On a global implicit function theorem for locally Lipschitz maps via nonsmooth critical point theory

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed

Family Abuse Against Minors

This study is a general analysis of data included in the statistics of the General Directorate of Romanian Penitentiaries with respect to violent crimes, sexual crimes and crimes related to deviations from social cohabitation rules in 2007. The study assesses, for this purpose the distribution, according to various indicators (sex, age, original environment, the relationship with the aggressor), of the victims of the abovementioned crimes, and also emphasizes the main risk factors which impact on, or cause, victimization rates

Suicide in Europe in the Context of the Global Economic Crisis

Starting from Durkheim's hypothesis according to which crisis periods cause a significant increase in the number of suicides, the author examines several variables associated with the recent economic crisis and the extent to which they influence suicide rates in Europe, including Romania. In this respect, an assessment is made of the findings of theoretical studies and empirical surveys conducted in several European countries, according to which the increase of unemployment rates, the decline of welfare standards and the deterioration of health conditions (including those related to mental health) are among the main factors associated with the increase of suicide rates in different countries

International and National Legislative Regulations Aiming at Preventing Sexual Abuse against Children, as Well as the Protection of Children

This article presents a short “history” of the evolution over time of internationallegal regulations concerning child rights and the protection of children against any actof physical, emotional or sexual abuse. The concerns of European bodies in this respectare also highlighted, as are the objectives of the legislation in different countries, whichare mainly aimed at sanctioning abuses committed against children and, in particular,the punishment of acts of sexual abuse

On Nash Equilibria in Normal-Form Games With Vectorial Payoffs

We provide an in-depth study of Nash equilibria in multi-objective normal form games (MONFGs), i.e., normal form games with vectorial payoffs. Taking a utility-based approach, we assume that each player's utility can be modelled with a utility function that maps a vector to a scalar utility. In the case of a mixed strategy, it is meaningful to apply such a scalarisation both before calculating the expectation of the payoff vector as well as after. This distinction leads to two optimisation criteria. With the first criterion, players aim to optimise the expected value of their utility function applied to the payoff vectors obtained in the game. With the second criterion, players aim to optimise the utility of expected payoff vectors given a joint strategy. Under this latter criterion, it was shown that Nash equilibria need not exist. Our first contribution is to provide a sufficient condition under which Nash equilibria are guaranteed to exist. Secondly, we show that when Nash equilibria do exist under both criteria, no equilibrium needs to be shared between the two criteria, and even the number of equilibria can differ. Thirdly, we contribute a study of pure strategy Nash equilibria under both criteria. We show that when assuming quasiconvex utility functions for players, the sets of pure strategy Nash equilibria under both optimisation criteria are equivalent. This result is further extended to games in which players adhere to different optimisation criteria. Finally, given these theoretical results, we construct an algorithm to compute all pure strategy Nash equilibria in MONFGs where players have a quasiconvex utility function

Opponent Learning Awareness and Modelling in Multi-Objective Normal Form Games

Many real-world multi-agent interactions consider multiple distinct criteria, i.e. the payoffs are multi-objective in nature. However, the same multi-objective payoff vector may lead to different utilities for each participant. Therefore, it is essential for an agent to learn about the behaviour of other agents in the system. In this work, we present the first study of the effects of such opponent modelling on multi-objective multi-agent interactions with non-linear utilities. Specifically, we consider two-player multi-objective normal form games with non-linear utility functions under the scalarised expected returns optimisation criterion. We contribute novel actor-critic and policy gradient formulations to allow reinforcement learning of mixed strategies in this setting, along with extensions that incorporate opponent policy reconstruction and learning with opponent learning awareness (i.e., learning while considering the impact of one's policy when anticipating the opponent's learning step). Empirical results in five different MONFGs demonstrate that opponent learning awareness and modelling can drastically alter the learning dynamics in this setting. When equilibria are present, opponent modelling can confer significant benefits on agents that implement it. When there are no Nash equilibria, opponent learning awareness and modelling allows agents to still converge to meaningful solutions that approximate equilibria.Comment: Under review since 14 November 202

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.Comment: To appear in DCDS-

A Practical Guide to Multi-Objective Reinforcement Learning and Planning

Real-world decision-making tasks are generally complex, requiring trade-offs between multiple, often conflicting, objectives. Despite this, the majority of research in reinforcement learning and decision-theoretic planning either assumes only a single objective, or that multiple objectives can be adequately handled via a simple linear combination. Such approaches may oversimplify the underlying problem and hence produce suboptimal results. This paper serves as a guide to the application of multi-objective methods to difficult problems, and is aimed at researchers who are already familiar with single-objective reinforcement learning and planning methods who wish to adopt a multi-objective perspective on their research, as well as practitioners who encounter multi-objective decision problems in practice. It identifies the factors that may influence the nature of the desired solution, and illustrates by example how these influence the design of multi-objective decision-making systems for complex problems