Let
The shape of 3
 Award ID(s):
 2145080
 NSFPAR ID:
 10376187
 Publisher / Repository:
 Nature Publishing Group
 Date Published:
 Journal Name:
 Nature Communications
 Volume:
 13
 Issue:
 1
 ISSN:
 20411723
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract denote the standard Haar system on [0, 1], indexed by$$(h_I)$$ $\left({h}_{I}\right)$ , the set of dyadic intervals and$$I\in \mathcal {D}$$ $I\in D$ denote the tensor product$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ ,$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto {h}_{I}\left(s\right){h}_{J}\left(t\right)$ . We consider a class of twoparameter function spaces which are completions of the linear span$$I,J\in \mathcal {D}$$ $I,J\in D$ of$$\mathcal {V}(\delta ^2)$$ $V\left({\delta}^{2}\right)$ ,$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ . This class contains all the spaces of the form$$I,J\in \mathcal {D}$$ $I,J\in D$X (Y ), whereX andY are either the Lebesgue spaces or the Hardy spaces$$L^p[0,1]$$ ${L}^{p}[0,1]$ ,$$H^p[0,1]$$ ${H}^{p}[0,1]$ . We say that$$1\le p < \infty $$ $1\le p<\infty $ is a Haar multiplier if$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ , where$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D({h}_{I}\otimes {h}_{J})={d}_{I,J}{h}_{I}\otimes {h}_{J}$ , and ask which more elementary operators factor through$$d_{I,J}\in \mathbb {R}$$ ${d}_{I,J}\in R$D . A decisive role is played by theCapon projection given by$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta}^{2}\right)\to V\left({\delta}^{2}\right)$ if$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$ , and$$I\le J$$ $\leftI\right\le \leftJ\right$ if$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_{I}\otimes {h}_{J}=0$ , as our main result highlights: Given any bounded Haar multiplier$$I > J$$ $\leftI\right>\leftJ\right$ , there exist$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ such that$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$ i.e., for all$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})\text { approximately 1projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)\phantom{\rule{0ex}{0ex}}\text{approximately 1projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$ , there exist bounded operators$$\eta > 0$$ $\eta >0$A ,B so thatAB is the identity operator ,$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$ and$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert \xb7\Vert B\Vert =1$ . Additionally, if$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})  ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)ADB\Vert <\eta $ is unbounded on$$\mathcal {C}$$ $C$X (Y ), then and then$$\lambda = \mu $$ $\lambda =\mu $ either factors through$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$D or .$${{\,\textrm{Id}\,}}D$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}D$ 
Abstract A study of possible superconducting phases of graphene has been constructed in detail. A realistic tight binding model, fit to ab initio calculations, accounts for the Lidecoration of graphene with broken lattice symmetry, and includes
s andd symmetry Bloch character that influences the gap symmetries that can arise. The resulting seven hybridized LiC orbitals that support nine possible bond pairing amplitudes. The gap equation is solved for all possible gap symmetries. One band is weakly dispersive near the Fermi energy along Γ →M where its Bloch wave function has linear combination of and$${d}_{{x}^{2}{y}^{2}}$$ ${d}_{{x}^{2}{y}^{2}}$d _{xy}character, and is responsible for and$${d}_{{x}^{2}{y}^{2}}$$ ${d}_{{x}^{2}{y}^{2}}$d _{xy}pairing with lowest pairing energy in our model. These symmetries almost preserve properties from a two band model of pristine graphene. Another part of this band, alongK → Γ, is nearly degenerate with uppers band that favors extendeds wave pairing which is not found in two band model. Upon electron doping to a critical chemical potentialμ _{1} = 0.22eV the pairing potential decreases, then increases until a second critical valueμ _{2} = 1.3 eV at which a phase transition to a distorteds wave occurs. The distortion ofd  or swave phases are a consequence of decoration which is not appear in two band pristine model. In the pristine graphene these phases convert to usuald wave or extendeds wave pairing. 
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ $\Omega $ 
Abstract Perovskite oxides (ternary chemical formula ABO_{3}) are a diverse class of materials with applications including heterogeneous catalysis, solidoxide fuel cells, thermochemical conversion, and oxygen transport membranes. However, their multicomponent (chemical formula
) chemical space is underexplored due to the immense number of possible compositions. To expand the number of computed$${A}_{x}{A}_{1x}^{\text{'}}{B}_{y}{B}_{1y}^{\text{'}}{O}_{3}$$ ${A}_{x}{A}_{1x}^{\text{'}}{B}_{y}{B}_{1y}^{\text{'}}{O}_{3}$ compounds we report a dataset of 66,516 theoretical multinary oxides, 59,708 of which are perovskites. First, 69,407$${A}_{x}{A}_{1x}^{{\prime} }{B}_{y}{B}_{1y}^{{\prime} }{O}_{3}$$ ${A}_{x}{A}_{1x}^{\prime}{B}_{y}{B}_{1y}^{\prime}{O}_{3}$ compositions were generated in the$${A}_{0.5}{A}_{0.5}^{{\prime} }{B}_{0.5}{B}_{0.5}^{{\prime} }{O}_{3}$$ ${A}_{0.5}{A}_{0.5}^{\prime}{B}_{0.5}{B}_{0.5}^{\prime}{O}_{3}$a ^{−}b ^{+}a ^{−}Glazer tilting mode using the computationallyinexpensive Structure Prediction and Diagnostic Software (SPuDS) program. Next, we optimized these structures with density functional theory (DFT) using parameters compatible with the Materials Project (MP) database. Our dataset contains these optimized structures and their formation (ΔH _{f}) and decomposition enthalpies (ΔH _{d}) computed relative to MP tabulated elemental references and competing phases, respectively. This dataset can be mined, used to train machine learning models, and rapidly and systematically expanded by optimizing more SPuDSgenerated perovskite structures using MPcompatible DFT calculations.$${A}_{0.5}{A}_{0.5}^{{\prime} }{B}_{0.5}{B}_{0.5}^{{\prime} }{O}_{3}$$ ${A}_{0.5}{A}_{0.5}^{\prime}{B}_{0.5}{B}_{0.5}^{\prime}{O}_{3}$ 
Abstract Approximate integer programming is the following: For a given convex body
, either determine whether$$K \subseteq {\mathbb {R}}^n$$ $K\subseteq {R}^{n}$ is empty, or find an integer point in the convex body$$K \cap {\mathbb {Z}}^n$$ $K\cap {Z}^{n}$ which is$$2\cdot (K  c) +c$$ $2\xb7(Kc)+c$K , scaled by 2 from its center of gravityc . Approximate integer programming can be solved in time while the fastest known methods for exact integer programming run in time$$2^{O(n)}$$ ${2}^{O\left(n\right)}$ . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$2^{O(n)} \cdot n^n$$ ${2}^{O\left(n\right)}\xb7{n}^{n}$ can be found in time$$x^* \in (K \cap {\mathbb {Z}}^n)$$ ${x}^{\ast}\in (K\cap {Z}^{n})$ , provided that the$$2^{O(n)}$$ ${2}^{O\left(n\right)}$remainders of each component for some arbitrarily fixed$$x_i^* \mod \ell $$ ${x}_{i}^{\ast}\phantom{\rule{0ex}{0ex}}mod\phantom{\rule{0ex}{0ex}}\ell $ of$$\ell \ge 5(n+1)$$ $\ell \ge 5(n+1)$ are given. The algorithm is based on a$$x^*$$ ${x}^{\ast}$cuttingplane technique , iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new$$2^{O(n)}n^n$$ ${2}^{O\left(n\right)}{n}^{n}$asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equationstandard form . Such a problem can be reduced to the solution of$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ $Ax=b,0\le x\le u,\phantom{\rule{0ex}{0ex}}x\in {Z}^{n}$ approximate integer programming problems. This implies, for example that$$\prod _i O(\log u_i +1)$$ ${\prod}_{i}O(log{u}_{i}+1)$knapsack orsubsetsum problems withpolynomial variable range can be solved in time$$0 \le x_i \le p(n)$$ $0\le {x}_{i}\le p\left(n\right)$ . For these problems, the best running time so far was$$(\log n)^{O(n)}$$ ${(logn)}^{O\left(n\right)}$ .$$n^n \cdot 2^{O(n)}$$ ${n}^{n}\xb7{2}^{O\left(n\right)}$