arXiv research

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient o…

The paper proves conditions for compact complex manifolds to be Kahler outside analytic subsets.

We construct a family of analytic discs attached to a real submanifold M \subset $\mathbb{C}^{N+1}$ of codimension $2$ defined near a CR singularity.

Study proves rigidity results for analytic spacetimes without boundary or timelike boundary.

The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $σZ_ω$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies t…

We show that every complete metric space is homeomorphic to the precise locus of zeros of an entire analytic map from a Hilbert space to a Banach space. As a corollary, every complete separable metric space is homeomorphic to the precise locus of zeros of an entire analytic map between two separable complex Hilbert spa…

Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.

This paper describes two real analytic symplectomorphisms defined on appropriate dense open subsets of any coadjoint orbit of a compact semisimple Lie algebra. The first symplectomorphism sends the open dense subset to a bounded subset of a standard cotangent bundle. The second symplectomorphism has target a bounded su…

Let $X$ be a real analytic orbifold. Then each stratum of $X$ is a subanalytic subset of $X$. We show that $X$ has a unique subanalytic triangulation compatible with the strata of $X$. We also show that every ${\rm C}^r$-orbifold, $1\leq r\leq \infty$, has a real analytic structure. This allows us to triangulate differ…

Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,μ}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $Γ$, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real-analytic in the case $N$ is real-anal…

The paper defines wave-front singularities using explicit analytic functions.

It is a theorem of S. Bando that if $g(t)$ is a solution to the Ricci flow on a compact manifold $M$, then $(M, g(t))$ is real-analytic for each $t >0$. In this note, we extend his result to smooth solutions on open domains $U\subset M$.

Abstract: Determines thermoelastic coefficients from boundary data.

Loss Data Analytics is an interactive, online, freely available text. The idea behind the name Loss Data Analytics is to integrate classical loss data models from applied probability with modern analytic tools. In particular, we seek to recognize that big data (including social media and usage based insurance) are here…

Proves Kähler-Ricci shrinkers are complex analytic varieties.

We prove that if $K$ is a compact subset of an affine variety O = P^n - D (where D is a projective hypersuface), and if K is a compact subset of a closed analytic subvariety V \subset O, then the projective hull K^ of K has the property that K^ \cap O is contained in V. If V is smooth and 1-dimensional, then K^ \cap O …

Let $Ω\subset R^2$ be a bounded piecewise smooth domain and $φ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{φ_λ} = {x \in Ω; φ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $n(λ,H):= \# (H \cap N_{φ_λ} ).$ We first prove that fo…

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have tubular neighbou…

For the link $M$ of a normal complex surface singularity $(X,0)$ we ask when a knot $K\subset M$ exists for which the answer to whether $K$ is the link of the zero set of some analytic germ $(X,0)\to (\mathbb C,0)$ affects the analytic structure on $(X,0)$. We show that if $M$ is an integral homology sphere then such a…

Equivalences between conformal foliations on Euclidean $3$-space, Hermitian structures on Euclidean $4$-space, shear-free ray congruences on Minkowski $4$-space, and holomorphic foliations on complex $4$-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …

Real analytic functions can be extended on manifolds with normal crossings.

We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere f…

We use twistor theory to identify the harmonic hull of an arbitrary connected open subset U of R^{2m} for m at least 2. It is the natural domain of analytic continuation in C^{2m} for harmonic functions on U.

Geometrically studies Moore-Penrose inverse and polar decomposition continuity.

Catenoids in de Sitter $3$-space $S^3_1$ belong to a certain class of space-like constant mean curvature one surfaces. In a previous work, the authors classified such catenoids, and found that two different classes of countably many exceptional elliptic catenoids are not realized as closed subsets in $S^3_1$. Here we s…

In Divide & Recombine (D&R), big data are divided into subsets, each analytic method is applied to subsets, and the outputs are recombined. This enables deep analysis and practical computational performance. An innovate D\&R procedure is proposed to compute likelihood functions of data-model (DM) parameters for big dat…

We prove that if $D\subset C^n$ is a bounded domain with real analytic boundary and D is pseudoconvex then the compact open topology in the group of holomorphic automorphisms of D is the topology of uniform convergence on D.

The study bounds quantum eigenfunctions on complex manifolds.

Null geodesics in Kerr spacetimes cannot be closed or bounded.

In this paper, we prove that if $g(t)$ is a smooth, complete solution to the Ricci flow of uniformly bounded curvature on $M\times[0, Ω]$, then the correspondence $t\mapsto g(t)$ is real-analytic at each $t_0\in (0, Ω)$. The analyticity is a consequence of classical Bernstein-type estimates on the temporal and spatial …

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \su…

With rapidly increasing data, clustering algorithms are important tools for data analytics in modern research. They have been successfully applied to a wide range of domains; for instance, bioinformatics, speech recognition, and financial analysis. Formally speaking, given a set of data instances, a clustering algorith…

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifo…

Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and Σa semianalytic subset of X. Then the closure of Σin X with respect to the canonical topology is again semianalytic. The proof uses Embedded Resolution of Singularities.

If (M,g) is a Riemannian manifold and x,y are points in M, then a subset P of M\{x,y} is said to be a blocking set for (x,y) if every geodesic from x to y passes through a point of P. If no pair (x,y) in M X M has a finite blocking set, then (M,g) is said to be totally insecure. We prove that there exist real analytic …

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex manifold) is uniquely determined. If M is regular and the…

Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two notions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle ${}^bTX$. The usual definition of smooth structure uses $TX$, as $f:X\to\mathbb{R}$ is defined to be …

We prove in this article that given a linearly concave domain $D$ in the projective space $\Bbb{CP}^{n}$, a 1-dimensional comlex analytic set $V$ in $D$, and a meromorphic 1-form $φ$ on $V$, $V$ is a subset of an algebraic variety of $\Bbb{CP}^{n}$ and $φ$ is the restriction to $V$ of an algebraic 1-form on $\Bbb{CP}^{…

We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between th…

EBBS integrates expert assessments into MIO best-subsets problem.

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the …

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where…

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f\colon (M,p)\to \mathbb S$ and $d < n/2$, then $f$ is {\em rigid} in the sense t…

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial ψ_{1} = (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{2} \\ &\bar{\partial} ψ_{2} =- (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{1} \end{split} \] descripti…

We explain a new phenomenon on non compact complete Riemannian four manifolds, where d^+ image of one forms can not exhaust densely on L^2 self dual forms on each compact subset, if a certain L^2 self dual harmonic form exists. This leads to construct a new functional analytic framework on the Seiberg-Witten map.

We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $…