Weak amenability of free products of hyperbolic and amenable groups

We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

Nanoparticle sterilization methods for biomedical applications in animals

Objective: To review different nanoparticle sterilization methods for their use in biomedical applications in animals. Approach: Sterilization is used to obtain a microorganism-free product without altering its physicochemical characteristics during its preparation, storage, or administration route. This review explores different sterilization methods and their advantages and disadvantages on the nanoparticle level. Study limitations/implications: Nanoparticles are used in animal production, including their parenteral administration. Therefore, establishing the characteristics of different technologies applied to sterilize nanoparticles is essential to ensure the delivery of sterile products preventing health risks. Conclusions: The use of nanotechnology in livestock production offers several advantages for animal nutrition, reproduction, and health, among other things. When nanoparticles must be sterilized, choosing the most suitable method is essential. This depends on the amount of product and its compound type because each technique has specific requirements that must be taken into account to be ready for potential changes in the structure and availability of the final product.

Aut-invariant quasimorphisms on free products

Let $$G=A *B$$ G = A ∗ B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

Economic efficiency of national governments: case study of Poland and Ukraine

The article proposes a new unique approach to assessing the economic efficiency of national governments. The assessment is based on the indicator of gross free product per capita, which is a difference between GDP and government size per capita. This method was used to analyze the situation in two post-communist states – Poland and Ukraine. The author studied their economic development in 2009–2019, and the received data was used to draw conclusions about economic policies in the two countries in the period in question. A forecast has been made about the possible impact of the Covid-19 pandemic on economic processes from the perspective of changes in the gross free product per capita.

Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

An Analysis of “Buy X, Get One Free” Reward Programs

Redemption hurdles, such as finite expiration terms and redemption thresholds, are common for customer reward programs. In “An Analysis of ‘Buy X, Get One Free’ Reward Programs,” Yan Liu, Yacheng Sun, and Dan Zhang study the economic rationale behind redemption hurdles and how they should be optimally set. They show analytically that redemption hurdles can be used as a price-discriminating vehicle that increases firm profitability. Redemption hurdles can facilitate the firm’s price discrimination on consumers whose valuations may vary over time. Redemption threshold alone cannot ensure profitability, unless it is coupled with a finite expiration term or a positive transaction utility from the rewarded free product. Optimal design of redemption hurdles is not straightforward, and the interdependence between the two types of redemption hurdles and the price is nontrivial. Optimally set redemption hurdles may not only increase firm profitability but also, increase the welfare of consumers who purchase frequently.