A solar wind-derived water reservoir on the Moon hosted by impact glass beads

The past two decades of lunar exploration have seen the detection of substantial quantities of water on the Moon’s surface. It has been proposed that a hydrated layer exists at depth in lunar soils, buffering a water cycle on the Moon globally. However, a reservoir has yet to be identified for this hydrated layer. Here we report the abundance, hydrogen isotope composition and core-to-rim variations of water measured in impact glass beads extracted from lunar soils returned by the Chang’e-5 mission. The impact glass beads preserve hydration signatures and display water abundance profiles consistent with the inward diffusion of solar wind-derived water. Diffusion modelling estimates diffusion timescales of less than 15 years at a temperature of 360 K. Such short diffusion timescales suggest an efficient water recharge mechanism that could sustain the lunar surface water cycle. We estimate that the amount of water hosted by impact glass beads in lunar soils may reach up to 2.7 × 1014 kg. Our direct measurements of this surface reservoir of lunar water show that impact glass beads can store substantial quantities of solar wind-derived water on the Moon and suggest that impact glass may be water reservoirs on other airless bodies

Dynamics of a Family of Nonlinear Delay Difference Equations

We study the global asymptotic stability of the following difference equation: xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…, where 0≤k1<k2<⋯<ks and 0≤m1<m2<⋯<mt with {k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅, the initial values are positive, and f∈C(Es+t,(0,+∞)) with E∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibrium x- of that equation is globally asymptotically stable

Eventually Periodic Solutions of a Max-Type Difference Equation

We study the following max-type difference equation xn=max⁡{An/xn-r,xn-k}, n=1,2,…, where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max⁡{r,k}. We show that if p=1 (or p≥2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p≥2 and k being even which has a well-defined solution that is not eventually periodic

Oscillation for Higher Order Dynamic Equations on Time Scales

We investigate the oscillation of the following higher order dynamic equation: {an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scale T, where n≥2, ak(t)   (1≤k≤n) and p(t) are positive rd-continuous functions on T and α,β are the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero

Stability of Rotation Pairs of Cycles for the Interval Maps

Let C0(I) be the set of all continuous self-maps of the closed interval I, and P(u,v)={f∈C0(I):f has a cycle with rotation pair (u,v)} for any positive integer v>u. In this paper, we prove that if (2mns,2mnt)⊣(γ,λ), then P(2mns,2mnt)⊂  int  P(γ,λ), where m≥0 is integer, n≥1 odd, 1≤s<t with s,t coprime, and 1≤γ<λ