Asymptotic boundary forms for tight Gabor frames and lattice localization domains

We consider Gabor localization operators $G_{\phi,\Omega}$ defined by two parameters, the generating function $\phi$ of a tight Gabor frame $\{\phi_\lambda\}_{\lambda \in \Lambda}$, parametrized by the elements of a given lattice $\Lambda \subset \Bbb{R}^2$, i.e. a discrete cocompact subgroup of $\Bbb{R}^2$, and a lattice localization domain $\Omega \subset \Bbb{R}^2$ with its boundary consisting of line segments connecting points of $\Lambda$. We find an explicit formula for the boundary form $BF(\phi,\Omega)=\text{A}_\Lambda \lim_{R\rightarrow \infty}\frac{PF(G_{\phi,R\Omega})}{R}$, the normalized limit of the projection functional $PF(G_{\phi,\Omega})=\sum_{i=0}^{\infty}\lambda_i(G_{\phi,\Omega})(1-\lambda_i(G_{\phi,\Omega}))$, where $\lambda_i(G_{\phi,\Omega})$ are the eigenvalues of the localization operators $G_{\phi,\Omega}$ applied to dilated domains $R\Omega$, $R$ is an integer and $\text{A}_\Lambda$ is the area of the fundamental domain of the lattice $\Lambda$.Comment: 35 page

A Resource-Constrained Optimal Control Model for Crackdown on Illicit Drug Markets

In this paper we present a budget-constrained optimal control model aimed at finding the optimal enforcement profile for a street-level, illicit drug crackdown operation. The objective is defined as minimizing the number of dealers dealing at the end of the crackdown operation, using this as a surrogate measure of residual criminal activity. Analytical results show that optimal enforcement policy will invariably use the budget resources completely. Numerical analysis using realistic estimates of parameters shows that crackdowns normally lead to significant results within a matter of a week, and if they do not, it is likely that they will be offering very limited success even if pursued for a much longer duration. We also show that a ramp-up enforcement policy will be most effective in collapsing a drug market if the drug dealers are risk-seeking, and the policy of using maximum enforcement as early as possible is usually optimal in the case when the dealers are risk averse or risk neutral. The work then goes on to argue that the underlying model has some general characteristics that are both reasonable and intuitive, allowing possible applications in focussed, local enforcement operations on other similar illegal activities.crackdown enforcement;illicit drug markets;optimal control

Dynamics of Drug Consumption: a Theoretical Model

A continuous time model is proposed to describe the dynamics of drug consumption in a given country. The model has two state variables, addicts and dealers, and eleven parameters, including the total effort exerted by the State, which is considered as control parameter. The model is highly nonlinear and the analysis shows that it is characterized by a transcritical and a fold bifurcation. This implies that for intermediate values of the State's effort the model has two stable equilibria, one trivial, corresponding to the absence of drugs, and one positive, corresponding to drug consumption. On the contrary, for low and high values of the effort only one of the two equilibria is stable. This suggests a two-step control policy. First, exert a very high effort for a few years, so that the system has the time to approach the trivial equilibrium, and then reduce the effort but maintain it sufficiently high so that drug consumption cannot rise anymore. Interesting results on the role played by the price of the drug and the severity of the punishment inflicted to dealers, as well as on the allocation of the effort between therapy and police, have also been obtained

A Deformation Quantization Theory for Non-Commutative Quantum Mechanics

We show that the deformation quantization of non-commutative quantum mechanics previously considered by Dias and Prata can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined, and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef.Comment: Submitted for publicatio

Complex Dynamics in Romantic Relationships

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and in synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as couples composed of nonsynergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for complex love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercitical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov-Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partner's love and appeal. It also explains why aging has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper

Optimal language policy for the preservation of a minority language

We develop a dynamic language competition model with dynamic state intervention. Parents choose the language(s) to raise their children based on the communicational value of each language as well as on their emotional attachment to the languages at hand. Languages are thus conceptualized as tools for communication as well as carriers of cultural identity. The model includes a high and a low status language, and children can be brought up as monolinguals or bilinguals. Through investment into language policies, the status of the minority language can be increased. The aim of the intervention is to preserve the minority language in a bilingual subpopulation at low costs. We investigate the dynamic structure of the optimally controlled system as well as the optimal policy, identify stable equilibria and provide numerical case studies

On the Matthew effect in research careers: Abnormality on the boundary

The observation that a socioeconomic agent with a high reputation gets a disproportionately higher recognition for the same work than an agent with lower reputation is typical in career development and wealth. This phenomenon, which is known as Matthew effect in the literature, leads to an increasing inequality over time. The present paper employs an optimal control model to study the implications of the Matthew effect on the optimal efforts of a scientist into reputation. The solution of the model exhibits, for suffiently low effort costs, a new type of unstable equilibrium at which effort is at its upper bound. This equilibrium, which we denote as Stalling Equilibrium, serves as a threshold level separating success and failure in academia. In addition we show that at the Stalling Equilibrium the solution can be abnormal. We provide a clear economic interpretation for this solution characteristic