This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation (...) to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman–Hilton duality. (shrink)

We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid’s and Euler’s word choices in their influential developments of geometry and, in particular, their use of the term ‘polyhedron’. Then, jumping to the twentieth century, we look at word choices surrounding the use of the term ‘polyhedron’ in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach between Grünbaum (...) and Shephard on the one hand and that of Hilton and Pedersen on the other, elucidating that the conflict was engendered by disagreement over the proper conceptualization, and so also the appropriate word choices, in the study of polyhedra. (shrink)

The hotel sector of the travel industry is the leader according to the indicator of economic growth, which is observed in both developed and developing countries. Even under the economic instability and global natural disasters, the industry has seen growth in recent years. The franchising model for expanding activities is central to all successful hotel chains. The article deals with the franchising model for the travel business expansion and the economic performance of hotel chains such as Marriott International, Wyndham Hotels, (...) and Hilton. They hold a prominent place in the global hotel business, have a steady tendency to expand their business, hold high positions in the ranking of the best franchises in the world and have been recognized by experts. The main risks for the franchisor and franchisee are determined when making a decision on the expansion of the international travel business, which must be taken into account when developing new markets for hotel chains. The main advantages of the franchising model of expansion, which promote understanding between the parties of the franchise agreement, dynamicize hotel chains expansion and allow for reducing their expenses and increasing incomes. It was found that the growth of net profit and total income in the indicated hotel chains was due to the increase in the number of rooms in franchising and the positive dynamics of franchise income. A more dynamic pace occurred after the crisis growth in revenues from the franchisee compared with the managerial model. A closer relationship between revenues and key financial indicators has been proved when using a franchising model. (shrink)