Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question. In this study, a first important step forward was performed in this direction; in fact, a differential equation was worked out which describes how entropy varies as a function of total energy, and this variation is driven by the total energy dependence of a topology-related quantity of the relevant submanifolds of the phase space. Hence, general conditions can be in principle defined for topology-driven loss of differentiability of the entropy.

A Bioeconomic Analysis of a Renewable Resource in the Presence of Illegal, Unreported and Unregulated Fishing

The issue of illegal, unreported and unregulated (IUU) fishing is of prime concern to fisheries in developing countries where the regulatory regimes are often weak. This study proposes a Gordon-Schaefer bioeconomic model with non-constant catchability and nonlinear cost to study the impact of IUU fishing on the stock size of a marine fishery in Ghana. The static equilibrium reference points of the model are established and discussed. Bifurcation analysis on the modified Schaefer model shows the existence of a transcritical bifurcation point were the model is structurally unstable. Pontryagin’s maximum principle is employed to investigate the necessary conditions of the model, and also established are the sufficiency conditions that guarantee the existence and uniqueness of the optimality system. The characterization of the optimal control gives rise to both the boundary and interior solutions, with the former indicating that the resource should be harvested if and only if the marginal revenue of harvest exceeds the marginal revenue of stock. Numerical simulations with empirical data on the Ghana sardinella are carried out to validate the theoretical results. It is shown that IUU fishing leads to excessive exploitation of the resource biomass to levels below 50% of the carrying capacity. This has the tendency of making the fishery unsustainable, with its concomitant loss of revenue to fishers.

Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

CHARACTERIZATION OF DISKCYCLIC OPERATORS

In this article we discuss an interesting idea related to characte rization of diskcyclicity phenomena. We use the Diskcyclic Crit erion to characterize the subsets Γ of ℂ for which the notion of Γ-supe rcyclicity coincides with the notion of diskcyclicity. In particular , we give a complete description of the subset Γ. The necessi ty and sufficiency conditions on the subset Γ were prese nted. Some examples were exhibited to illustrate the proofs and the relationship among different classes of the operators.

Jordan Form-Based Algebraic Conditions for Controllability of Multiagent Systems under Directed Graphs

Based on the Jordan form of system matrix, this paper discusses algebraic conditions for the controllability of the multiagent network system with directed graph from two aspects: leader-follower network attribute and coupling input disturbance. Leader-follower network attribute refers to the topology structure and information communication among agents. Coupling input disturbance includes the number of external coupling inputs and the selection of leader nodes. When the leader-follower network attribute is fixed, the selection method of coupling input disturbance is studied for the controllability, and when the coupling input disturbance is known, we derive necessity and sufficiency conditions to determine the controllability. The reliability of theoretical results is verified by numerical examples and model simulation. Besides, the generally perfect controllability is introduced, that is, the system is always controllable regardless of the number and the locations of leaders. In practical engineering applications, the perfectly controllable topology can improve the system fault tolerance and accelerate the commercialization process, which has a profound significance for promoting the modernization process.