It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
Given a suitable solution
 Award ID(s):
 1856755
 NSFPAR ID:
 10381012
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Communications in Mathematical Physics
 Volume:
 397
 Issue:
 3
 ISSN:
 00103616
 Page Range / eLocation ID:
 p. 13871439
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 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 In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator
are raised to the power$$\Delta +V(x)$$ $\Delta +V\left(x\right)$ is never given by the onebound state case when$$\kappa $$ $\kappa $ in space dimension$$\kappa >\max (0,2d/2)$$ $\kappa >max(0,2d/2)$ . When in addition$$d\ge 1$$ $d\ge 1$ we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021.$$\kappa \ge 1$$ $\kappa \ge 1$https://doi.org/10.1007/s00205021016347 ). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function. 
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$$(\log n)^{O(n)}$$ ${(logn)}^{O\left(n\right)}$ .$$n^n \cdot 2^{O(n)}$$ ${n}^{n}\xb7{2}^{O\left(n\right)}$ 
Abstract Let
be an elliptically fibered$$X\rightarrow {{\mathbb {P}}}^1$$ $X\to {P}^{1}$K 3 surface, admitting a sequence of Ricciflat metrics collapsing the fibers. Let$$\omega _{i}$$ ${\omega}_{i}$V be a holomorphicSU (n ) bundle overX , stable with respect to . Given the corresponding sequence$$\omega _i$$ ${\omega}_{i}$ of Hermitian–Yang–Mills connections on$$\Xi _i$$ ${\Xi}_{i}$V , we prove that, ifE is a generic fiber, the restricted sequence converges to a flat connection$$\Xi _i_{E}$$ ${\Xi}_{i}{}_{E}$ . Furthermore, if the restriction$$A_0$$ ${A}_{0}$ is of the form$$V_E$$ ${V}_{E}$ for$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j0)$$ ${\oplus}_{j=1}^{n}{O}_{E}({q}_{j}0)$n distinct points , then these points uniquely determine$$q_j\in E$$ ${q}_{j}\in E$ .$$A_0$$ ${A}_{0}$ 
Abstract We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting
S (a ,b ) be the number of families observed in (a ,b ), we study the expected sample variance and its asymptotics forp consecutive sequential samples , for$$S_p =(S(t_0,t_1),\dots , S(t_{p1},t_p))$$ ${S}_{p}=(S({t}_{0},{t}_{1}),\cdots ,S({t}_{p1},{t}_{p}))$ . By conditioning on the sizes of the samples, we provide a connection between$$0=t_0 $0={t}_{0}<{t}_{1}<\cdots <{t}_{p}$ and$$S_p$$ ${S}_{p}$p sequential samples of sizes , drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166–1194, 2023.$$n_1,n_2,\dots ,n_p$$ ${n}_{1},{n}_{2},\cdots ,{n}_{p}$https://doi.org/10.3150/22BEJ1494 ). We show how the continuoustime framework helps to make asymptotic calculations easier than its discretetime counterpart. As an application, for a specific choice of , where the lengths of intervals are logarithmically equal, we revisit Fisher’s 1943 multisampling problem and give another explanation of what Fisher’s model could have meant in the world of sequential samples drawn from a BI process.$$t_1,t_2,\dots , t_p$$ ${t}_{1},{t}_{2},\cdots ,{t}_{p}$