The Topological Correctness of PL Approximations of Isomanifolds

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$ f : R d → R d - n . A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ T of the ambient space $${\mathbb {R}}^d$$ R d . In this paper, we give conditions under which the PL approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine and thick triangulation $$\mathcal {T}$$ T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary.

Extreme Values of Euler-Kronecker Constants

In a family of Sn -fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log dK + O(log log log dK ) and loglog dK + O(log log log dK ), resp., where dK is the absolute value of the discriminant of a number field K.

A note on extensions of multilinear maps defined on multilinear varieties

Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

Low-Degree Approximation of Random Polynomials

We prove that with “high probability” a random Kostlan polynomial in $$n+1$$ n + 1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere $$\mathbb {S}^n$$ S n . The dependence between the “low degree” of the approximation and the “high probability” is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree $$O(\sqrt{d \log d})$$ O ( d log d ) . The proof is based on a probabilistic study of the size of $$C^1$$ C 1 -stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in $${\mathbb {R}}^3$$ R 3 , there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue $$+1$$ + 1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 1 / 3 involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ ϵ → 0 + , where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .

On the spectrality of self-affine measures with four digits on ℝ2

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

<p style='text-indent:20px;'>We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" <i>Analysis &amp; PDE</i>, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.</p>