Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets

Over the last few years, needlets have a emerged as a useful tool for the analysis of Cosmic Microwave Background (CMB) data. Our aim in this paper is first to introduce in the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller and Mayeli (2009a,b). We then proceed with an extensive study of the properties of both standard and Mexican needlets; these properties depend on some parameters which can be tuned in order to optimize the performance for a given application. Our second aim in this paper is then to give practical advice on how to adjust these parameters in order to achieve the best properties for a given problem in CMB data analysis. In particular we investigate localization properties in real and harmonic spaces and propose a recipe on how to quantify the influence of galactic and point source masks on the needlet coefficients. We also show that for certain parameter values, the Mexican needlets provide a close approximation to the Spherical Mexican Hat Wavelets (whence their name), with some advantages concerning their numerical implementation and the derivation of their statistical properties.Comment: 40 pages, 11 figures, published version, main modification: added section on more realistic galactic and point source mask

Adaptive Density Estimation on the Circle by Nearly-Tight Frames

This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the $L^{2}$-risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.Comment: 30 pages, 3 figure

Constraining the WMAP9 bispectrum and trispectrum with needlets

We develop a needlet approach to estimate the amplitude of general (including non-separable) bispectra and trispectra in the cosmic microwave background, and apply this to the WMAP 9-year data. We obtain estimates for the `orthogonal' bispectrum mode, yielding results which are consistent with the WMAP 7-year data. We do not observe the frequency-dependence suggested by the WMAP team's analysis of the 9-year data. We present 1-$\sigma$ constraints on the `local' trispectrum shape \gnl/10^5= -4.1\pm 2.3, the `$c1$' equilateral model \gnl^{c_1}/10^6= -0.8\pm 2.9, and the constant model \gnl^{\rm{const}}/10^6= -0.2\pm 1.8, together with a $95\%$ confidence-level upper bound on the multifield local parameter \taunl<22000. We estimate the bias on these parameters produced by point sources. The techniques developed in this paper should prove useful for other datasets such as Planck.Comment: 21 pages - matches published versio

A multi-level solver for Gaussian constrained CMB realizations

Updated to match published version (no major changes)Updated to match published version (no major changes)Updated to match published version (no major changes)We present a multi-level solver for drawing constrained Gaussian realizations or finding the maximum likelihood estimate of the CMB sky, given noisy sky maps with partial sky coverage. The method converges substantially faster than existing Conjugate Gradient (CG) methods for the same problem. For instance, for the 143 GHz Planck frequency channel, only 3 multi-level W-cycles result in an absolute error smaller than 1 microKelvin in any pixel. Using 16 CPU cores, this translates to a computational expense of 6 minutes wall time per realization, plus 8 minutes wall time for a power spectrum-dependent precomputation. Each additional W-cycle reduces the error by more than an order of magnitude, at an additional computational cost of 2 minutes. For comparison, we have never been able to achieve similar absolute convergence with conventional CG methods for this high signal-to-noise data set, even after thousands of CG iterations and employing expensive preconditioners. The solver is part of the Commander 2 code, which is available with an open source license at http://commander.bitbucket.org/

Paclitaxel-loaded Cationic Fluid Lipid Nanodiscs and Liposomes with Brush-Conformation PEG Chains Penetrate Breast Tumors and Trigger Caspase-3 Activation

Novel approaches are required to address the urgent need to develop lipid-based carriers of paclitaxel (PTX) and other hydrophobic drugs for cancer chemotherapy. Carriers based on cationic liposomes (CLs) with fluid (i.e., chain-melted) membranes (e.g., EndoTAG-1®) have shown promise in preclinical and late-stage clinical studies. Recent work found that the addition of a cone-shaped poly(ethylene glycol)-lipid (PEG-lipid) to PTX-loaded CLs (CLsPTX) promotes a transition to sterically stabilized, higher-curvature (smaller) nanoparticles consisting of a mixture of PEGylated CLsPTX and PTX-containing fluid lipid nanodiscs (nanodiscsPTX). These CLsPTX and nanodiscsPTX show significantly improved uptake and cytotoxicity in cultured human cancer cells at PEG coverage in the brush regime (10 mol% PEG-lipid). Here, we studied the PTX loading, in vivo circulation half-life, and biodistribution of systemically administered CLsPTX and nanodiscsPTX and assessed their ability to induce apoptosis in triple-negative breast cancer-bearing immunocompetent mice. We focused on fluid rather than solid lipid nanodiscs because of the significantly higher solubility of PTX in fluid membranes. At 5 and 10 mol% of a PEG-lipid (PEG5K-lipid, molecular weight of PEG 5000 g/mol), the mixture of PEGylated CLsPTX and nanodiscsPTX was able to incorporate up to 2.5 mol% PTX without crystallization for at least 20 h. Remarkably, compared to preparations containing 2 and 5 mol% PEG5K-lipid (with the PEG chains in the mushroom regime), the particles at 10 mol% (with PEG chains in the brush regime) showed significantly higher blood half-life, tumor penetration and proapoptotic activity. Our study suggests that increasing the PEG coverage of CL-based drug nanoformulations can improve their pharmacokinetics and therapeutic efficacy.This research study was supported by the National Institutes of Health under award R01GM130769 (CRS, KKE, WF; mechanistic studies on developing lipid nanoparticles for drug delivery), the European Regional Development Fund (TT, Project No. 2014-2020.4.01.15-0012), the Estonian Research Council (TT, grants PRG230 and EAG79; PS, grant PSG38; LSG, grant MOBJD11), EuronanomedII projects ECM-CART and iNanoGun (TT), H2020 MSCA-RISE project Oxigenated (TT), and the Spanish Ministry of Science and Innovation grants RYC2020-028754-I and PID2021-122364OA-I00 (PS). Partial support was provided by the US National Science Foundation (NSF) under Award DMR-1807327 (CRS; kinetic phase behavior of cationic vesicles with incorporated hydrophobic molecules).N