We calculate the exact values of the Følner function of the lamplighter group for the standard generating set, as well as the Følner sets that give rise to it. Følner functions encode the isoperimetric properties of amenable groups and have previously been studied up to asymptotic equivalence (that is to say, independently of the choice of finite generating set). What is more, we prove an isoperimetric result concerning the edge boundary on the Baumslag-Solitar group BSp1, 2q for the standard generating set.

We prove an inequality, valid on any finitely generated group with a fixed finite symmetric generating set, involving the growth of successive balls, and the average length of an element in a ball. It generalizes recent improvements of the Coulhon Saloff-Coste inequality. We reformulate the inequality in terms of the Følner function; in the case the finitely generated group is amenable with exponential growth, this allows us to express the best possible (outer) constant in the Coulhon Saloff-Coste isoperimetric inequality with the help of a formula involving the growth rate and the asymptotic behavior of the Følner function.

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on HpZq and its subgroups. The group HpZq is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup H of HpZq, we prove that either H is solvable, or every measure on H with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group F that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich.

Follow-up to the survey report [1], containing in-depth study of this problem, basically, since 1965, in the present report the almost 100-year’s experience of usage of the ultimate plastic strength criterion (U.S.C) in the Russian and Ukrainian practice of design of hulls of ship of different destination is reviewed. On the basis of comparison of results of computational and experimental estimations of an ultimate bending moments (Mult) and characteristics of their variability the expediency of usage of the (Mult) lower boundary by Boobnov-Papkovich-Shimansky method with a more precise estimation of carrying capacity of longitudinals by the generalized method of C. Smith is substantiated. These recommendations may be used also for estimation of the hull ultimate strength of FPSO (Floating Production Storage and Off-loading Units), the structure of which is similar to the hulls of modern tankers with the double bottom and double sides.

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