A Framework for Modular Properties of False Theta Functions

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.Comment: 20 page

Covariant Symplectic Structure and Conserved Charges of Topologically Massive Gravity

We present the covariant symplectic structure of the Topologically Massive Gravity and find a compact expression for the conserved charges of generic spacetimes with Killing symmetries.Comment: 10 pages, Dedicated to the memory of Yavuz Nutku (1943-2010), References added, Conserved charges of non-Einstein solutions of TMG are added, To appear in Phys. Rev.

Wireless Network Simplification: the Gaussian N-Relay Diamond Network

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 independent of the channel configurations and the operating SNR, we can always find a subset of k relays which alone provide a rate (kC/(k+1))-G, where C is the information theoretic cutset upper bound on the capacity of the whole network and G is a constant that depends only on N and k (logarithmic in N and linear in k). In particular, for k = 1, this means that half of the capacity of any N-relay diamond network can be approximately achieved by routing information over a single relay. We also show that this fraction is tight: there are configurations of the N-relay diamond network where every subset of k relays alone can at most provide approximately a fraction k/(k+1) of the total capacity. These high-capacity k-relay subnetworks can be also discovered efficiently. We propose an algorithm that computes a constant gap approximation to the capacity of the Gaussian N-relay diamond network in O(N log N) running time and discovers a high-capacity k-relay subnetwork in O(kN) running time. This result also provides a new approximation to the capacity of the Gaussian N-relay diamond network which is hybrid in nature: it has both multiplicative and additive gaps. In the intermediate SNR regime, this hybrid approximation is tighter than existing purely additive or purely multiplicative approximations to the capacity of this network.Comment: Submitted to Transactions on Information Theory in October 2012. The new version includes discussions on the algorithmic complexity of discovering a high-capacity subnetwork and on the performance of amplify-and-forwar

ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine

We consider double scaled little string theory on $K3$. These theories are labelled by a positive integer $k \ge 2$ 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 $K3 \times \left( \frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/ \mathbb{Z}_k$. 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.Comment: 1+56 page

Quantum Modular Forms from Real Quadratic Double Sums

In 2015, Lovejoy and Osburn discovered twelve $q$-hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and quantum modular properties and show that they yield three vector-valued quantum modular forms on the group $\Gamma_0 (2)$.Comment: 26 pages; v2: Brief comments added. To appear in the Quarterly Journal of Mathematic

On the asymptotic behavior for partitions separated by parity

The study of partitions with parts separated by parity was initiated by Andrews in connection with Ramanujan's mock theta functions, and his variations on this theme have produced generating functions with a large variety of different modular properties. In this paper, we use Ingham's Tauberian theorem to compute the asymptotic main term for each of the eight functions studied by Andrews

Higher depth false modular forms

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invariants of $3$-manifolds associated with gauge group $\mathrm{SU}(3)$

Accelerated Tooth Movement with Orthodontic Mini-Screws

This case report outlines the possibility of accelerated tooth movement with the combination of microosteoperforation and mini-screws. A 14-year-old male patient presented Class II malocclusion with maxillary incisor protrusion. Upper first premolars were extracted, and after leveling, accelerated canine distalization started. For pre- and postdistalization times, amount of distalization, periodontal health, and root resorption were assessed. Within the limitations of this case report, micro-osteoperforations with mini-screw have a potential for shortening the treatment time