On Minimal Trajectories for Mobile Sampling of Bandlimited Fields

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focussing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.Comment: 28 pages, 8 figure

Gravitational dynamics in s+1+1 dimensions II. Hamiltonian theory

We develop a Hamiltonian formalism of brane-world gravity, which singles out two preferred, mutually orthogonal directions. One is a unit twist-free field of spatial vectors with integral lines intersecting perpendicularly the brane. The other is a temporal vector field with respect to which we perform the Arnowitt-Deser-Misner decomposition of the Einstein-Hilbert Lagrangian. The gravitational variables arise from the projections of the spatial metric and their canonically conjugated momenta as tensorial, vectorial and scalar quantities defined on the family of hypersurfaces containing the brane. They represent the gravitons, a gravi-photon and a gravi-scalar, respectively. From the action we derive the canonical evolution equations and the constraints for these gravitational degrees of freedom both on the brane and outside it. By integrating across the brane, the dynamics also generates the tensorial and scalar projection of the Lanczos equation. The vectorial projection of the Lanczos equation arises in a similar way from the diffeomorphism constraint. Both the graviton and the gravi-scalar are continuous across the brane, however the momentum of the gravi-vector has a jump, related to the energy transport (heat flow) on the brane.Comment: 13 page

Network and psychological effects in urban movement

Correlations are regularly found in space syntax studies between graph-based configurational measures of street networks, represented as lines, and observed movement patterns. This suggests that topological and geometric complexity are critically involved in how people navigate urban grids. This has caused difficulties with orthodox urban modelling, since it has always been assumed that insofar as spatial factors play a role in navigation, it will be on the basis of metric distance. In spite of much experimental evidence from cognitive science that geometric and topological factors are involved in navigation, and that metric distance is unlikely to be the best criterion for navigational choices, the matter has not been convincingly resolved since no method has existed for extracting cognitive information from aggregate flows. Within the space syntax literature it has also remained unclear how far the correlations that are found with syntactic variables at the level of aggregate flows are due to cognitive factors operating at the level of individual movers, or they are simply mathematically probable network effects, that is emergent statistical effects from the structure of line networks, independent of the psychology of navigational choices. Here we suggest how both problems can be resolved, by showing three things: first, how cognitive inferences can be made from aggregate urban flow data and distinguished from network effects; second by showing that urban movement, both vehicular and pedestrian, are shaped far more by the geometrical and topological properties of the grid than by its metric properties; and third by demonstrating that the influence of these factors on movement is a cognitive, not network, effect

A Qualitative Representation of Spatial Scenes in R2 with Regions and Lines

Regions and lines are common geographic abstractions for geographic objects. Collections of regions, lines, and other representations of spatial objects form a spatial scene, along with their relations. For instance, the states of Maine and New Hampshire can be represented by a pair of regions and related based on their topological properties. These two states are adjacent (i.e., they meet along their shared boundary), whereas Maine and Florida are not adjacent (i.e., they are disjoint). A detailed model for qualitatively describing spatial scenes should capture the essential properties of a configuration such that a description of the represented objects and their relations can be generated. Such a description should then be able to reproduce a scene in a way that preserves all topological relationships, but without regards to metric details. Coarse approaches to qualitative spatial reasoning may underspecify certain relations. For example, if two objects meet, it is unclear if they meet along an edge, at a single point, or multiple times along their boundaries. Where the boundaries of spatial objects converge, this is called a spatial intersection. This thesis develops a model for spatial scene descriptions primarily through sequences of detailed spatial intersections and object containment, capturing how complex spatial objects relate. With a theory of complex spatial scenes developed, a tool that will automatically generate a formal description of a spatial scene is prototyped, enabling the described objects to be analyzed. The strengths and weaknesses of the provided model will be discussed relative to other models of spatial scene description, along with further refinements

Optical Tomography for Media with Variable Index of Refraction

Optical tomography is the use of near-infrared light to determine the optical absorption and scattering properties of a medium M ⊂ Rn. If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric and photons which do not get absorbed or scatter travel along straight lines. In this expository article we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. The data one has is the measurement of the out-going flux of photons leaving the body at the boundary. This may be knowledge of both the locations and directions of the exiting photons (fully angularly resolved measurements) or some kind of average over direction (angularly averaged measurements). We discuss the results known for both types of measurements in all spatial dimensions

First Acetic Acid Survey with CARMA in Hot Molecular Cores

Acetic acid (CH$_3$COOH) has been detected mainly in hot molecular cores where the distribution between oxygen (O) and nitrogen (N) containing molecular species is co-spatial within the telescope beam. Previous work has presumed that similar cores with co-spatial O and N species may be an indicator for detecting acetic acid. However, does this presumption hold as higher spatial resolution observations become available of large O and N-containing molecules? As the number of detected acetic acid sources is still low, more observations are needed to support this postulate. In this paper, we report the first acetic acid survey conducted with the Combined Array for Research in Millimeter-wave Astronomy (CARMA) at 3 mm wavelengths towards G19.61-0.23, G29.96-0.02 and IRAS 16293-2422. We have successfully detected CH$_3$COOH via two transitions toward G19.61-0.23 and tentatively confirmed the detection toward IRAS 16293-2422 A. The determined column density of CH$_3$COOH is 2.0(1.0)$\times 10^{16}$ cm$^{-2}$ and the abundance ratio of CH$_3$COOH to methyl formate (HCOOCH$_3$) is 2.2(0.1)$\times 10^{-1}$ toward G19.61-0.23. Toward IRAS 16293 A, the determined column density of CH$_3$COOH is $\sim$ 1.6 $\times 10^{15}$ cm$^{-2}$ and the abundance ratio of CH$_3$COOH to methyl formate (HCOOCH$_3$) is $\sim$ 1.0 $\times 10^{-1}$ both of which are consistent with abundance ratios determined toward other hot cores. Finally, we model all known line emission in our passband to determine physical conditions in the regions and introduce a new metric to better reveal weak spectral features that are blended with stronger lines or that may be near the 1-2$\sigma$ detection limit.Comment: 28 pages, 8 figures, accepted for publication in the ApJ; Revised citation in session 2, references remove

Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.Comment: 29 pages, 5 figure

The Consequences of the modulation instabilities

There are various cases of the development of the modulation instability of intense periodic structures in wave and non-wave media (see, for example [1]). The peculiarity of the modulation instability is the appearance of the perturbation spectrum, which is practically sym-metric with respect to large amplitude wave vector [2 - 6]. The modes of the perturbation spectrum are improper for a given medium, as a rule. The cases of different dis-sipation levels of large amplitude wave, in the presence of a source that supports it existence, are considered. In the case of a large dissipation level, near and above the threshold, the instability leads to the excitation of spectra whose width narrows, forming narrow spectral lines [7]. The line spectrum creates the conditions for the development of a more large-scale modulation [8]. Thus, the modulation instabilities near the threshold represent a cascade of processes with an increasing characteristic time of development and a larger characteristic scale [9, 10].The paper demonstrates the consequences of modulation instability of intense periodic structures in wave and non-wave media. In the case of a large dissipation level, near and above the threshold, the instability leads to the excitation of spectra whose width narrows, forming narrow spectral lines and self-similar structure of the big spatial clearness. At an insignificant level of dissipation, far from the threshold of modulation instability, the wave motion (initiated by the source) forms anomalous amplitude waves and envelopes exceeding the average amplitude by at three times. The shape of the envelope or wave packet is similar to the shape of Peregrine breather, and the dynam-ics over time is also similar. The formation of self-similar spatial structures in the developed convection of a thin liquid or gas layer due to the development of modulation instability is presented. In this case, toroidal convection vortices generate poloidal vortices of large scale − the effect of a hydrodynamic dynamo. Experimental results of the investigation of emerging self-similar structures on the graphite surface are presented. The features of the develop-ment of parametric instabilities are discussed

Cognitive processing of spatial relations in Euclidean diagrams

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, we tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philosophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed