We show that for any given n, there exists a sequence of words {ak}k≥1 in the generators σ1, . . . , σn−1 of the braid group Bn, representing the identity element of Bn, such that the number of braid relations of the form σiσi+1σi = σi+1σiσi+1 needed to pass from ak to the empty word is quadratic with respect to the length of ak.

If, for a morphism σ : ∆∗ → Σ∗ (where ∆ and Σ are arbitrary alphabets) and a word α ∈ ∆∗, there exists another morphism τ mapping α to σ(α), then σ is called ambiguous with respect to α ; if such a τ does not exist, then σ is unambiguous. For example, the morphism σ0 : {A,B,C}∗ → {a,b}∗ – given by σ0(A) := a, σ0(B) := a, σ0(C) := b – is ambiguous with respect to the word α0 := ABCACB, since the...

Let x, y be sets. Observe that {〈x, y〉} is one-to-one. In the sequel n denotes a natural number. One can prove the following two propositions: (1) For every non empty topological space T holds T and T ΩT are homeomorphic. (2) Let X be a non empty subspace of En T and f be a function from X into R1. Suppose f is continuous. Then there exists a function g from X into En T such that (i) for every ...

A finite non-empty word z is said to be a border of a finite non-empty word w if w = uz = zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form uvu, where u and v are finite words, in terms of its unbordered factors. The...

This paper considers the equilibrium balking behavior of customers in a single-server Markovian queue with variable vacation and interruption, where server can switch across four states: vacation, working idle period, busy period. Once becomes empty, commences slows down its service rate. However, this period may be interrupted anytime by interruption. Upon completion takes probability-based ma...

In this paper we look into the management of empty maritime containers by making use of reverse logistics theory. We map the reverse logistics system, identifying the drivers for recovery, the characteristics of the containers, why they are being returned, how they are being recovered and by whom. We identify that empty maritime container management has to address mainly the management of the c...

The glucan-rich fraction, hemicellulosic compounds-rich and a mixture of both fractions obtained from organosolv pretreatment oil palm empty fruit bunch (OPEFB) were used as substrates to produce volatile fatty acids (VFAs) in acidogenic fermentation. In this study, the effects medium adjustment (carbon nitrogen ratio trace elements supplementation) methanogenesis inhibition (through addition 2...