Abstract-We consider the Gaussian N -relay diamond network, where a source wants to communicate to a destination node through a layer of N -relay nodes. We investigate the following question: What fraction of the capacity can we maintain by using only k out of the N available relays? We show that in every Gaussian N -relay diamond network, there exists a subset of k relays which alone provide approximately k k+1 of the total capacity. The result holds independent of the number of available relay nodes N , the channel configurations and the operating SNR. The result is tight in the sense that there exists channel configurations for N -relay diamond networks, where every subset of k relays can provide at most k k+1 of the total capacity. The approximation is within 3 log N + 3k bits/s/Hz to the capacity.This result also provides a new approximation to the capacity of the Gaussian N -relay diamond network which is up to a multiplicative gap of 1 k+1 and additive gap of 3 log N + 3k. The current approximation results in the literature either aim to characterize the capacity within an additive gap by allowing no multiplicative gap or vice a versa. Our result suggests a new approximation approach where multiplicative and additive gaps are allowed simultaneously and are traded through an auxiliary parameter.1

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False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the Circle Method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.

We consider double scaled little string theory on K3. These theories are labelled by a positive integer k ≥ 2 and an ADE root lattice with Coxeter number k. We count BPS fundamental string states in the holographic dual of this theory using the superconformal field theory K3 × SL(2,R) k U (1)We show that the BPS fundamental string states that are counted by the second helicity supertrace of this theory give rise to weight two mixed mock modular forms. We compute the helicity supertraces using two separate techniques: a path integral analysis that leads to a modular invariant but non-holomorphic answer, and a Hamiltonian analysis of the contribution from discrete states which leads to a holomorphic but not modular invariant answer. From a mathematical point of view the Hamiltonian analysis leads to a mixed mock modular form while the path integral gives the completion of this mixed mock modular form. We also compare these weight two mixed mock modular forms to those that appear in instances of Umbral Moonshine labelled by Niemeier root lattices X that are powers of ADE root lattices and find that they are equal up to a constant factor that we determine. In the course of the analysis we encounter an interesting generalization of Appell-Lerch sums and generalizations of the Riemann relations of Jacobi theta functions that they obey.where the trace is over all space-time states, that is including both Ramond (R) and Neveu-Schwarz (NS) sectors of the SCFT with GSO projection. J s is the generator of a U (1) rotation in space-time which we take to be rotation about the asymptotic S 1 in the cigar CFT and F s is 1 In the ADE classification of modular invariant partition functions at level k the Coxeter number of the ADE root system must equal k.