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In this paper, we extend the work by Sato devoted to the development of economic growth models within the framework of the Lie group theory. We propose a new growth model based on the assumption of logistic growth in factors and derive the corresponding production functions, as well as a compatible notion of wage share. In the process, it is shown that the new functions compare reasonably well against relevant economic data. The corresponding problem of maximisation of profit under conditions of perfect competition is solved with the aid of one of these functions. In addition, it is explained in reasonably rigorous mathematical terms why Bowley’s law no longer holds true in the post-1960 data.
Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the
-body problem: periodic motions where the
bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part, we classify all possible symmetry groups of planar
-body collision-free choreographies. These symmetry groups fall into two infinite families and, if
is odd, three exceptional groups. In the second part, we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in
-body systems governed by a strong force potential.
We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.
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