We consider the N -relay Gaussian diamond network where a source node communicates to a destination node via N parallel relays through a cascade of a Gaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000 by Schein and Gallager, the capacity of this relay network is unknown in general. The best currently available capacity approximation, independent of the coefficients and the SNR's of the constituent channels, is within an additive gap of 1.3N bits, which follows from the recent capacity approximations for general Gaussian relay networks with arbitrary topology.In this paper, we approximate the capacity of this network within 2 log N bits. We show that two strategies can be used to achieve the information-theoretic cutset upper bound on the capacity of the network up to an additive gap of O(log N ) bits, independent of the channel configurations and the SNR's. The first of these strategies is simple partial decode-and-forward. Here, the source node uses a superposition codebook to broadcast independent messages to the relays at appropriately chosen rates; each relay decodes its intended message and then forwards it to the destination over the MAC channel. A similar performance can be also achieved with compress-and-forward type strategies (such as quantize-map-and-forward and noisy network coding) that provide the 1.3N -bit approximation for general Gaussian networks, but only if the relays quantize their observed signals at a resolution inversely proportional to the number of relay nodes N . This suggest that the rule-of-thumb to quantize the received signals at the noise level in the current literature can be highly suboptimal.I. INTRODUCTION Consider a Gaussian relay network where a source node communicates to a destination with the help of intermediate relay nodes. Characterizing the capacity of this network is a long-standing open problem in network information theory. The seminal work of Cover and El-Gamal [2] has established several basic achievability schemes for the single relay channel, such as decode-and-forward and compress-andforward. Recently, significant progress has been made by generalizing the compress-and-forward strategy to achieve the capacity of any Gaussian relay network within an additive gap that depends on the network only through the total number of relay nodes N (or the total number of transmit and receive antennas when nodes are equipped with multiple antennas) [3], [4], [5], [6], [7]. The fact that the gap to capacity is independent of the channel gains, the SNR's and the exact topology of the network suggests that compressand-forward can be universally good for relaying across different channel configurations, SNR regimes and topologies. However, the dependence of the gap to N limits the applicability of these results to small networks with few relays. The best currently available capacity approximation in [4] is within 1.3N bits (per second per Hz) of the information-theoretic cutset upper bound on the capacity of the network. For typical spectral...

In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions of [n] = {1, 2, · · · , n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.

Abstract-DNA sequencing technology has advanced to a point where storage is becoming the central bottleneck in the acquisition and mining of more data. Large amounts of data are vital for genomics research, and generic compression tools, while viable, cannot offer the same savings as approaches tuned to inherent biological properties. We propose an algorithm to compress a target genome given a known reference genome. The proposed algorithm first generates a mapping from the reference to the target genome, and then compresses this mapping with an entropy coder. As an illustration of the performance: applying our algorithm to James Watson's genome with hg18 as a reference, we are able to reduce the 2991 megabyte (MB) genome down to 6.99 MB, while Gzip compresses it to 834.8 MB.

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1, 2, . . . , n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

It is explained how the Mordell integral R e πiτ x 2 −2πzx cosh(π x) dx unifies the mock theta functions, partial (or false) theta functions, and some of Zagier's quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z. Mathematics Subject Classification: 11P55, 05A17Keywords: Jacobi forms; mock Jacobi Form; partial Jacobi theta function; False theta function; Partial theta function; Mock theta function; Mordell integral; Ramanujan The Mordell integralThe Mordell integral isThe integral was studied by Kronecker [19,20] and in the late 1800s. A few years later, the integral underwent an intensive investigation by Mordell [28,29]. He established relationships between this integral and class number formulas. Later, Siegel [37] used properties of the integral to determine the approximate functional equation of the Riemman zeta function as well as asymptotics for its first moment. Very recently, the Mordell integral has been used to efficiently compute values of the Riemann zeta function [21]. The integral also appears in the solution of the 1-dimensional heat equation, see [32] and the references therein. Andrews [1] gave an early investigation of the Mordell integral in the study of the mock theta functions of Ramanujan.The Mordell integral plays a central role in Sander Zwegers Ph.D. thesis [42] on the mock theta functions. His work shows that the Mordell integral may be viewed as the obstruction to modularity of the mock theta functions. Moreover, he showed how to reinterpret the integral in terms of a period integral of a certain weight 3/2 modular form. His thesis has paved the way for many applications of mock theta functions to the study