Related papers: Analytical Gomboc
Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
End sum is a natural operation for combining two noncompact manifolds and has been used to construct various manifolds with interesting properties. The uniqueness of end sum has been well-studied in dimensions three and higher. We study end…
We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…
We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a M\"obius automorphism group of dimension at least two. Our theorem…
This work deals with the presence of topological structures in models of two real scalar fields in the two-dimensional spacetime. The subject concerns the presence of a geometric constriction, which appears with a modification of the…
I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian…
We consider a holomorphic 1-form $\omega$ with an isolated zero on an isolated complete intersection singularity $(V,0)$. We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair…
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…
Given recipe of qualitative, kinetic modelling by geometric methods of three-dimensional dendritic crystals. Characteristic features of the perturbations appearing on the surface of a spherical body, leading to different scenarios of the…
I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a…
In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to…
The structure equations for a surface are introduced and two required results based on the Codazzi equations are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem of Bonnet is obtained. A…
The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…
We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second…
We determine an explicit triangular integral basis for any separable cubic extension of a rational function field over a finite field in any characteristic. We obtain a formula for the discriminant of every such extension in terms of a…
The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue…
This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in arXiv:1304.0708 to global function fields of odd characteristics. First, we present algorithm for checking if a given…
The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied…
We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.