When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in $\ell_1(\mathbb{N}_0)$ is open. By appealing to ultraproduct techniques, we demonstrate that neither in $\ell_1(\mathbb{Z})$ nor in $\ell_1(\mathbb Q)$ convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.Comment: 15 p

Viscous fingering and dendritic growth under an elastic membrane

We investigate the viscous fingering instability that arises when air is injected from the end of an oil-filled, compliant channel. We show that induced axial and transverse depth gradients foster novel pattern formation. Moreover, the steady propagation of the interface allows us to elucidate the nonlinear saturation of a fingering pattern first observed in a time-evolving system (Pihler-Puzovic et al. PRL 108, 074502, 2012): the wavelength is set by the viscous fingering mechanism, but the amplitude is inversely proportional to the tangent of the compliant wall's inclination angle

Serbia: Part of the south zone of the range of distribution of the European grayling (Thymallus thymallus L.)

The grayling (Thymallus thymallus L.) inhabits only a small number of hill-mountain rivers in Serbia. All these rivers belong to the Danube river basin. The Republic of Serbia represents a part of the southern zone of this fish species' range of distribution in Europe. The eastern, southwestern and western zones of the grayling's range of distribution have been determined. During the second half of the 20th century, a decline in the number of its populations has been observed, primarily because of the construction of hydroenergetic objects, intensified fishing and inadequate protection. This can result in a shift in the grayling limits of distribution range in South Europe.Evropski lipljen (Thymallus thymallus L.) naseljava samo mali broj brdsko-planinskih reka u Srbiji. Sve pripadaju slivu Dunava. Srbija je deo južne zone rasprostranjenja ove vrste riba u Evropi. Utvrđene su tri zone rasprostranjenja lipljena: istočna, jugozapadna i zapadna. U drugoj polovini 20-og veka zapaženo je opadanje brojnosti njegovih populacija. Uzrok tome je pre svega izgradnja hidroenergetskih objekata, pojačan izlov i neadekvatna zaštita. Sve to može dovesti do pomeranja južne granice njegovog rasprostranjenja u Evropi.nul