Abstract: Abstract In this paper, by using the \(G_{2(2)}\)-structure on \(\hbox {Im}(\mathbf{Ca'})\cong {\mathbb {R}}^{3,4}\) of the purely imaginary Cayley’s split-octaves \(\mathbf{Ca'}\), the \(G_{2(2)}\)-bi-normal motion of curves \(\gamma _t(s)\) in the pseudo-Euclidean space \({\mathbb {R}}^{3,4}\) is studied. The motion is closely related to Schrödinger flows into homogeneous pseudo-Riemannian manifolds \({\mathbb {S}}^{2,4}(1)=G_{2(2)}/SU(1,2)\) of signature (2, 4) and \({\mathbb {S}}^{3,3}(\sqrt{-1})=G_{2(2)}/SL(3,{\mathbb {R}})\) of signature (3, 3), in which \({\mathbb {S}}^{2,4}(1)\) (resp. \({\mathbb {S}}^{3,3}(\sqrt{-1}))\) admits the almost complex (resp. para-complex) structure. Furthermore, the motion of spacelike (resp. timelike) curves is also shown to be equivalent to the nonlinear Schrödinger-like system (resp. the nonlinear coupled heat equations) in three unknown functions, which generalizes the correspondence between the bi-normal motion of timelike (resp. spacelike) curves in \({\mathbb {R}}^{2,1}\) and the defocusing nonlinear Schrödinger equation (resp. the nonlinear heat equation). To show this correspondence, \(G_{2(2)}\)-frame field on the curve is used. PubDate: 2020-03-12

Abstract: Abstract We develop a link between degree estimates for rational sphere maps and compressed sensing. We provide several new ideas and many examples, both old and new, that amplify connections with linear programming. We close with a list of ten open problems. PubDate: 2020-02-11

Abstract: Abstract We solve the Levi-flat Plateau problem in the following case. Let \(M \subset {\mathbb {C}}^{n+1}\), \(n \ge 2\), be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose M is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in \({\mathbb {C}}^n \times {\mathbb {R}}\) with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary M. We also study boundary regularity of CR automorphisms of domains in \({\mathbb {C}}^n \times {\mathbb {R}}\). PubDate: 2020-01-13

Abstract: Abstract Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. More precisely, there are three different, but related, types of envelopes associated to most families of curves. In this paper, we illustrate how one can use these envelopes to approach problems appearing in fields like matrix theory and hyperbolic geometry. As a first example, we use envelopes to fill in the details of a proof by Donoghue of the elliptical range theorem; this use of envelopes provides insight into how certain numerical ranges arise from families of circles. More generally, envelopes can be used to compute boundaries of families of curves. To demonstrate how such arguments work, we first deduce a general relationship between the envelopes and boundaries of families of circles and then use this to compute the boundaries of several families of pseudohyperbolic disks. PubDate: 2020-01-08

Abstract: Abstract We present an explicit formula for the leading coefficient in the asymptotic expansion of the eigenvalue counting function of the Kohn Laplacian on the unit sphere \({\mathbb {S}}^{2n-1}\). PubDate: 2020-01-02

Abstract: Abstract We prove that for every \(p>1\) the set $$\begin{aligned} D_p=\left\{ c\in {{\mathbb{C}}}\;\big \; \forall z\in \overline{{{\mathbb{D}}}},\; \left \frac{1+z}{2}+ c\left( \frac{1-z}{2}\right) ^p\right \le 1\right\} \end{aligned}$$is an intersection of closed disks, in particular a closed convex set, and exactly a disk for \(p=2\). It is also shown that $$\begin{aligned} {{\mathbb{D}}}\cup \{1\}\subseteq \bigcup _{p\ge 2} D_p\subseteq \overline{{{\mathbb{D}}}}. \end{aligned}$$ PubDate: 2019-11-19

Abstract: Abstract We prove characterizations of Goldberg’s local Hardy space \(h^1({\mathbb {R}})\) by means of a local Hilbert transform and a molecular decomposition. We use this decomposition to prove a version of Hardy’s inequality for the Fourier transform of functions in this space. PubDate: 2019-04-05

Abstract: Abstract In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank. PubDate: 2019-03-25

Abstract: Abstract The Hall-magnetohydrodynamics system has a long history of various applications in physics and engineering, in particular magnetic reconnection, star formation and the study of the sun. Mathematically, the additional Hall term elevates the level of non-linearity to the quasi-linear type while similar systems of equations such as the Navier–Stokes equations, as well as the viscous and magnetically diffusive magnetohydrodynamics system are of the semi-linear type. Consequently, the mathematical analysis on the Hall-magnetohydrodynamics system had been relatively absent for more than half a century, and only started seeing rapid developments in the last few years. In this manuscript, we survey recent progress on the mathematical analysis of the Hall-magnetohydrodynamics system, both in the deterministic and stochastic cases, elaborating on the structure of the Hall term. We list some remaining open problems, and discuss their difficulty. As a byproduct of our discussion, it is deduced that the two and a half dimensional Hall-magnetohydrodynamics system with viscous diffusion in the form of a Laplacian and the magnetic diffusion in the form of a fractional Laplacian with power \(\frac{3}{2}\) admits a unique smooth solution for all time. PubDate: 2019-03-25

Abstract: Abstract In this paper we consider the following initial value problem $$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t - \Delta _x + w\cdot \nabla _x +\alpha I) u(t,x) - \beta u(t-\tau ,x) = 0, &{} (t,x)\in {\mathbb {R}}_+\times {\mathbb {R}}^n \\ u(t,x) = \phi (t,x)\in C([-\tau ,0],L^1({\mathbb {R}}^n)) &{} \end{array} \right. \end{aligned}$$ We describe the ranges of the parameters \(\alpha , \beta ,\tau \) that guarantee that every solution u(t, x) vanishes as \(t\rightarrow \infty \) . PubDate: 2019-03-25

Abstract: Abstract For a locally compact Abelian group G, the algebra of Rajchman measures, denoted by \({M}_{0}(G)\) , is the set of all bounded regular Borel measures on G Fourier transform of which vanish at infinity. In this paper, we investigate the spectral structure of the algebra of Rajchman measures, and illustrate aspects of the residual analytic structure of its maximal ideal space. In particular, we show that \({M}_{0}(G)\) has a nonzero continuous point derivation, whenever G is a nondiscrete locally compact Abelian group. We then give the definition of the Rajchman algebra for a general (not necessarily Abelian) locally compact group, and prove that for a noncompact connected SIN group, the Rajchman algebra admits a nonzero continuous point derivation. Moreover, we discuss the analytic behavior of the spectrum of \({M}_{0}(G)\) . Namely, we show that for every nondiscrete metrizable locally compact Abelian group G, the maximal ideal space of \({M}_{0}(G)\) contains analytic disks. PubDate: 2019-03-25

Abstract: Abstract Regularity of a compact set is important in Pluripotential Theory, Approximation Theory, and in the study of separately analytic functions. Recently, in his paper Topics in Several Complex Variables, Contemporary Mathematics, vol. 662, pp. 145–156. American Mathematical Society, Providence, [16], Sadullaev introduced regularity notions in projective spaces, and weighted regularity notions in \({\mathbb {C}}^{n}\) . In this paper, we discuss weighted regularity notions in Lelong classes and answer two open problems. PubDate: 2019-03-25

Abstract: Abstract We obtain asymptotics of sequences of the holomorphic sections of the pluricanonical bundles on ball quotients associated to closed geodesics. A nonvanishing result for relative Poincaré series follows. PubDate: 2019-03-25

Abstract: Abstract One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or \(C^{1,\alpha }\) images of Euclidean subspaces, including theorems of Azzam–Tolsa, Badger–Schul, Badger–Vellis, Edelen–Naber–Valtorta, Ghinassi, and Tolsa–Toro. PubDate: 2019-03-06

Abstract: Abstract The geometric mean of a pair of oriented, meromorphic foliations on a Riemann surface is an unoriented foliation; the quadratic differential defining the latter is the product of the linear differentials defining the former. For instance, geometric means of pairs of oriented circle pencils in the Riemann sphere may be used to represent confocal families of (Euclidean or non-Euclidean) conics. PubDate: 2018-11-08

Abstract: Abstract We prove some characterizations of Schatten class of Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents. Our results are applied to obtain Schatten class characterization of some Cesàro-type operators. PubDate: 2018-08-20

Abstract: Abstract In this paper, we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics and the number of zeros of harmonic polynomials. PubDate: 2018-03-07

Abstract: Abstract We study the existence of Fatou components on parabolic skew product maps. We focus on skew products in which each coordinate has a fixed point that is parabolic. As in the geometrically attracting case, we prove that there exists maps F that have one-dimensional disks that are mapped to a point in the Julia set of the restriction of F to an invariant one-dimensional fiber. We first prove a linearization theorem for a one-dimensional map, then for a parabolic skew product. Finally, we apply this result to construct the skew product map described above. PubDate: 2018-02-09