We prove that for two connected sets $E,F\subset \mathbb {R}^2$ with cardinalities greater than $1$ , if one of E and F is compact and not a line segment, then the arithmetic sum $E+F$ has nonempty interior. This improves a recent result of Banakh et al. [‘The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb {R}^n$ ’, Topol. Methods Nonlinear Anal. 54(1)(2019), 247–256] in dimension two by relaxing their assumption that E and F are both compact.

In the spring of 1893, the Austrian writer and critic Hermann Bahr began interviewing various people on antisemitism, a subject of heated discussion in the European feuilleton around 1900. “Once again, I am travelling the world sounding out people’s opinions and listening to what they have to say,” he wrote in his introduction to a series of articles on that issue that appeared in the feuilleton of the Deutsche Zeitung between March and September 1893. A year later, the Berlin publishing house S. Fischer turned Bahr’s articles into a book. Bahr conducted a total of thirty-eight interviews with prominent personages, such as August Bebel, Theodor Mommsen, Ernst Haeckel, Henrik Ibsen and Jules Simon. Bahr did not focus on the arguments in favour or against antisemitism. Instead, he set out explicitly to investigate the sentiments, perceptions and opinions on this topic within the cultured classes. Yet, as I will show in this article, Bahr tried to capture not only the “sentiments” [Empfindungen] aired by his interviewees, but also the settings and interiors in which the interviews took place. I argue that these descriptions of physical space served Bahr as authentication, as a three-dimensional certificate for the “facts of opinion” [Meinungstatsachen] he recorded.

One of the major scientific objectives of ESA's JUICE (JUpiter ICy moons Explorer) mission, which is scheduled for launch in 2022 and planned to arrive at the Jovian system in 2030, is to characterise the internal water ocean and overlying ice shell of Jupiter's largest moon Ganymede. As part of the strategy developed to realise this objective, the tidal response of Ganymede's interior will be constrained by JUICE's measurements of surface displacements (by the Ganymede Laser Altimeter (GALA) instrument) and variations in the gravitational potential (by the 3GM radio science package) due to the acting diurnal tides. Here we calculate the tidal response at the surface of Ganymede for several plausible internal configurations in order to analyse the relation between the tidal response and the geophysical parameters that characterise Ganymede's interior. Similarly to the case of Jupiter's smallest icy satellite Europa, the tidal response of Ganymede in the presence of a subsurface ocean, which could be as large as about 3.5 m in terms of the induced radial deformation, mostly depends on the structural (thickness, density) and rheological (rigidity, viscosity) properties of the ice-I shell. Nevertheless, the dependence of the tidal response on several geophysical parameters of the interior, in particular on the thickness and rigidity of the ice-I shell, does not allow for the unambiguous determination of the shell thickness from tidal measurements alone. Additional constraints could be provided by the measurement of forced longitudinal librations at the surface, as their amplitude is more sensitive to the rigidity than to the thickness of the shell.

The helicity is an important quantity to present the basic topological configuration of magnetic field transferred form the solar subatmosphere into the interplanetary space. In this paper, we present the observational solar magnetic field and the relationship with the magnetic helicity.

The Sun and solar-like stars possess intense and cyclic magnetic activity. In order to understand how this comes about we have developed series of 2-D and 3-D models in order to simulate their global dynamics and magnetism. We here report on our latest findings.

Time-distance helioseismology and data from SOHO/MDI are used for obtaining 3D images of subsurface sound-speed perturbations and maps of plasma flows, associated with emerging magnetic flux and flaring activity of large active regions in October 2003. The results reveal extremely complicated dynamical processes in the upper convection zone and indicate that subsurface shear flows may play an important role in magnetic energy release in solar flares. Strong X-class flares generated impulsive seismic waves (“sunquakes”), traveling through surrounding sunspots, thus providing new insight into the interaction of seismic waves with magnetic fields.