Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known.
more »
« less
On the arithmetic of a family of degree - two K3 surfaces
Abstract Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*}X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2.\end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ( t ). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$ , together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X .
more »
« less
- Award ID(s):
- 1439786
- NSF-PAR ID:
- 10302686
- Date Published:
- Journal Name:
- Mathematical Proceedings of the Cambridge Philosophical Society
- Volume:
- 166
- Issue:
- 3
- ISSN:
- 0305-0041
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Ever-increasing demands for energy, particularly being environmentally friendly have promoted the transition from fossil fuels to renewable energy.1Lithium-ion batteries (LIBs), arguably the most well-studied energy storage system, have dominated the energy market since their advent in the 1990s.2However, challenging issues regarding safety, supply of lithium, and high price of lithium resources limit the further advancement of LIBs for large-scale energy storage applications.3Therefore, attention is being concentrated on an alternative electrochemical energy storage device that features high safety, low cost, and long cycle life. Rechargeable aqueous zinc-ion batteries (ZIBs) is considered one of the most promising alternative energy storage systems due to the high theoretical energy and power densities where the multiple electrons (Zn2+) . In addition, aqueous ZIBs are safer due to non-flammable electrolyte (e.g., typically aqueous solution) and can be manufactured since they can be assembled in ambient air conditions.4As an essential component in aqueous Zn-based batteries, the Zn metal anode generally suffers from the growth of dendrites, which would affect battery performance in several ways. Second, the led by the loose structure of Zn dendrite may reduce the coulombic efficiency and shorten the battery lifespan.5
Several approaches were suggested to improve the electrochemical stability of ZIBs, such as implementing an interfacial buffer layer that separates the active Zn from the bulk electrolyte.6However, the and thick thickness of the conventional Zn metal foils remain a critical challenge in this field, which may diminish the energy density of the battery drastically. According to a theretical calculation, the thickness of a Zn metal anode with an areal capacity of 1 mAh cm-2is about 1.7 μm. However, existing extrusion-based fabrication technologies are not capable of downscaling the thickness Zn metal foils below 20 μm.
Herein, we demonstrate a thickness controllable coating approach to fabricate an ultrathin Zn metal anode as well as a thin dielectric oxide separator. First, a 1.7 μm Zn layer was uniformly thermally evaporated onto a Cu foil. Then, Al2O3, the separator was deposited through sputtering on the Zn layer to a thickness of 10 nm. The inert and high hardness Al2O3layer is expected to lower the polarization and restrain the growth of Zn dendrites. Atomic force microscopy was employed to evaluate the roughness of the surface of the deposited Zn and Al2O3/Zn anode structures. Long-term cycling stability was gauged under the symmetrical cells at 0.5 mA cm-2for 1 mAh cm-2. Then the fabricated Zn anode was paired with MnO2as a full cell for further electrochemical performance testing. To investigate the evolution of the interface between the Zn anode and the electrolyte, a home-developed in-situ optical observation battery cage was employed to record and compare the process of Zn deposition on the anodes of the Al2O3/Zn (demonstrated in this study) and the procured thick Zn anode. The surface morphology of the two Zn anodes after circulation was characterized and compared through scanning electron microscopy. The tunable ultrathin Zn metal anode with enhanced anode stability provides a pathway for future high-energy-density Zn-ion batteries.
Obama, B., The irreversible momentum of clean energy.
Science 2017, 355 (6321), 126-129.Goodenough, J. B.; Park, K. S., The Li-ion rechargeable battery: a perspective.
J Am Chem Soc 2013, 135 (4), 1167-76.Li, C.; Xie, X.; Liang, S.; Zhou, J., Issues and Future Perspective on Zinc Metal Anode for Rechargeable Aqueous Zinc‐ion Batteries.
Energy & Environmental Materials 2020, 3 (2), 146-159.Jia, H.; Wang, Z.; Tawiah, B.; Wang, Y.; Chan, C.-Y.; Fei, B.; Pan, F., Recent advances in zinc anodes for high-performance aqueous Zn-ion batteries.
Nano Energy 2020, 70 .Yang, J.; Yin, B.; Sun, Y.; Pan, H.; Sun, W.; Jia, B.; Zhang, S.; Ma, T., Zinc Anode for Mild Aqueous Zinc-Ion Batteries: Challenges, Strategies, and Perspectives.
Nanomicro Lett 2022, 14 (1), 42.Yang, Q.; Li, Q.; Liu, Z.; Wang, D.; Guo, Y.; Li, X.; Tang, Y.; Li, H.; Dong, B.; Zhi, C., Dendrites in Zn-Based Batteries.
Adv Mater 2020, 32 (48), e2001854.Acknowledgment
This work was partially supported by the U.S. National Science Foundation (NSF) Award No. ECCS-1931088. S.L. and H.W.S. acknowledge the support from the Improvement of Measurement Standards and Technology for Mechanical Metrology (Grant No. 22011044) by KRISS.
Figure 1
-
A gr e at d e al of i nt er e st s urr o u n d s t h e u s e of tr a n s cr a ni al dir e ct c urr e nt sti m ul ati o n (t D C S) t o a u g m e nt c o g niti v e tr ai ni n g. H o w e v er, eff e ct s ar e i n c o n si st e nt a cr o s s st u di e s, a n d m et aa n al yti c e vi d e n c e i s mi x e d, e s p e ci all y f o r h e alt h y, y o u n g a d ult s. O n e m aj or s o ur c e of t hi s i n c o n si st e n c y i s i n di vi d u al diff er e n c e s a m o n g t h e p arti ci p a nt s, b ut t h e s e diff er e n c e s ar e r ar el y e x a mi n e d i n t h e c o nt e xt of c o m bi n e d tr ai ni n g/ sti m ul ati o n st u di e s. I n a d diti o n, it i s u n cl e ar h o w l o n g t h e eff e ct s of sti m ul ati o n l a st, e v e n i n s u c c e s sf ul i nt er v e nti o n s. S o m e st u di e s m a k e u s e of f oll o w- u p a s s e s s m e nt s, b ut v er y f e w h a v e m e a s ur e d p erf or m a n c e m or e t h a n a f e w m o nt hs aft er a n i nt er v e nti o n. H er e, w e utili z e d d at a fr o m a pr e vi o u s st u d y of t D C S a n d c o g niti v e tr ai ni n g [ A u, J., K at z, B., B u s c h k u e hl, M., B u n arj o, K., S e n g er, T., Z a b el, C., et al. E n h a n ci n g w or ki n g m e m or y tr ai ni n g wit h tr a n scr a ni al dir e ct c urr e nt sti m ul ati o n. J o u r n al of C o g niti v e N e u r os ci e n c e, 2 8, 1 4 1 9 – 1 4 3 2, 2 0 1 6] i n w hi c h p arti ci p a nts tr ai n e d o n a w or ki n g m e m or y t as k o v er 7 d a y s w hil e r e c ei vi n g a cti v e or s h a m t D C S. A n e w, l o n g er-t er m f oll o w- u p t o a ss es s l at er p erf or m a n c e w a s c o n d u ct e d, a n d a d diti o n al p arti ci p a nt s w er e a d d e d s o t h at t h e s h a m c o n diti o n w a s b ett er p o w er e d. W e a s s e s s e d b a s eli n e c o g niti v e a bilit y, g e n d er, tr ai ni n g sit e, a n d m oti v ati o n l e v el a n d f o u n d si g nifi c a nt i nt er a cti o ns b et w e e n b ot h b as eli n e a bilit y a n d m oti v ati o n wit h c o n diti o n ( a cti v e or s h a m) i n m o d els pr e di cti n g tr ai ni n g g ai n. I n a d diti o n, t h e i m pr o v e m e nt s i n t h e a cti v e c o nditi o n v er s u s s h a m c o n diti o n a p p e ar t o b e st a bl e e v e n a s l o n g a s a y e ar aft er t h e ori gi n al i nt er v e nti o n. ■more » « less
-
Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M , g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.more » « less
-
Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t ) - h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0305004118000087
