Counting rational points on a Grassmannian

We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical result of Thunder ([21]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.Comment: Fixed more minor errors, plus a proof for the counting formula on a flag variet

Bounds on the gaps in Kronecker sequences (and a little bit more)

We provide bounds on the sizes of the gaps -- defined broadly -- in the set \{k_1\vbeta_1 + \ldots + k_n\vbeta_n \mbox{ (mod 1)} : k_i \in \Z \cap (0,Q^\frac{1}{n}]\} for generic \vbeta_1, \ldots, \vbeta_n \in \R^m and all sufficiently large $Q$. We also introduce a related problem in Diophantine approximation, which we believe is of independent interest

LLL and stochastic sandpile models

Theaimofthepresentpaperistosuggestthatstatisticalphysicsprovides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory

A physical study of the LLL algorithm

This paper presents a study of the LLL algorithm from the perspective of statistical physics. Based on our experimental and theoretical results, we suggest that interpreting LLL as a sandpile model may help understand much of its mysterious behavior. In the language of physics, our work presents evidence that LLL and certain 1-d sandpile models with simpler toppling rules belong to the same universality class. This paper consists of three parts. First, we introduce sandpile models whose statistics imitate those of LLL with compelling accuracy, which leads to the idea that there must exist a meaningful connection between the two. Indeed, on those sandpile models, we are able to prove the analogues of some of the most desired statements for LLL, such as the existence of the gap between the theoretical and the experimental RHF bounds. Furthermore, we test the formulas from the finite-size scaling theory (FSS) against the LLL algorithm itself, and find that they are in excellent agreement. This in particular explains and refines the geometric series assumption (GSA), and allows one to extrapolate various quantities of interest to the dimension limit. In particular, we predict the empirical average RHF converges to $\approx 1.02265$ as dimension goes to infinity.Comment: Augmented version of 1804.03285; expect some overlap

A Facile Fabrication and Transfer Method of Vertically Aligned Carbon Nanotubes on a Mo/Ni Bilayer for Wearable Energy Devices

Carbon nanotubes are a promising material for flexible/wearable electrochemical device due to their mechanical softness, chemical stability, and high conductivity. Furthermore, the vertically aligned form of carbon nanotubes (VACNTs) have a large surface area due to their unique three-dimensional (3D) nanostructure. Thus, VACNTs are particularly useful for wearable electrochemical sensors and/or energy devices. However, VACNTs are generally grown via a high-temperature chemical vapor deposition process, which requires a rigid substrate. As a flexible/wearable device platform, therefore, VACNTs should be transferred from rigid substrates to soft substrates. Here, a facile fabrication and transfer method of a unique 3D nanostructure, that is, VACNTs on the Mo/Ni bilayer, for high performance flexible/wearable devices is reported. After growth of VACNTs on a Mo/Ni bilayer, VACNTs with the Mo/Ni bilayer can be easily peeled-off from the SiO2 wafer by using weak adhesion of Ni to SiO2 for transfer printing onto polymeric/elastomeric substrates. Moreover, the Mo layer helps facile growth of VACNTs, and the Mo/Ni bilayer underneath VACNTs maximizes the lateral current flow. The proposed 3D nanostructure (VACNTs on the Mo/Ni bilayer) is successfully applied as flexible electrodes for high-performance wearable asymmetric supercapacitors.

Wearable Fall Detector using Integrated Sensors and Energy Devices

Wearable devices have attracted great attentions as next-generation electronic devices. For the comfortable, portable, and easy-to-use system platform in wearable electronics, a key requirement is to replace conventional bulky and rigid energy devices into thin and deformable ones accompanying the capability of long-term energy supply. Here, we demonstrate a wearable fall detection system composed of a wristband-type deformable triboelectric generator and lithium ion battery in conjunction with integrated sensors, controllers, and wireless units. A stretchable conductive nylon is used as electrodes of the triboelectric generator and the interconnection between battery cells. Ethoxylated polyethylenimine, coated on the surface of the conductive nylon electrode, tunes the work function of a triboelectric generator and maximizes its performance. The electrical energy harvested from the triboelectric generator through human body motions continuously recharges the stretchable battery and prolongs hours of its use. The integrated energy supply system runs the 3-axis accelerometer and related electronics that record human body motions and send the data wirelessly. Upon the unexpected fall occurring, a custom-made software discriminates the fall signal and an emergency alert is immediately sent to an external mobile device. This wearable fall detection system would provide new opportunities in the mobile electronics and wearable healthcare.

LLL and stochastic sandpile models

We introduce stochastic sandpile models which imitate numerous aspects of the practical behavior of the LLL algorithm with compelling accuracy. In addition, we argue that the physics and mathematics of sandpile models provide satisfactory heuristic explanations to much of the mysteries of LLL, and pleasant implications for lattice-based cryptography as a whole. Based on these successes, we suggest a paradigm in which one regards blockwise reduction algorithms as 1-d stochastic self-organized criticality(SOC) models and study them as such

Development of theranostic dual-layered Au-liposome for effective tumor targeting and photothermal therapy

Abstract Background Photothermal therapy (PTT) is an emerging anti-cancer therapeutic strategy that generates hyperthermia to ablate cancer cells under laser irradiation. Gold (Au) coated liposome (AL) was reported as an effective PTT agent with good biocompatibility and excretory property. However, exposed Au components on liposomes can cause instability in vivo and difficulty in further functionalization. Results Herein, we developed a theranostic dual-layered nanomaterial by adding liposomal layer to AL (LAL), followed by attaching polyethylene glycol (PEG) and radiolabeling. Functionalization with PEG improves the in vivo stability of LAL, and radioisotope labeling enables in vivo imaging of LAL. Functionalized LAL is stable in physiological conditions, and 64Cu labeled LAL (64Cu-LAL) shows a sufficient blood circulation property and an effective tumor targeting ability of 16.4%ID g−1 from in vivo positron emission tomography (PET) imaging. Also, intravenously injected LAL shows higher tumor targeting, temperature elevation in vivo, and better PTT effect in orthotopic breast cancer mouse model compared to AL. The tumor growth inhibition rate of LAL was 3.9-fold higher than AL. Conclusion Based on these high stability, in vivo imaging ability, and tumor targeting efficiency, LAL could be a promising theranostic PTT agent. Graphic Abstrac