Ecological niche partitioning between baleen whales inhabiting Icelandic waters

The highly productive waters off Iceland are an important feeding ground for baleen whales. Five balaenopterid species coexist there during the summer feeding season: the blue whale, the fin whale, the sei whale, the humpback whale and the common minke whale. For capital breeders such as baleen whales, niche partitioning and reduced interspecific competition during their stay in the feeding grounds may be critical for the completion of their annual cycles and the long-term stability of populations. Coexistence often entails spatio-temporal or trophic segregation to avoid competitive exclusion. With the aim of studying how these species share habitat and trophic resources, we analyzed the δ13C, δ15N and δ34S values in skin samples. Bayesian stable isotope mixing models to calculate compositional mixture of food sources showed that most species segregate by consuming different prey. Segregation was further enhanced by some degree of spatio-temporal exclusion. Overall, clear ecological niche partitioning was apparent between all species except between blue and fin whales. All the species consumed krill and, except for the common minke whale, this was the dominant prey. Among baleen whales, common minke whales and humpback whales were the major predators of sand eel, capelin and herring. In humpback whales, a strong reliance on krill may explain the apparently low rates of local entanglement in fishing nets as compared to other areas. Except for the blue whale, all species have shown evidence of adapting to shifts in prey availability and thus suggested capacity to cope with variability. However, in a scenario of increasing environmental variability associated to global warming, the overlap between ecological niches may have to decrease to allow long-term coexistence

Factorizing the time evolution operator

There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator Exp{-itH/h} on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function f(x,0), for it implies to apply terms operators like (a+b)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize, i. e. Exp{a+b} is not equal to Exp{a}Exp{b}. In this work we present a useful procedure for factorizing the time evolution operator when the argument of the exponential is a sum of two operators, which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the particle subject to a constant force and the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.Comment: 24 pages; added references; one figure change

Mean Sombor index

A Special Volume on Chemical Graph Theory in Memory of Nenad TrinajsticWe introduce a degree–based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: SOα(G) = P uv∈E(G) [(d α u + d α v ) /2]1/α. Here, uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0}. We also consider the limit cases mSOα→0(G) and SOα→±∞(G). Indeed, for given values of α, the mean Sombor index is related to well-known opological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky–Puebla index and several ´Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that mSOα(G) correlates well with several physicochemical properties of octane isomers. Some mathematical properties of the mean Sombor index as well as bounds and new relationships with known topological indices are also discussed.J.A.M.-B. acknowledges financial support from CONACyT (Grant No. A1-S-22706) and BUAP (Grant No. 100405811VIEP2021) .E.D.M. and J.M.R. were supported by a grant from Agencia Estatal de Investigación (PID 2019-106433GBI00 / AEI / 10.13039 / 501100011033), Spain. J.M.R. was supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the VPRICIT (Regional Programme of Research and Technological Innovation)

Soft breaking of the μ↔τ\mu\leftrightarrow \tau symmetry by S4⊗Z2\mathbf{S}_{4}\otimes \mathbf{Z}_{2}

The $\mu \leftrightarrow \tau$ symmetry has been ruled out by its predictions on the reactor and atmospheric angles, nevertheless, a breaking of this symmetry might provide correct values. For that reason, we build a non-renormalizable lepton model where the mixings arise from the spontaneous breaking of the $\mathbf{S}_{4}\otimes \mathbf{Z}_{2}$ discrete group, subsequently the $\mu \leftrightarrow \tau$ symmetry is broken in the effective neutrino mass matrix, that comes from the type II see-saw mechanism. As main result, the reactor and atmospheric angles are corrected and their values are in good agreement with the experimental data for the inverted hierarchy. Furthermore, we point out a link between the atmospheric angle and reactor one. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed.Comment: 19 pages, 6 figures. Minor changes, accepted in Revista Mexicana de F\'isic

A white dwarf cooling age of 8 Gyr for NGC 6791 from physical separation processes

NGC 6791 is a well studied open cluster1 that it is so close to us that can be imaged down to very faint luminosities. The main sequence turn-off age (~8 Gyr) and the age derived from the termination of the white dwarf cooling sequence (~6 Gyr) are significantly different. One possible explanation is that as white dwarfs cool, one of the ashes of helium burning, 22Ne, sinks in the deep interior of these stars. At lower temperatures, white dwarfs are expected to crystallise and phase separation of the main constituents of the core of a typical white dwarf, 12C and 16O, is expected to occur. This sequence of events is expected to introduce significant delays in the cooling times, but has not hitherto been proven. Here we report that, as theoretically anticipated, physical separation processes occur in the cores of white dwarfs, solving the age discrepancy for NGC 6791.Comment: 3 pages, 2 figures, published in Natur