Related papers: Weak Approximation for Cubic Hypersurfaces of Larg…

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic…

We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard…

The sparse representation of signals defined on Euclidean domains has been successfully applied in signal processing. Bringing the power of sparse representations to non-regular domains is still a challenge, but promising approaches have…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we…

Generalizing the well-known relations on characteristic functions on a plane to the case of a one-dimensional regular surface (curve) with compact support, we establish implicit equations for these functions. Introducing an approximation,…

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci…

In this paper, by using the atomic decomposition theory of weighted Hardy spaces, we will give some weighted weak type estimates for intrinsic square functions including the Lusin area function, Littlewood-Paley $g$-function and…

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently…

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ in dimensions $n\ge 3$. These conjectures ask whether a complete minimal hypersurface must be unbounded, and more…

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…

We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite…

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, the authors introduce the weak Hardy-type space $WH_X({\mathbb R}^n)$, associated with $X$, via the radial maximal function. Assuming that the powered…

We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension $3\leq n+1\leq 7$. Furthermore, we show that bumpy metrics with positive Ricci…

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces…

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(t\xi)=t\varrho(\xi)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $\delta>0$, we consider convolution operator ${\Cal…

We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the…

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that…