Regularity and relaxed problems of minimizing biharmonic maps into spheres

For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).

Removable singularities for p-harmonic maps: The subquadratic case

We prove a removable singularity theorem for p-harmonic maps in the subquadratic case. The theorem states that an isolated singularity of a weakly p-harmonic map is removable if the energy is sufficiently small in a neighbourhood of the singularity.

The development of retinotectal maps: A review of models based on molecular gradients

Information about the world is often represented in the brain in the form of topographic maps. A paradigm example is the topographic representation of the visual world in the optic tectum/superior colliculus. This map initially forms during neural development using activity-independent molecular cues, most notably some type of chemospecific matching between molecular gradients in the retina and corresponding gradients in the tectum/superior colliculus. Exactly how this process might work has been studied both experimentally and theoretically for several decades. This review discusses the experimental data briefly, and then in more detail the theoretical models proposed. The principal conclusions are that (1) theoretical models have helped clarify several important ideas in the field, (2) earlier models were often more sophisticated than more recent models, and (3) substantial revisions to current modelling approaches are probably required to account for more than isolated subsets of the experimental data.

Brain maps, great and small: lessons from comparative studies of primate visual cortical organization

In this paper, we review evidence from comparative studies of primate cortical organization, highlighting recent findings and hypotheses that may help us to understand the rules governing evolutionary changes of the cortical map and the process of formation of areas during development. We argue that clear unequivocal views of cortical areas and their homologies are more likely to emerge for 'core' fields, including the primary sensory areas, which are specified early in development by precise molecular identification steps. In primates, the middle temporal area is probably one of these primordial cortical fields. Areas that form at progressively later stages of development correspond to progressively more recent evolutionary events, their development being less firmly anchored in molecular specification. The certainty with which areal boundaries can be delimited, and likely homologies can be assigned, becomes increasingly blurred in parallel with this evolutionary/developmental sequence. For example, while current concepts for the definition of cortical areas have been vindicated in allowing a clarification of the organization of the New World monkey 'third tier' visual cortex (the third and dorsomedial areas, V3 and DM), our analyses suggest that more flexible mapping criteria may be needed to unravel the organization of higher-order visual association and polysensory areas.

Combining influence maps and cellular automata for reactive game agents

Agents make up an important part of game worlds, ranging from the characters and monsters that live in the world to the armies that the player controls. Despite their importance, agents in current games rarely display an awareness of their environment or react appropriately, which severely detracts from the believability of the game. Some games have included agents with a basic awareness of other agents, but they are still unaware of important game events or environmental conditions. This paper presents an agent design we have developed, which combines cellular automata for environmental modeling with influence maps for agent decision-making. The agents were implemented into a 3D game environment we have developed, the EmerGEnT system, and tuned through three experiments. The result is simple, flexible game agents that are able to respond to natural phenomena (e.g. rain or fire), while pursuing a goal.

Dynamics and topographic organization of recursive self-organizing maps

Recently there has been an outburst of interest in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, there is no general consensus as to how best to process sequences using topographicmaps, and this topic remains an active focus of neurocomputational research. The representational capabilities and internal representations of the models are not well understood. Here, we rigorously analyze a generalization of the self-organizingmap (SOM) for processing sequential data, recursive SOM (RecSOM) (Voegtlin, 2002), as a nonautonomous dynamical system consisting of a set of fixed input maps. We argue that contractive fixed-input maps are likely to produce Markovian organizations of receptive fields on the RecSOM map. We derive bounds on parameter β (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed-input maps is guaranteed. Some generalizations of SOM contain a dynamic module responsible for processing temporal contexts as an integral part of the model. We show that Markovian topographic maps of sequential data can be produced using a simple fixed (nonadaptable) dynamic module externally feeding a standard topographic model designed to process static vectorial data of fixed dimensionality (e.g., SOM). However, by allowing trainable feedback connections, one can obtain Markovian maps with superior memory depth and topography preservation. We elaborate on the importance of non-Markovian organizations in topographic maps of sequential data. © 2006 Massachusetts Institute of Technology.

"Magic" dispersion maps with distributed Raman amplification

We analyze pulse propagation in an optical fiber with a periodic dispersion map and distributed amplification. Using an asymptotic theory and a momentum method, we identify a family of dispersion management schemes that are advantageous for massive multichannel soliton transmission. For the case of two-step dispersion maps with distributed Raman amplification to compensate for the fiber loss, we find special schemes that have optimal (chirp-free) launch point locations that are independent of the fiber dispersion. Despite the variation of dispersion with wavelength due to the fiber dispersion slope, the transmission in several different channels can be optimized simultaneously using the same optimal launch point. The theoretical predictions are verified by direct numerical simulations. The obtained results are applied to a practical multichannel transmission system.