On the lower bound of the inner radius of nodal domains

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions \varphi _{\lambda} on a closed Riemannian manifold (M, g) . In the real-analytic case, we present an improvement of the currently best-known bounds, due to Mangoubi (Commun Partial Differ Equ 33:1611–1621, 2008; Can Math Bull 51(2):249–260, 2008). Furthermore, using recent results of Hezari (P Am Math Soc, 2016, https://doi.org/10.1090/proc/13766; Anal PDE 11(4):855–871, 2018), we obtain log-type improvements in the case of negative curvature and improved bounds for (M, g) possessing an ergodic geodesic flow

BVR photometry of the resolved dwarf galaxy Ho IX

We present BVR CCD photometry down to limiting magnitude B=23.5 mag for 232 starlike objects and 11 diffuse objects in a 5.4' x 5.4' field of Ho IX. The galaxy is a gas-rich irregular dwarf galaxy possibly very close to M 81, which makes it especially interesting in the context of the evolution of satellite galaxies and the accretion of dwarf galaxies. Investigations of Ho IX were hampered by relatively large contradictions in the magnitude scale between earlier studies. With our new photometry we resolved these discrepancies. The color magnitude diagram (CMD) of Ho IX is fairly typical of a star-forming dwarf irregular, consistent with earlier results. Distance estimates from our new CMD are consistent with Ho IX being very close to M 81 and therefore being a definite member of the M 81 group, apparently in very close physical proximity to M 81.Comment: 9 pages, 8 figures, uses aa.cls, A&A in pres

Some remarks on nodal geometry in the smooth setting

We consider a Laplace eigenfunction $\varphi_\lambda$ on a smooth closed Riemannian manifold, that is, satisfying $-\Delta \varphi_\lambda = \lambda\varphi_\lambda$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $\varphi_\lambda$. This extends previous work of Jakobson and Mangoubi in case $(M, g)$ is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. Ann Math (2) 187(1):241–262, 2018). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $\Omega_\lambda$ on which $\varphi_\lambda$ satisfies exponentially small $L^\infty$ bounds, we give some quantitative estimates for radii of inscribed balls

Some applications of heat flow to Laplace eigenfunctions

We consider mass concentration properties of Laplace eigenfunctions $\varphi_\lambda$, that is, smooth functions satisfying the equation $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$, on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" part and "thin" elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets

Positive solutions for a class of non-autonomous second order difference equations via a new functional fixed point theorem

summary:In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators $T+S$ where $I-T$ is Lipschitz invertible and $S$ a $k$-set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation

Anomalous nucleation far from equilibrium

We present precision Monte Carlo data and analytic arguments for an asymmetric exclusion process, involving two species of particles driven in opposite directions on a $2 \times L$ lattice. We propose a scenario which resolves a stark discrepancy between earlier simulation data, suggesting the existence of an ordered phase, and an analytic conjecture according to which the system should revert to a disordered state in the thermodynamic limit. By analyzing the finite size effects in detail, we argue that the presence of a single, seemingly macroscopic, cluster is an intermediate stage of a complex nucleation process: In smaller systems, this cluster is destabilized while larger systems allow the formation of multiple clusters. Both limits lead to exponential cluster size distributions which are, however, controlled by very different length scales.Comment: 5 pages, 3 figures, one colum

Chemo-dynamical evolution of Globular Cluster Systems

We studied the relation between the ratio of rotational velocity to velocity dispersion and the metallicity (/\sigma_{v}-metallicity relation) of globular cluster systems (GCS) of disk galaxies by comparing the relation predicted from simple chemo-dynamical models for the formation and evolution of disk galaxies with the observed kinematical and chemical properties of their GCSs. We conclude that proto disk galaxies underwent a slow initial collapse that was followed by a rapid contraction and derive that the ratio of the initial collapse time scale to the active star formation time scale is \sim 6 for our Galaxy and \sim 15 for M31. The fundamental formation process of disk galaxies was simulated based on simple chemo-dynamical models assuming the conservation of their angular momentum. We suggest that there is a typical universal pattern in the /\sigma_{v}-metallicity relation of the GCS of disk galaxies. This picture is supported by the observed properties of GCSs in the Galaxy and in M31. This relation would deviate from the universal pattern, however, if large-scale merging events took major role in chemo-dynamical evolution of galaxies and will reflect the epoch of such merging events. We discuss the properties of the GCS of M81 and suggest the presence of past major merging event.Comment: 25 pages, 8 figures, Accepted for publication in the Astrophysical Journa

Structural-acoustic properties of flexible rectangular boxes

Rectangular box-like structures are used widely in a large number of engineering applications, e.g. as elements of railway carriages, heavy goods vehicles, buildings, civil engineering constructions, etc. Although flexible rectangular boxes represent one of the geometrically simple types of engineering structures, their structural-acoustic properties cannot be described by closed-form analytical solutions. In the present study, a comprehensive numerical investigation of typical all-flexible rectangular box structures has been carried out to elucidate the physics of structural-acoustic interaction in them and to explore the possibilities of reduction of the associated structure-borne interior noise. Finite element method has been used to compute the resonant frequencies, the mode shapes and the structural-acoustic frequency response functions of different rectangular box models. The obtained results could assist in better understanding of structural-acoustic properties of flexible rectangular boxes as well as of numerous more complex structures using rectangular boxes as their building elements