Related papers: $l^2$ decoupling theorem for surfaces in $\mathbb{…

Let $\alpha\in(0,1)\setminus{\Bbb Q}$ and $K=\{(e^z,e^{\alpha z}):\,|z|\leq1\}\subset{\Bbb C}^2$. If $P$ is a polynomial of degree $n$ in ${\Bbb C}^2$, normalized by $\|P\|_K=1$, we obtain sharp estimates for $\|P\|_{\Delta^2}$ in terms of…

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the…

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations…

Accurate and fast 3D imaging of specular surfaces still poses major challenges for state-of-the-art optical measurement principles. Frequently used methods, such as phase-measuring deflectometry (PMD) or shape-from-polarization (SfP), rely…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle…

Generative models that produce point clouds have emerged as a powerful tool to represent 3D surfaces, and the best current ones rely on learning an ensemble of parametric representations. Unfortunately, they offer no control over the…

Elastic scattering governed by the Lame system associated with the third-type or fourth-type boundary condition is considered. It was shown in [8] by two of the authors that under suitable geometric conditions on the boundary surface of the…

Using a high/low argument, we prove a universal $\ell^2L^6$ decoupling estimate with constant $C_\epsilon R^{\epsilon}$ for general convex curves in the plane. These curves have no additional regularity assumptions, and the constant…

In the list-decodable learning setup, an overwhelming majority (say a $1-\beta$-fraction) of the input data consists of outliers and the goal of an algorithm is to output a small list $\mathcal{L}$ of hypotheses such that one of them agrees…

Sparse decomposition has been widely used for different applications, such as source separation, image classification and image denoising. This paper presents a new algorithm for segmentation of an image into background and foreground text…

The surface partition of large fragments is derived analytically within a simple statistical model by the Laplace-Fourier transformation method. In the limit of small amplitude deformations, a suggested Hills and Dales Model reproduces the…

We compute the entanglement entropy for some quantum field theories on de Sitter space. We consider a superhorizon size spherical surface that divides the spatial slice into two regions, with the field theory in the standard vacuum state.…

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p >…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting…

This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the…

We prove sharp $L^p-L^q$ estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve…