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Concept# Entire function

Summary

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a
root at w, then f(z)/(z-w), taking the limit value at w, is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.
A transcendental entire function is an entire function that is not a polynomial.
Just as meromorphic functions can be viewed as a generalization of rational fractions, entire

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Astronomy is one of the oldest sciences known to humanity. We have been studying celestial objects for millennia, and continue to peer deeper into space in our thirst for knowledge about our origins and the universe that surrounds us. Radio astronomy -- observing celestial objects at radio frequencies -- has helped push the boundaries on the kind of objects we can study. Indeed, some of the most important discoveries about the structure of our universe, like the cosmic microwave background, and entire classes of objects like quasars and pulsars, were made using radio astronomy. Radio interferometers are telescopes made of multiple antennas spread over a distance. Signals detected at different antennas are combined to provide images with much higher resolution and sensitivity than with a traditional single-dish radio telescope. The Square Kilometre Array (SKA) is one such radio interferometer, with plans to have antennas separated by as much as 3000,km. In its quest for ever-higher resolution and ever-wider coverage of the sky, the SKA heralds a data explosion, with an expected acquisition rate of 5,terabits per second. The high data rate fed into the pipeline can be handled with a two-pronged approach -- (i) scalable, parallel imaging algorithms that fully utilize the latest computing technologies like accelerators and distributed clusters, and (ii) dimensionality reduction methods that embed the high-dimensional telescope data to much smaller sizes without losing information and guaranteeing accurate recovery of the images, thereby enabling imaging methods to scale to big data sizes and alleviating heavy loads on pipeline buffers without compromising on the science goals of the SKA.
In this thesis we propose fast and robust dimensionality reduction methods that embed data to very low sizes while preserving information present in the original data. These methods are presented in the context of compressed sensing theory and related signal recovery techniques. The effectiveness of the reduction methods is illustrated by coupling them with advanced convex optimization algorithms to solve a sparse recovery problem. Images thus reconstructed from extremely low-sized embedded data are shown to have quality comparable to those obtained from full data without any reduction. Comparisons with other standard `data compression' techniques in radio interferometry (like averaging) show a clear advantage in using our methods which provide higher quality images from much lower data sizes. We confirm these claims on both synthetic data simulating SKA data patterns as well as actual telescope data from a state-of-the-art radio interferometer. Additionally, imaging with reduced data is shown to have a lighter computational load -- smaller memory footprint owing to the size and faster iterative image recovery owing to the fast embedding.
Extensions to the work presented in this thesis are already underway. We propose an `on-line' version of our reduction methods that works on blocks of data and thus can be applied on-the-fly on data as they are being acquired by telescopes in real-time. This is of immediate interest to the SKA where large buffers in the data acquisition pipeline are very expensive and thus undesirable. Some directions to be probed in the immediate future are in transient imaging, and imaging hyperspectral data to test computational load while in a high resolution, multi-frequency setting.

We give a Liouville theorem for entire solutions and Laurent series expansions for solutions with isolated singularities of the heat equation.

Michael Christoph Gastpar, Sameer Pawar

Download Citation Email Print Request Permissions Save to Project In this paper we study the problem of data exchange, where each node in the system has a number of linear combinations of the data packets. Communicating over a public channel, the goal is for all nodes to reconstruct the entire set of the data packets in minimal total number of bits exchanged over the channel. We present a novel divide and conquer based architecture that determines the number of bits each node should transmit. This along with the well known fact, that it is sufficient for the nodes to broadcast linear combinations of their local information, provides a polynomial time deterministic algorithm for reconstructing the entire set of the data packets at all nodes in minimal amount of total communication.