Related papers: Stone-Weierstrass Theorem
The Lovelock gravity extends the theory of general relativity to higher dimensions in such a way that the field equations remain of second order. The theory has many constant coefficients with no a priori meaning. Nevertheless it is…
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature.…
We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erd\H{o}s-Stone theorem. We also provide means…
We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$…
The uncertainty principle constitutes one of the famous physical concepts which continues to attract researchers from different related fields since its discovery due to its utility in many applications. Among the classical (Fourier-based)…
Under Gromov--Hausdorff convergence, and equivariant Gromov--Hausdorff convergence, we prove stability results of Wasserstein spaces over certain classes of singular and non-singular spaces. For example, we obtain an analogue of Perelman's…
A Liouville type theorem is proven for the steady-state Navier-Stokes equations. It follows from the corresponding theorem on the Stokes equations with the drift. The drift is supposed to belong to a certain Morrey space.
A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional…
Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra…
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…
In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this…
The disc property is formulated for domains in $\mathbb{C}^n$. Holomorphic Lipschitz functions enjoy a gain in the order of Lipschitz regularity along the complex tangential direction on domains with disc property. Disc property is studied…
We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose $h : \mathbb{R}^2 \to\mathbb{R}^2$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$…
A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all…
We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class…
At higher energies the present complex quantum theory with its unitary group might expand into a real quantum theory with an orthogonal group, broken by an approximate $i$ operator at lower energies. Implementing this possibility requires a…
In this paper we give a metric construction of a tree which correctly identifies connected components of superlevel sets of $\mathbb{R}$-valued continuous functions $f$ on $X$ and show that it is possible to retrieve the $H_0$-persistent…
Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity.…
Stokes theorem holds for Lipschitz forms on a smooth manifold.
The possibility of physics in multiple time dimensions is investigated. Drawing on recent work by Walter Craig and myself, I show that, contrary to conventional wisdom, there is a well-posed initial value problem--deterministic, stable…