Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C: K--> 2(Y) a point-to-set mapping such that for any x is an element of K, C(x) is a pointed, closed, and convex cone in Y and int C(x) not equal 0. Given a mapping g : K --> K and a vector valued bifunction f : K x K - Y, we consider the implicit vector equilibrium problem (IVEP) of finding x* is an element of K such that f (g(x*), y) is not an element of - int C(x) for all y is an element of K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. (C) 2003 Elsevier Science Ltd. All rights reserved.

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

Dynamic equilibrium through reinforcement learning

Reinforcement Learning is an area of Machine Learning that deals with how an agent should take actions in an environment such as to maximize the notion of accumulated reward. This type of learning is inspired by the way humans learn and has led to the creation of various algorithms for reinforcement learning. These algorithms focus on the way in which an agent’s behaviour can be improved, assuming independence as to their surroundings. The current work studies the application of reinforcement learning methods to solve the inverted pendulum problem. The importance of the variability of the environment (factors that are external to the agent) on the execution of reinforcement learning agents is studied by using a model that seeks to obtain equilibrium (stability) through dynamism – a Cart-Pole system or inverted pendulum. We sought to improve the behaviour of the autonomous agents by changing the information passed to them, while maintaining the agent’s internal parameters constant (learning rate, discount factors, decay rate, etc.), instead of the classical approach of tuning the agent’s internal parameters. The influence of changes on the state set and the action set on an agent’s capability to solve the Cart-pole problem was studied. We have studied typical behaviour of reinforcement learning agents applied to the classic BOXES model and a new form of characterizing the environment was proposed using the notion of convergence towards a reference value. We demonstrate the gain in performance of this new method applied to a Q-Learning agent.

Basic dyestuffs removal from textile effluents using feathers: equilibrium, kinetic and column studies

This research work has been focused in the study of gallinaceous feathers, a waste that may be valorised as sorbent, to remove the Dark Blue Astrazon 2RN (DBA) from Dystar. This study was focused on the following aspects: optimization of experimental conditions through factorial design methodology, kinetic studies into a continuous stirred tank adsorber (at pH 7 and 20ºC), equilibrium isotherms (at pH 5, 7 and 9 at 20 and 45ºC) and column studies (at 20ºC, at pH 5, 7 and 9). In order to evaluate the influence of the presence of other components in the sorption of the dyestuff, all experiments were performed both for the dyestuff in aqueous solution and in real textile effluent. The pseudo-first and pseudo-second order kinetic models were fitted to the experimental data, being the latter the best fit for the aqueous solution of dyestuff. For the real effluent both models fit the experimental results and there is no statistical difference between them. The Central Composite Design (CCD) was used to evaluate the effects of temperature (15 - 45ºC) and pH (5 - 9) over the sorption in aqueous solution. The influence of pH was more significant than temperature. The optimal conditions selected were 45ºC and pH 9. Both Langmuir and Freundlich models could fit the equilibrium data. In the concentration range studied, the highest sorbent capacity was obtained for the optimal conditions in aqueous solution, which corresponds to a maximum capacity of 47± 4 mg g-1. The Yoon-Nelson, Thomas and Yan’s models fitted well the column experimental data. The highest breakthrough time for 50% removal, 170 min, was obtained at pH 9 in aqueous solution. The presence of the dyeing agents in the real wastewater decreased the sorption of the dyestuff mostly for pH 9, which is the optimal pH. The effect of pH is less pronounced in the real effluent than in aqueous solution. This work shows that feathers can be used as sorbent in the treatment of textile wastewaters containing DBA.

Copper and lead removal by peanut hulls: equilibrium and kinetic studies

This research work aims to study the use of peanut hulls, an agricultural and food industry waste, for copper and lead removal through equilibrium and kinetic parameters evaluation. Equilibrium batch studies were performed in a batch adsorber. The influence of initial pH was evaluated (3–5) and it was selected between 4.0 and 4.5. The maximum sorption capacities obtained for the Langmuir model were 0.21 ± 0.03 and 0.18 ± 0.02 mmol/g, respectively for copper and lead. In bi-component systems, competitive sorption of copper and lead was verified, the total amount adsorbed being around 0.21 mmol of metal per gram of material in both mono and bi-component systems. In the kinetic studies equilibrium was reached after 200 min contact time using a 400 rpm stirring rate, achieving 78% and 58% removal, in mono-component system, for copper and lead respectively. Their removal follows a pseudo-second-order kinetics. These studies show that most of the metals removal occurred in the first 20 min of contact, which shows a good uptake rate in all systems.

Equilibrium distributions of discrete non-autonomous graphs

We introduce the notions of equilibrium distribution and time of convergence in discrete non-autonomous graphs. Under some conditions we give an estimate to the convergence time to the equilibrium distribution using the second largest eigenvalue of some matrices associated with the system.

A physical packing sequence algorithm for the container loading problem with static mechanical equilibrium conditions

The container loading problem (CLP) is a combinatorial optimization problem for the spatial arrangement of cargo inside containers so as to maximize the usage of space. The algorithms for this problem are of limited practical applicability if real-world constraints are not considered, one of the most important of which is deemed to be stability. This paper addresses static stability, as opposed to dynamic stability, looking at the stability of the cargo during container loading. This paper proposes two algorithms. The first is a static stability algorithm based on static mechanical equilibrium conditions that can be used as a stability evaluation function embedded in CLP algorithms (e.g. constructive heuristics, metaheuristics). The second proposed algorithm is a physical packing sequence algorithm that, given a container loading arrangement, generates the actual sequence by which each box is placed inside the container, considering static stability and loading operation efficiency constraints.

Separating sequential equilibrium in a Bayesian international duopoly

We consider two Cournot firms, one located in the home country and the other in the foreign country, producing substitute goods for consumption in a third country. We suppose that neither the home government nor the foreign firm know the costs of the home firm, while the foreign firm cost is common knowledge. We determine the separating sequential equilibrium outputs.

Collateral, default penalties and infinite horizon equilibrium

Pascoa and Seghir (2009) noticed that when collateralized promises become subject to utility penalties on default, Ponzi schemes may occur. However, equilibrium exists in some interesting cases. Under low penalties, equilibrium exists if the collateral does not yield utility (for example, when it is a productive asset or a security). Equilibrium exists also under more severe penalties and collateral utility gains, when the promise or the collateral are nominal assets and the margin requirements are endogenous: relative inflation rates and margin coefficients can make the income effects dominate the penalty effects. An equilibrium refinement avoids no-trade equilibria with unduly repayment beliefs. Our refinement differs from the one used by Dubey, Geanakoplos and Shubik (2005) as it does not eliminate no trade equilibria whose low delivery rates are consistent with the propensity to default of agents that are on the verge of selling.

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin (1986) and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou (1993) and Reny (1999). Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.

Corporate finance in general equilibrium models with incomplete markets: A survey

A Masters Thesis, presented as part of the requirements for the award of a Research Masters Degree in Economics from NOVA – School of Business and Economics

Three essays on game theory and bargaining

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

Experimental Gravimetric Adsorption Equilibrium of n-Alkanes and Alkenes, Carbon Dioxide and Nitrogen in MIL-53(Al) and Zeolite 5A

Dissertação para obtenção do Grau de Mestre em Engenharia Química e Bioquímica