The Shadows of a Cycle Cannot All Be Paths

A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).Comment: 6 pages, 10 figure

Routing for analog chip designs at NXP Semiconductors

During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP. This resulted in an heuristic approach, which we presented at the end of the week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach

Rectangle expansion A∗ pathfinding for grid maps

AbstractSearch speed, quality of resulting paths and the cost of pre-processing are the principle evaluation metrics of a pathfinding algorithm. In this paper, a new algorithm for grid-based maps, rectangle expansion A∗ (REA∗), is presented that improves the performance of A∗ significantly. REA∗ explores maps in units of unblocked rectangles. All unnecessary points inside the rectangles are pruned and boundaries of the rectangles (instead of individual points within those boundaries) are used as search nodes. This makes the algorithm plot fewer points and have a much shorter open list than A∗. REA∗ returns jump and grid-optimal path points, but since the line of sight between jump points is protected by the unblocked rectangles, the resulting path of REA∗ is usually better than grid-optimal. The algorithm is entirely online and requires no offline pre-processing. Experimental results for typical benchmark problem sets show that REA∗ can speed up a highly optimized A∗ by an order of magnitude and more while preserving completeness and optimality. This new algorithm is competitive with other highly successful variants of A∗

Mouse Cognition-Related Behavior in the Open-Field: Emergence of Places of Attraction

Spatial memory is often studied in the Morris Water Maze, where the animal's spatial orientation has been shown to be mainly shaped by distal visual cues. Cognition-related behavior has also been described along “well-trodden paths”—spatial habits established by animals in the wild and in captivity reflecting a form of spatial memory. In the present study we combine the study of Open Field behavior with the study of behavior on well-trodden paths, revealing a form of locational memory that appears to correlate with spatial memory. The tracked path of the mouse is used to examine the dynamics of visiting behavior to locations. A visit is defined as either progressing through a location or stopping there, where progressing and stopping are computationally defined. We then estimate the probability of stopping at a location as a function of the number of previous visits to that location, i.e., we measure the effect of visiting history to a location on stopping in it. This can be regarded as an estimate of the familiarity of the mouse with locations. The recently wild-derived inbred strain CZECHII shows the highest effect of visiting history on stopping, C57 inbred mice show a lower effect, and DBA mice show no effect. We employ a rarely used, bottom-to-top computational approach, starting from simple kinematics of movement and gradually building our way up until we end with (emergent) locational memory. The effect of visiting history to a location on stopping in it can be regarded as an estimate of the familiarity of the mouse with locations, implying memory of these locations. We show that the magnitude of this estimate is strain-specific, implying a genetic influence. The dynamics of this process reveal that locations along the mouse's trodden path gradually become places of attraction, where the mouse stops habitually

Qualitative Analysis of POMDPs with Temporal Logic Specifications for Robotics Applications

We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty

Challenges for identifying the neural mechanisms that support spatial navigation: the impact of spatial scale.

Spatial navigation is a fascinating behavior that is essential for our everyday lives. It involves nearly all sensory systems, it requires numerous parallel computations, and it engages multiple memory systems. One of the key problems in this field pertains to the question of reference frames: spatial information such as direction or distance can be coded egocentrically-relative to an observer-or allocentrically-in a reference frame independent of the observer. While many studies have associated striatal and parietal circuits with egocentric coding and entorhinal/hippocampal circuits with allocentric coding, this strict dissociation is not in line with a growing body of experimental data. In this review, we discuss some of the problems that can arise when studying the neural mechanisms that are presumed to support different spatial reference frames. We argue that the scale of space in which a navigation task takes place plays a crucial role in determining the processes that are being recruited. This has important implications, particularly for the inferences that can be made from animal studies in small scale space about the neural mechanisms supporting human spatial navigation in large (environmental) spaces. Furthermore, we argue that many of the commonly used tasks to study spatial navigation and the underlying neuronal mechanisms involve different types of reference frames, which can complicate the interpretation of neurophysiological data