Related papers: Spherical rectangles

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

We construct new examples of bispectral dual Hahn polynomials, i.e., orthogonal polynomials with respect to certain superposition of Christoffel and Geronimus transforms of the dual Hahn measure and which are also eigenfunctions of a higher…

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families…

We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem…

We classify those curvature-homogeneous Einstein four-manifolds, of all metric signatures, which have a complex-diagonalizable curvature operator. They all turn out to be locally homogeneous. More precisely, any such manifold must be either…

In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…

A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…

We consider the steady Swift - Hohenberg partial differential equation. It is a one-parameter family of PDE on the plane, modeling for example Rayleigh - B\'enard convection. For values of the parameter near its critical value, we look for…

We prove \emph{uniform solvability estimates} for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2 = x3+/alpha x^2-n^2x.$ This…

The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional…

In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct…

We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean…

We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space corresponding to complex horseshoes. We discuss some conjectures about the global topology of the complex…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which…

We exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also analyze the…

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this…