We compare and contrast from a geometric perspective a number of low-dimensional signal models that support stable information-preserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal models, point clouds, and manifold signal models. Each model has a particular geometrical structure that enables signal information in to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space whose dimension either is independent of the ambient dimension at best or grows logarithmically with it at worst. As a bonus, we point out a common misconception related to probabilistic compressible signal models, that is, that the generalized Gaussian and Laplacian random models do not support stable linear dimensionality reduction.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (x m /(x)) (m+1) is completely monotonic. We conjecture that (x m / (m\Gamma1 (x)) (m) is completely monotonic for m 2, we prove it, with help from Maple, for 2 m 16. We give a very useful Maple proceedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive. Running Title: Gamma Function Inequalities Mathematics Subject Classification. Primary 33B15. Secondary 26D07, 26D10. Key words and phrases. gamma function, digamma function, inequalities, complete monotonicity. 1. Introduction. Inequalities of functions involving gamma functions have been of interest since the 1950's when inequalities by Gautchi, Erber and Kershaw were established. For references and generalizations we refer the interested reader to [5], [13], [14], [15], [16], and to Alzer's p...

Abstract — We define a near-orthogonal set of channel vectors as one that meets certain SIR and SNR guarantees. The probability of finding a near-orthogonal set in a pool of n users is characterized. We identify a phase transition phenomenon in channel geometry whereby the probability transitions from 0 to 1 quite sharply. In particular, it is shown that the probability of failing to find such a set as a function of the number of users k that have been examined passes through a sharp threshold at k ∼ Θ(1), after which it behaves like Θ(k −m). The rate at which SNR and SIR can be scaled while we remain above this threshold is also characterized. The existence results we provide are not specific to the MIMO scheduling problem, but apply to the more general setting of finding a near-orthogonal set in a random collection of isotropic vectors. The proofs make use of new tight bounds we develop to bound the surface content of spherical caps in arbitrary dimensions. Broader implications of these results are discussed. Specifically, in the case of zero-forcing the rate increases at a rate on the order of log log n. I.

Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z |> c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z) | is of order |z | a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.

Transmit beamforming is a powerful technique for enhancing performance of wireless communication systems. Most existing transmit beamforming techniques require perfect channel state information at the transmitter (CSIT), which is typically not available in practice. In such situations, the design should take errors in CSIT into account to avoid performance degradation. Among two popular robust designs, the stochastic approach exploits channel statistics and optimizes the average system performance. The maximin approach considers errors as deterministic and optimizes the worst-case performance. The latter usually leads to conservative results as the extreme (but rare) conditions may occur at a very low probability. In this work, we propose a more flexible approach that maximizes the average signal-to-noise ratio (SNR) and takes the extreme conditions into account proportionally. Simulation results show that the proposed beamformer offers higher robustness against channel estimation errors than several popular transmit beamformers. 1.

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