The Tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point $\mathfrak{sl}_2$ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra $\boxtimes$ by generators and relations and displayed an isomorphism from $\boxtimes$ to the three-point $\mathfrak{sl}_2$ loop algebra. In this paper we classify the finite-dimensional irreducible $\boxtimes$-modules.Comment: 18 page

Statistical predictions for the first black holes

The recent observations of supermassive black holes (SMBHs) at high redshift challenge our understanding of their formation and growth. There are different proposed pathways to form black hole (BH) seeds, such as the remnants of the first stars (chapter 4), gas-dynamical processes (chapter 5), direct collapse (chapter 6), or stellar collisions in dense nuclear clusters (chapter 7). In this chapter, we discuss the probability of forming supermassive black holes (SMBHs) via these channels and determine the expected number density of the BH seeds. We start with a brief discussion of the observational constraints on SMBHs at low and high redshift that theoretical models have to reproduce (a more detailed account is provided in chapter 12). We further present the most popular formation channels of SMBHs, discuss under which conditions they can reproduce the observations, and compare various estimates in the literature on the expected number density of SMBHs. To account for the density of quasars at $z>6$ requires very efficient gas accretion mechanisms or high BH seeds masses. The bottleneck to obtain sufficiently high number densities of seed BHs with masses $>10^5$M$_\odot$ is the interplay between radiative and chemical feedback, which constrains the conditions for primordial, isothermal gas collapse.Comment: Preprint of the chapter "Statistical predictions for the first black holes", to be published in the review volume "Formation of the First Black Holes", Latif, M. and Schleicher, D. R. G., eds., World Scientific Publishing Company, 2018, pp 161-175 [see https://www.worldscientific.com/worldscibooks/10.1142/10652

Three Mutually Adjacent Leonard Pairs

Let (A,B) and (C,D) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for (A,B) (resp. (C,D)) is split for (C,D) (resp. (A,B)). Our main results are as follows: Theorem 1. There exists at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2. Theorem 2. Let (A,B), (C,D), and (E,F) denote three mutually adjacent Leonard pairs on V. There for each of these pairs, the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Theorem 3. Let (A,B) denote a Leonard pair on V whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Then there exist Leonard pairs (C,D) and (E,F) on V such that (A,B), (C,D), and (E,F) are mutually adjacent.Comment: 19 pages. To be published in Linear Algebra and it Application

[Review of] Werner Sollors, cd. The Invention of Ethnicity

The Invention of Ethnicity is obviously and admittedly shaped after Hobsbawm\u27s and Ranger\u27s Invention of Tradition, published in 1986, when the essays in this volume were finished. In the meantime, much of what Sollors argues for in his editor\u27s introduction has become accepted knowledge under the general heading of the constructedness of identity and subjectivity. If, however, the book has not, as might appear probable at first sight, been overtaken and made obsolete by the very success of the views it advocates -- if the reverse, rather, seems to be true and to justify a belated discussion in these pages -- this is so because it also affords one an opportunity to reopen questions that may have been prematurely closed

Maximum evaporation rates for nonisothermal droplets

Calculations have been carried out in order to determine the rate of evaporation of a liquid droplet surrounded by hot gases. The present study represents an extension of earlier work by Penner on evaporation rates for isothermal droplets. In particular, allowance was made for temperature gradients within the droplet by considering a droplet composed of an isothermal core and an isothermal shell. The results obtained in the present investigation were found to be in satisfactory agreement with the known data for evaporation of isothermal droplets, thus justifying calculations for isothermal droplets as a reasonable first approximation

SOCIAL RESPONSIBILITY: SHOULDTHE NATIONS CARE ABOUT IT?

In the article the global trends in the social responsibility on the national level were presented. The analysis of the global trends was made on the basis of comparing the national level of social responsibility and components of the economic freedom and social progress. It was detected, thatGDP per capita as basic indicator of economic wealth is strongly correlated to level of nation’s socialresponsibility. The influence of corruption on the social development was defined also. The basic direction for further development of the social responsibility in context of providing the sustainable development such as integrated matrixes of distribution of social responsibility were proposed