Forest diagrams for elements of Thompson's group F

We introduce forest diagrams to represent elements of Thompson's group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x_0,x_1} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.Comment: 44 pages, 70 figure

The effect of the electric field on the nucleation of the nanometer periodic structure of adatoms in GaAs semiconductor under the action of laser irradiation

In the paper, the effect of the electric field on the conditions of formation and on the period of the surface superlattice of adatoms in $n$-GaAs semiconductor is investigated. It is established that in GaAs semiconductor, an increase in the electric field strength, depending on the direction, leads to an increase or decrease of the critical temperature (the critical concentration of adatoms), at which the formation of self-organized nanostructure is possible. It is shown that in strongly alloyed $n$-GaAs semiconductor, an increase of the electric field strength leads to a monotonous change (decrease or increase depending on the direction of the electric field) of the period of self-organized surface nanostructures of adatoms.Comment: 9 pages, 4 figures. arXiv admin note: text overlap with arXiv:1512.0780

Growth rates of amenable groups

Let Fm be a free group with m generators and let R be a normal subgroup such that Fm /R projects onto ℤ. We give a lower bound for the growth rate of the group Fm / R′(where R′ is the derived subgroup of R) in terms of the length ρ= ρ(R) of the shortest non-trivial relation in R. It follows that the growth rate of Fm / R′ approaches 2m-1 as ρ approaches infinity. This implies that the growth rate of an m-generated amenable group can be arbitrarily close to the maximum value 2m- 1. This answers an open question of P. de la Harpe. We prove that such groups can be found in the class of abelian-by-nilpotent groups as well as in the class of virtually metabelian group

Strategies and innovations in modern trade marketing

Purpose: The article is devoted to the critical analysis of existing theoretical and methodological approaches to determining the content and nature of sales marketing as a factor and element of companies’ competitive strategy. Design/Methodology/Approach: Based on a critical retrospective analysis, authors presented the features of the trade marketing toolkit providing substantial increase in market sales. Authors considered key existing functional marketing strategies, as well as its innovative trends and ideas. Findings: The modern market of trade enterprises is characterized by fundamental changes in economic conditions caused by the transformation of the economic system, the market priority of consumers, the formation of the information society, and the integration of economic processes. The economic phenomena escalate the struggle in the trading business for its share of the competitive market. In order to maintain and strengthen their positions, trade enterprises should build their trade and technological processes more and more efficiently. Practical implications: Economic growth of a commercial enterprise could be carried out in line with intensive and extensive factors. With the growing level of concentration of the trade industry, factors of intensive economic growth of trade enterprises become relevant. Originality/Value: The author's approach could be used in the development of promising corporate marketing strategies in terms of the development and implementation of innovative marketing solutions available for implementation throughout the entire cycle of the strategic marketing process.peer-reviewe

Elementary amenable subgroups of R. Thompson's group F

The subgroup structure of Thompson's group F is not yet fully understood. The group F is a subgroup of the group PL(I) of orientation preserving, piecewise linear self homeomorphisms of the unit interval and this larger group thus also has a poorly understood subgroup structure. It is reasonable to guess that F is the "only" subgroup of PL(I) that is not elementary amenable. In this paper, we explore the complexity of the elementary amenable subgroups of F in an attempt to understand the boundary between the elementary amenable subgroups and the non-elementary amenable. We construct an example of an elementary amenable subgroup up to class (height) omega squared, where omega is the first infinite ordinal.Comment: 20 page

Metrics on diagram groups and uniform embeddings in a Hilbert space

"Vegeu el resum a l'inici del document del fitxer adjunt"

Interfering Doorway States and Giant Resonances. I: Resonance Spectrum and Multipole Strengths

A phenomenological schematic model of multipole giant resonances (GR) is considered which treats the external interaction via common decay channels on the same footing as the coherent part of the internal residual interaction. The damping due to the coupling to the sea of complicated states is neglected. As a result, the formation of GR is governed by the interplay and competition of two kinds of collectivity, the internal and the external one. The mixing of the doorway components of a GR due to the external interaction influences significantly their multipole strengths, widths and positions in energy. In particular, a narrow resonance state with an appreciable multipole strength is formed when the doorway components strongly overlap.Comment: 20 pages, LaTeX, 3 ps-figures, to appear in PRC (July 1997

Polymerization of Lactic Acid Using Microwave and Conventional Heating

The polymerization of lactic acid (LA) has been investigated by two methods: conventional heating and under microwave irradiation. The reactions of polymerization have been carried out in two stages: at the beginning water is removed and oligomer is obtained; then, the catalysts/co-catalysts are added and reactions are carried out. Tin octoate, toluene sulfonic acid, 2-aminopropanoic acid (alanine) have been investigated as polymerization catalysts and the derivatives of 2,4,6,8- tetramethilol -2,4,6,8- tetraazabicyclo[3.3.0]octane -3,7-dion (Tetraol), comprising atoms of Mg, Zn, Al have been synthesized for the first time. The structure of the synthesized catalyst has been investigated using the method of IR, {1}H NMR. It has been shown that the process of obtaining polylactic acid (PLA) by microwave irradiation proceeds hundreds of times faster. PLA samples synthesized by this method have the same optical characteristics as the PLA obtained by conventional heating