSTRACT The enigmatic type IIP SN 2016X demonstrates the unprecedented asphericity in the nebular Hα line profile, the absence of nebular [O i] emission, and the unusual occultation effect due to the internal dust. The hydrodynamic modelling of the bolometric light curve and expansion velocities suggests that the event is an outcome of the massive star explosion that ejected 28 M$\odot$ with the kinetic energy of 1.7 × 1051 erg and 0.03 M$\odot$ of radioactive 56Ni. We recover the bipolar distribution of 56Ni from the Hα profile via the simulation of the emissivity produced by non-spherical 56Ni ejecta. The conspicuous effect of the dust absorption in the Hα profile rules out the occultation by the dusty sphere or dusty thick disc, but turns out consistent with the thin dusty disc-like structure in the plane perpendicular to the bipolar axis. We speculate that the absence of the nebular [O i] emission might originate from the significant cooling of the oxygen-rich matter mediated by CO and SiO molecules.

Umbilicaria subpolyphylla Oxner: the correct name for U. iberica Sancho & Krzewicka and its bipolar distribution pattern

The Umbilicaria polyphylla aggregate (U. polyphylla (L.) Baumg., U. subpolyphylla Oxner and U. iberica Sancho & Krzewicka) is discussed based on morphological, chemical and molecular data. Umbilicaria iberica is proposed to be a later synonym of U. subpolyphylla. The constructed nrITS + mtLSU phylogeny, which includes specimens with wide geographical ranges, shows that both U. polyphylla and U. subpolyphylla are monophyletic and closely related. Both species have the same type of thalloconidia and identical secondary metabolites. Umbilicaria subpolyphylla has prominent phenotypic differences when compared to U. polyphylla including the monophyllous thallus with a dull upper surface and an elevated, slightly wrinkled centre, often covered with white pruina, and a medulla of the ‘U. havaasii’ type. Phylogenetic evidence for the bipolar distribution of both U. polyphylla and U. subpolyphylla is provided. Sympatric speciation in one region followed by long-distance dispersal seems to be the most plausible phylogeographical explanation for the observed patterns. Umbilicaria subpolyphylla is found in southern temperate-subtropical (Mediterranean) mountains, at least in Europe.

Revision of Testechiniscus Kristensen, 1987 (Heterotardigrada: Echiniscidae) refutes the polar-temperate distribution of the genus

The family Echiniscidae comprises limno-terrestrial heterotardigrades with a strongly sclerotised dorsum, typically covered with plates. Among other members of the Echiniscus evolutionary line, the genus Testechiniscus Kristensen, 1987 stands out with well-developed ventral armature and polygonal sculpturing of the dorsal plates. It has alleged bipolar distribution (with satellite alpine records in the Holarctic). Thanks to fresh material from terra typica (Svalbard), we integratively redescribe (i.e. using light and electron microscopy imaging, morphometry, and molecular methods) the nominal species for the genus, Testechiniscus spitsbergensis (Scourfield, 1897). A comparison of the neotype series with a number of Holarctic records revealed morphological variability suggesting that the species may encompass several taxa, which, most likely, will be possible to delineate primarily with molecular tools. Moreover, based on material from Simien Mountains (Northern Ethiopia), we describe a new sibling subspecies, Testechiniscus spitsbergensis tropicalis ssp. nov. Extensive morphometric datasets are provided for the genus members for the first time. A new generic definition is proposed, embracing the two subspecies of T. spitsbergensis and T. laterculus (Schuster et al., 1980), but excluding two circum-Antarctic species, T. macronyx (Richters, 1907) and T. meridionalis (Murray, 1906). The later species are likely to be erected in the future as separate genera, and their autapomorphies are described here. In the light of our findings, the genus Testechiniscus should be recognised as a native element of the Northern Hemisphere, with mainly circum-Arctic distribution and additional, insular alpine records from the Nearctic, Palearctic and Eastern Afrotropic.