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1. Bathysciadium pacificum Dall, 1908 from the bathyal of Oregon: range extension and anatomy (Gastropoda: Cocculinida: Bathysciadiidae).

   Geiger, Daniel L. ; Thacker, Christine E. ( August 2023 , The Nautilus)

   Leal, José (Ed.)

   Three new records of Bathysciadium pacificum Dall, 1908 from off Oregon and Washington extend the range of the species from Peru by ~7,300 km to the temperate northeastern Pacific. The preserved and relaxed animal shows that the mantle exten-sions are not sensory papillae, but mantle folds forming the periostracum rays on the shell. Examination of radula, mantle, and mantle cavity by scanning electron microscopy (SEM), including environmental SEM (ESEM) of the fluid-preserved animal, adds a novel dimension to the understanding of the anatomy of these rare deep-sea snails. ESEM is a suitable tool for the morphological investigations of rare natural history specimens. The monotypic genera Bathypelta Moskalev, 1971 and Bonus Moskalev, 1973 are re-synonymized under Bathy-sciadium Dautzenberg and H. Fischer, 1900. No mineralization of the radula was detected by SEM-Energy Dispersive Spec-troscopy (EDS/EDAX) mapping. 

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2. Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

   https://doi.org/10.1007/s12220-020-00477-0  

   Björn, Anders ; Björn, Jana ; Shanmugalingam, Nageswari ( May 2021 , The Journal of Geometric Analysis)

   null (Ed.)

   Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p -Poincaré inequality, then the transformed measure is globally doubling and supports a global p -Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p -Poincaré inequality, carries nonconstant globally defined p -harmonic functions with finite p -energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. 

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3. Large-amplitude modulation of periodic traveling waves

   https://doi.org/10.3934/dcdss.2022070  

   Métivier, Guy ; Zumbrun, Kevin ( January 2022 , Discrete and Continuous Dynamical Systems - S)

   We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

    

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4. Proper mappings between indefinite hyperbolic spaces and type I classical domains

   https://doi.org/10.1090/tran/8618  

   Huang, Xiaojun ; Lu, Jin ; Tang, Xiaomin ; Xiao, Ming ( December 2022 , Transactions of the American Mathematical Society)

   In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. 

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5. An analogue of polynomially integrable bodies in even-dimensional spaces

   https://doi.org/10.1016/j.jmaa.2023.127071  

   Agranovsky, M ; Koldobsky, A ; Ryabogin, D ; Yaskin, V. ( January 2024 , Journal of mathematical analysis and applications)

   A bounded domain $K \subset R^n$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $\Pi$ polynomially depends on the distance from $\Pi$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $n$ is even, and if $n$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $n$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property. 

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