Lie theory for asymptotic symmetries in general relativity: The BMS Group

We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman--Unti group is highlighted, which will be studied in future work.

$$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups

We define and investigate $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$ M p , q s , α ( R n ) that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$ α = 0 ) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$ α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$ α = 0 , 1 where the spaces $$M_{p,q}^{s}(G)$$ M p , q s ( G ) and $${\mathcal {B}}_{p,q}^{s}(G)$$ B p , q s ( G ) have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$ Q ( G ) underlying the $$\alpha = 0$$ α = 0 case $$M_{p,q}^{s}(G)$$ M p , q s ( G ) allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$ F : M p , q s ( R k ) ⟶ M p , q s ( G ) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.

Symmetries in Generalized Kaprekar’s Routine

In this paper, algebraic relations were established that determined the invariance of a transformed number after several transformations. The restrictions that determine the group structure of these relationships were analyzed, as was the case of the Klein group. Parametric Kr functions associated with the existence of cycles were presented, as well as the role of the number of their links in the grouping of numbers in higher-order equivalence classes. For this, we developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

In this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

Molecular Structure and Hybridization Patterns of Abramis brama × Rutilus rutilus Hybrids from Modrac Reservoir, Bosnia and Herzegovina

Interspecific hybridization in the Cyprinidae family has been recorded worldwide, with Abramis brama (bream) and Rutilus rutilus (roach) as one of the often-reported hybridizing pairs. The only account of such an event in Bosnia and Herzegovina has been in Modrac Reservoir. Using morphological and molecular markers, the presence of hybrids was surveyed, the hybridization direction was determined and the hybrid group structure in this ecosystem was evaluated. Our findings confirmed unhindered natural hybridization between roach and bream in Modrac Reservoir. Over 50% of the hybrid specimens were classified as F2 hybrids by the NewHybrids software, while the rest were categorized as pure parental form, making it the first such finding in Europe. The analysis of mitochondrial cytochrome b showed that 90% of hybrid individuals were of bream maternal origin. The hybrid group expressed higher mean values of observed heterozygosity and gene diversity than both parental species. Signs of introgressive hybridization between parental species were detected. The hybrid zone of Modrac Reservoir appears to follow the intermediate or “flat” hybrid model based on the balanced distribution of parental and hybrid genotypes. Further investigation is needed to elucidate the factors that enable the survival and mating success of post-F1 individuals.

Arbitrage spaces in the offshore world: Layering, ‘fuses’ and partitioning of the legal structure of modern firms

In this article, we discuss the way offshore financial centres are used by the multi-subsidiary, multi-jurisdictional group structure known as the ‘multinational enterprise’ to arbitrage between social geographies of political jurisdictions. We define arbitrage as the use of corporate legal entities located in diverse jurisdictions to arbitrate a third country's rules and regulations. Using a new method to categorize firm-level data from Van Dijk’s Orbis, we operationalize the notion of arbitrage to systematically detail and compare the structural sequencing choices firms are making, likely in part for reasons of arbitrage. We base our techniques on legal theory of the firm, acknowledging the underpinning of social technologies of law and accounting by which business enterprises are constructed and maintained. We conclude that two specific types of entities, ‘standalones’ versus ‘in-betweeners’, are qualitatively different from others in the activities they perform. We also highlight the existence of liability structures, or ‘fuses’, which typically take the form of a split ownership arrangement. Ultimately, we demonstrate that the position of a firm’s subsidiary within the overall network ecology of that firm is as important as its jurisdictional registration location.

Cohesion in male singing behavior predicts group reproductive output in a social songbird

All social groups require organization to function optimally. Group organization is often shaped by social 'rules', which function to manage conflict, discourage cheating, or promote cooperation. If social rules promote effective social living, then the ability to learn and follow these rules may be expected to influence individual and group-level fitness. However, such links can rarely be tested, due to the complexity of the factors mediating social systems and the difficulty of gathering data across multiple groups. Songbirds offer an opportunity to investigate the link between social rules and reproductive output because most of their social interactions are mediated by song, a well-studied and readily quantifiable behavior. Using observations from 19 groups of brown-headed cowbirds (Molothrus ater) studied across 15 years, we find evidence for a previously undocumented social rule: cohesive group transitions between dominance- and courtship-related singing. Comparing across groups, the degree of cohesion in male singing behavior predicts the reproductive output of their group. Experimental manipulation of group structure via the introduction of juvenile males to captive flocks reduced group cohesion and adult male reproductive success. Taken together, these results demonstrate that cohesion in group behavioral states can affect both individual and group-level reproductive success, suggesting that selection can act not only on individual-level traits, but also on an individual's ability and opportunity to participate effectively in organized social interactions. Social cohesion could therefore be an unappreciated force affecting social evolution in many diverse systems.