Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have $q$-expansions resembling modular theta functions, is not well understood. Here we consider families of $q$-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.Comment: 13 page

Constraints on small-scale cosmological perturbations from gamma-ray searches for dark matter

Events like inflation or phase transitions can produce large density perturbations on very small scales in the early Universe. Probes of small scales are therefore useful for e.g. discriminating between inflationary models. Until recently, the only such constraint came from non-observation of primordial black holes (PBHs), associated with the largest perturbations. Moderate-amplitude perturbations can collapse shortly after matter-radiation equality to form ultracompact minihalos (UCMHs) of dark matter, in far greater abundance than PBHs. If dark matter self-annihilates, UCMHs become excellent targets for indirect detection. Here we discuss the gamma-ray fluxes expected from UCMHs, the prospects of observing them with gamma-ray telescopes, and limits upon the primordial power spectrum derived from their non-observation by the Fermi Large Area Space Telescope.Comment: 4 pages, 3 figures. To appear in J Phys Conf Series (Proceedings of TAUP 2011, Munich

Disentangling Instrumental Features of the 130 GeV Fermi Line

We study the instrumental features of photons from the peak observed at $E_\gamma=130$ GeV in the spectrum of Fermi-LAT data. We use the {\sc sPlots} algorithm to reconstruct -- seperately for the photons in the peak and for background photons -- the distributions of incident angles, the recorded time, features of the spacecraft position, the zenith angles, the conversion type and details of the energy and direction reconstruction. The presence of a striking feature or cluster in such a variable would suggest an instrumental cause for the peak. In the publically available data, we find several suggestive features which may inform further studies by instrumental experts, though the size of the signal sample is too small to draw statistically significant conclusions.Comment: 9 pages, 22 figures; this version includes additional variables, study of stat sensitivity, and modification to the chi-sq calculatio

Bringing Order to Special Cases of Klee's Measure Problem

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].Comment: 17 page

Thermal decoupling and the smallest subhalo mass in dark matter models with Sommerfeld-enhanced annihilation rates

We consider dark matter consisting of weakly interacting massive particles (WIMPs) and revisit in detail its thermal evolution in the early universe, with a particular focus on models where the annihilation rate is enhanced by the Sommerfeld effect. After chemical decoupling, or freeze-out, dark matter no longer annihilates but is still kept in local thermal equilibrium due to scattering events with the much more abundant standard model particles. During kinetic decoupling, even these processes stop to be effective, which eventually sets the scale for a small-scale cutoff in the matter density fluctuations. Afterwards, the WIMP temperature decreases more quickly than the heat bath temperature, which causes dark matter to reenter an era of annihilation if the cross-section is enhanced by the Sommerfeld effect. Here, we give a detailed and self-consistent description of these effects. As an application, we consider the phenomenology of simple leptophilic models that have been discussed in the literature and find that the relic abundance can be affected by as much two orders of magnitude or more. We also compute the mass of the smallest dark matter subhalos in these models and find it to be in the range of about 10^{-10} to 10 solar masses; even much larger cutoff values are possible if the WIMPs couple to force carriers lighter than about 100 MeV. We point out that a precise determination of the cutoff mass allows to infer new limits on the model parameters, in particular from gamma-ray observations of galaxy clusters, that are highly complementary to existing constraints from g-2 or beam dump experiments.Comment: minor changes to match published versio

LET-99, GOA-1/GPA-16, and GPR-1/2 are required for aster-positioned cytokinesis.

At anaphase, the mitotic spindle positions the cytokinesis furrow [1]. Two populations of spindle microtubules are implicated in cytokinesis: radial microtubule arrays called asters and bundled nonkinetochore microtubules called the spindle midzone [2-4]. In C. elegans embryos, these two populations of microtubules provide two consecutive signals that position the cytokinesis furrow: The first signal is positioned midway between the microtubule asters; the second signal is positioned over the spindle midzone [5]. Evidence for two cytokinesis signals came from the identification of molecules that block midzone-positioned cytokinesis [5-7]. However, no molecules that are only required for, and thus define, the molecular pathway of aster-positioned cytokinesis have been identified. With RNAi screening, we identify LET-99 and the heterotrimeric G proteins GOA-1/GPA-16 and their regulator GPR-1/2 [10-12] in aster-positioned cytokinesis. By using mechanical spindle displacement, we show that the anaphase spindle positions cortical LET-99, at the site of the presumptive cytokinesis furrow. LET-99 enrichment at the furrow depends on the G proteins. GPR-1 is locally reduced at the site of cytokinesis-furrow formation by LET-99, which prevents accumulation of GPR-1 at this site. We conclude that LET-99 and the G proteins define a molecular pathway required for aster-positioned cytokinesis

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 $\mathrm{SU}(3)$

Impact of Meditation–Based Lifestyle Modification on HRV in Outpatients With Mild to Moderate Depression: An Exploratory Study

Background: The scientific evaluation of mind-body-interventions (MBI), including yoga and meditation, has increased significantly in recent decades. However, evidence of MBI's efficacy on biological parameters is still insufficient. Objectives: In this study, we used HRV analysis to evaluate a novel MBI as a treatment of outpatients with mild to moderate depressive disorder. The Meditation-Based Lifestyle Modification (MBLM) program incorporates all major elements of classical yoga, including ethical principles of yoga philosophy, breathing exercises, postural yoga, and meditation. Methods: In this exploratory randomized controlled trial, we compared the changes in HRV indices of a MBLM group (N = 22) and a minimal treatment group (MINIMAL, drugs only, N = 17) with those of a multimodal treatment-as-usual group (TAU, according to best clinical practice, N = 22). Electrocardiogram (ECG) recordings were derived from a Holter monitoring device, and HRV indices have been extracted from nearly stationary 20-min periods. Results: Short-term HRV analysis revealed statistically significant differences in the pre-to-post changes between MBLM and TAU. In particular, the vagal tone mediating RMSSD and the Renyi entropy of symbolic dynamics indicated HRV gains in MBLM participants compared with TAU. Almost no alterations were observed in the MINIMAL group. Conclusions: Our results suggest a benefit in selected HRV parameters for outpatients with mild to moderate depression participating in the MBLM program. For further investigations, we propose analysis of complete 24-h HRV recordings and additional continuous pulse wave or blood pressure analysis to assess long-term modulations and cardiovascular effects

Fast and Simple Modular Subset Sum

We revisit the Subset Sum problem over the finite cyclic group $\mathbb{Z}_m$ for some given integer $m$. A series of recent works has provided asymptotically optimal algorithms for this problem under the Strong Exponential Time Hypothesis. Koiliaris and Xu (SODA'17, TALG'19) gave a deterministic algorithm running in time $\tilde{O}(m^{5/4})$, which was later improved to $O(m \log^7 m)$ randomized time by Axiotis et al. (SODA'19). In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in $m$, both efficiently implementing Bellman's iteration over $\mathbb{Z}_m$. The first one is a randomized algorithm running in time $O(m\log^2 m)$, that is based solely on rolling hash and an elementary data-structure for prefix sums; to illustrate its simplicity we provide a short and efficient implementation of the algorithm in Python. Our second solution is a deterministic algorithm running in time $O(m\ \mathrm{polylog}\ m)$, that uses dynamic data structures for string manipulation. We further show that the techniques developed in this work can also lead to simple algorithms for the All Pairs Non-Decreasing Paths Problem (APNP) on undirected graphs, matching the asymptotically optimal running time of $\tilde{O}(n^2)$ provided in the recent work of Duan et al. (ICALP'19)