Related papers: Chebyshev-type cubature formulas for doubling weig…
The double copy relates scattering amplitudes and classical solutions in non-abelian gauge theories and gravity. As such, it is usually expressed in the conventional second-order formalisms in both theories corresponding to standard…
73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than…
We prove scattering for the 2D cubic derivative Schr\"odinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate…
We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $\R^d$, equipped with power weights $w(x) = |x|^\gamma$, $\gamma>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we…
The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this…
Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating…
The Dvoretzky--Kiefer--Wolfowitz (DKW) inequality says that if $F_n$ is an empirical distribution function for variables i.i.d.\ with a distribution function $F$, and $K_n$ is the Kolmogorov statistic $\sqrt{n}\sup_x|(F_n-F)(x)|$, then…
We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the…
Spherical needlets are highly localized radial polynomials on the sphere $\mathbb{S}^{d}\subset \mathbb{R}^{d+1}$, $d\ge 2$, with centers at the nodes of a suitable cubature rule. The original semidiscrete spherical needlet approximation of…
The role of charge order in the phase diagram of high temperature cuprate superconductors has been recently re-emphasized by the experimental discovery of an incipient bi-directional charge density wave (CDW) phase in a class of underdoped…
Suggestive, but indirect evidence of the existence of pair-density wave (PDW) order in several high-$T_{\rm c}$ cuprates has been reported. As this constitutes a new quantum phase of matter, it is important to {\it establish} its existence…
Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_\sigma(v)= Kv^{\frac{n+2\sigma}{n-2\sigma}}$ on…
We introduce a new type of cubature formula for the evaluation of an integral over the disk with respect to a weight function. The method is based on an analysis of the Fourier series of the weight function and a reduction of the bivariate…
Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[…
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct…
We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with…
Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the…
We consider the Sobolev space over $\mathbb{R}^d$ of square integrable functions whose gradient is also square integrable with respect to some positive weight. Tt is well known that smooth functions are dense in the weighted Sobolev space…
Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf C_{\mathsf S}$…
We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least…