A Catalan word of length n is defined as a word w=w1w2⋯wn over the set of positive integers such that w1=1 and 1≤wj+1≤wj+1 for j=1,2,…,n−1. Catalan words may be represented geometrically as bargraphs. An ℓ-peak (resp. ℓ-valley) is an occurrence of a subword abℓc, where ℓ≥1 and b is greater (resp. less) than both a and c. In this paper, we introduce the notion of order i for peaks and valleys, defined as the difference i=a−c, and study (ℓ,i)-peaks and (ℓ,i)-valleys in Catalan words. We derive generating functions tracking the number of (ℓ,i)-peaks and (ℓ,i)-valleys for arbitrary sets of values of ℓ and i, with respect to word length. The method employs the (standard) bijection from these Catalan words to Dyck paths of semilength n.
Cruise tourism sustainability discourse prioritises vessel level innovation, marginalising destination level social ecological impacts. To address this, we develop a structured literature based conceptual model of cruise mobility as a coupled global and local social ecological system. Supported by AI assisted extraction and synthesis of multidisciplinary literature, this paper constructs a governance-oriented framework. The analysis identifies two interacting dynamics: a reinforcing expansion loop where revenues stimulate infrastructure and deepen dependency, and a degrading acceptance loop where cumulative stress erodes tolerance. Presented as heuristic formalisation devices, these loops help explain why vessel efficiency alone cannot prevent destination level degradation. We translate these findings into a five-pillar governance framework featuring capacity thresholds, social monitoring, economic diversification, power rebalancing, and multi-level coordination. This shifts attention from ship focused compliance towards destination embedded sustainability governance.
The integration of social sustainability clauses into port terminal concession agreements is essential for fostering a more inclusive, balanced, and resilient framework for port operation and development. Such integration addresses the needs of both external and internal stakeholders, namely local communities and society at large, as well as port employees who seek to safeguard their interests. This paper examines how negative externalities can be mitigated or minimized, and how positive externalities can be reinforced, through the inclusion of social sustainability measures and monitoring mechanisms within concession frameworks. Building upon an analytical review of existing literature, available concession agreements and guidelines for port concessions, as well as relevant best practices, the paper identifies and classifies a comprehensive set of social sustainability indicators applicable to the port sector. The results of the analysis are presented through two categories of indicators: external indicators, encompassing stakeholder engagement, community development, and participation of society in economic benefits; and internal indicators, covering employment conditions, training, diversity, occupational health and safety, and the working environment. These indicators, aggregated and structured for the first time within the context of port literature, form a scientific foundation for embedding social sustainability considerations into concession agreements.
In this paper, we introduce the dynamic cumulative residual interval Tsallis entropy (DCRITE) of order a, an information measure that generalizes both cumulative residual Tsallis entropy and interval (doubly truncated) entropies to quantify uncertainty for a lifetime (or loss) known to fall inside (t_1, t_2) . We provide its formal definition and show how DCRITE reduces to known measures. Several equivalent representations and probabilistic interpretations are derived, linking DCRITE to doubly truncated mean residual lifetimes and to normalized versions useful for comparisons across intervals. We also derive analytical bounds and monotonicity results, and characterize distributions by functional relations of DCRITE. Actuarial applications to loss data are provided to demonstrate model fitting and the practical computation of DCRITE for a range of parameter choices. A simulation study investigates the sampling properties of the DCRITE estimator and compares its performance with a competing entropy measure. Finally, we introduce the dynamic cumulative residual interval entropy generating function and present its basic properties and connections with DCRITE.
We investigate the problem of pricing perpetual American put and call options under the assumption that the option can be exercised only at random inspection times which are formulated by a Poisson process with constant intensity \lambda . More specifically, we are interested in the expected payoff of the option, the optimal exercise policy, the probability of exercising, and the distribution of the time until exercise, under the real-world and the risk-neutral measure of the market. The main results are valid when the log-price of the underlying asset follows a Le'\vy jump-diffusion process (with two-sided jumps). Initially we present key identities to express the option's expected payoff in terms of the distribution of the undershoot/overshoot for the put/call scenarios. We further study in more detail the case of pure diffusion (Brownian motion) and the case of Brownian motion with double-exponential jumps by offering explicit formulae for the quantities of interest. By assuming that \lambda \rightarrow \infty we also confirm well-established results that apply to previously studied continuous inspection models. We also include several numerical examples in order to illustrate the impact of the inspection intensity on pricing outcomes for both put and call options.
