Relative CC"-Numerical Ranges for Applications in Quantum Control and Quantum Information

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.Comment: accompanying paper to math-ph/070103

The Problem of Signal and Symbol Integration: A Study of Cooperative Mobile Autonomous Agent Behaviors

This paper explores and reasons about the interplay between symbolic and continuous representations. We first provide some historical perspective on signal and symbol integration as viewed by the Artificial Intelligence (AI), Robotics and Computer Vision communities. The domain of autonomous robotic agents residing in dynamically changing environments anchors well different aspects of this integration and allows us to look at the problem in its entirety. Models of reasoning, sensing and control actions of such agents determine three different dimensions for discretization of the agent-world behavioral state space. The design and modeling of robotic agents, where these three aspects have to be closely tied together, provide a good experimental platform for addressing the signal-to-symbol transformation problem. We present some experimental results from the domain of cooperating mobile agents involved in tasks of navigation and manipulation

Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

The effects of migrant remittances on population–environment dynamics in migrant origin areas: international migration, fertility, and consumption in highland Guatemala

International migration impacts origin regions in many ways. As examples, remittances from distant migrants may alter consumption patterns within sending communities, while exposure to different cultural norms may alter other behaviors. This paper combines these insights to offer a unique lens on migration’s environmental impact. From an environmental perspective, we ask the following question: is the likely rise in consumption brought about by remittances counterbalanced by a reduction in fertility in migrant households following exposure to lower fertility cultures? Based on ethnographic case studies in two western highland Guatemalan communities, we argue that the near-term rise in consumption due to remittances is not counterbalanced by rapid decline in migrant household fertility. However, over time, the environmental cost of consumption may be mitigated at the community level through diffusion of contraception and family planning norms yielding lower family size