Ballistic trajectory: parabola, ellipse, or what?

Mechanics texts tell us that a particle in a bound orbit under gravitational central force moves on an ellipse, while introductory physics texts approximate the earth as flat, and tell us that the particle moves in a parabola. The uniform-gravity, flat-earth parabola is clearly meant to be an approximation to a small segment of the true central-force/ellipse orbit. To look more deeply into this connection we convert earth-centered polar coordinates to ``flat-earth coordinates'' by treating radial lines as vertical, and by treating lines of constant radial distance as horizontal. With the exact trajectory and dynamics in this system, we consider such questions as whether gravity is purely vertical in this picture, and whether the central force nature of gravity is important only when the height or range of a ballistic trajectory is comparable to the earth radius. Somewhat surprisingly, the answers to both questions is ``no,'' and therein lie some interesting lessons.Comment: 7 pages, 3 figure

MAC with Action-Dependent State Information at One Encoder

Problems dealing with the ability to take an action that affects the states of state-dependent communication channels are of timely interest and importance. Therefore, we extend the study of action-dependent channels, which until now focused on point-to-point models, to multiple-access channels (MAC). In this paper, we consider a two-user, state-dependent MAC, in which one of the encoders, called the informed encoder, is allowed to take an action that affects the formation of the channel states. Two independent messages are to be sent through the channel: a common message known to both encoders and a private message known only to the informed encoder. In addition, the informed encoder has access to the sequence of channel states in a non-causal manner. Our framework generalizes previously evaluated settings of state dependent point-to-point channels with actions and MACs with common messages. We derive a single letter characterization of the capacity region for this setting. Using this general result, we obtain and compute the capacity region for the Gaussian action-dependent MAC. The unique methods used in solving the Gaussian case are then applied to obtain the capacity of the Gaussian action-dependent point-to-point channel; a problem was left open until this work. Finally, we establish some dualities between action-dependent channel coding and source coding problems. Specifically, we obtain a duality between the considered MAC setting and the rate distortion model known as "Successive Refinement with Actions". This is done by developing a set of simple duality principles that enable us to successfully evaluate the outcome of one problem given the other.Comment: 1. Parts of this paper appeared in the IEEE International Symposium on Information Theory (ISIT 2012),Cambridge, MA, US, July 2012 and at the IEEE 27th Convention of Electrical and Electronics Engineers in Israel (IEEEI 2012), Nov. 2012. 2. This work has been supported by the CORNET Consortium Israel Ministry for Industry and Commerc

Late time tails from momentarily stationary, compact initial data in Schwarzschild spacetimes

An L-pole perturbation in Schwarzschild spacetime generally falls off at late times t as t^{-2L-3}. It has recently been pointed out by Karkowski, Swierczynski and Malec, that for initial data that is of compact support, and is initially momentarily static, the late-time behavior is different, going as t^{-2L-4}. By considering the Laplace transforms of the fields, we show here why the momentarily stationary case is exceptional. We also explain, using a time-domain description, the special features of the time development in this exceptional case.Comment: 7 pages, 5 figure

Analyses of deep mammalian sequence alignments and constraint predictions for 1% of the human genome

A key component of the ongoing ENCODE project involves rigorous comparative sequence analyses for the initially targeted 1% of the human genome. Here, we present orthologous sequence generation, alignment, and evolutionary constraint analyses of 23 mammalian species for all ENCODE targets. Alignments were generated using four different methods; comparisons of these methods reveal large-scale consistency but substantial differences in terms of small genomic rearrangements, sensitivity (sequence coverage), and specificity (alignment accuracy). We describe the quantitative and qualitative trade-offs concomitant with alignment method choice and the levels of technical error that need to be accounted for in applications that require multisequence alignments. Using the generated alignments, we identified constrained regions using three different methods. While the different constraint-detecting methods are in general agreement, there are important discrepancies relating to both the underlying alignments and the specific algorithms. However, by integrating the results across the alignments and constraint-detecting methods, we produced constraint annotations that were found to be robust based on multiple independent measures. Analyses of these annotations illustrate that most classes of experimentally annotated functional elements are enriched for constrained sequences; however, large portions of each class (with the exception of protein-coding sequences) do not overlap constrained regions. The latter elements might not be under primary sequence constraint, might not be constrained across all mammals, or might have expendable molecular functions. Conversely, 40% of the constrained sequences do not overlap any of the functional elements that have been experimentally identified. Together, these findings demonstrate and quantify how many genomic functional elements await basic molecular characterization

Computational Efficiency of Frequency-- and Time--Domain Calculations of Extreme Mass--Ratio Binaries: Equatorial Orbits

Gravitational waveforms and fluxes from extreme mass--ratio inspirals can be computed using time--domain methods with accuracy that is fast approaching that of frequency--domain methods. We study in detail the computational efficiency of these methods for equatorial orbits of fast spinning Kerr black holes, and find the number of modes needed in either method --as functions of the orbital parameters-- in order to achieve a desired accuracy level. We then estimate the total computation time and argue that for high eccentricity orbits the time--domain approach is more efficient computationally. We suggest that in practice low--$m$ modes are computed using the frequency--domain approach, and high--$m$ modes are computed using the time--domain approach, where $m$ is the azimuthal mode number.Comment: 19 figures, 6 table

Universality of massive scalar field late-time tails in black-hole spacetimes

The late-time tails of a massive scalar field in the spacetime of black holes are studied numerically. Previous analytical results for a Schwarzschild black hole are confirmed: The late-time behavior of the field as recorded by a static observer is given by $\psi(t)\sim t^{-5/6}\sin [\omega (t)\times t]$, where $\omega(t)$ depends weakly on time. This result is carried over to the case of a Kerr black hole. In particular, it is found that the power-law index of -5/6 depends on neither the multipole mode $\ell$ nor on the spin rate of the black hole $a/M$. In all black hole spacetimes, massive scalar fields have the same late-time behavior irrespective of their initial data (i.e., angular distribution). Their late-time behavior is universal.Comment: 11 pages, 14 figures, published versio