Related papers: Normal form of swallowtail and its applications

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties,…

We use a factorization technique and representation of canonical transformations to construct globally valid closed form expressions without singularities of semi-classical wave functions for arbitrary smooth potentials over a…

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…

We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

Let f : X -> Y be a morphism between normal complex varieties, and assume that Y is Kawamata log terminal. Given any differential form, defined on the smooth locus of Y, we construct a "pull-back form" on X. The pull-back map obtained by…

Moduli space of the Bogomolny equation on R^3 with certain asymptotic conditions at infinity has been well studied for a long time. This paper studies the moduli space of solutions to the Bogomolny equation on R^3 with a knot singularity.…

In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves…

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha_p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable)…

This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a…

In this note, we extend to the singular case some results on the birational geometry of irreducible holomorphic symplectic manifolds.

We discuss a particular class of rational Gorenstein singularities, which we call symplectic. A normal variety V has symplectic singularities if its smooth part carries a closed symplectic 2-form whose pull-back in any resolution X --> V…

We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal…

This paper is the second part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a…

We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a…

Characterizing face-number-related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regard, one of the well-known invariants is $g_2$. Let $K$ be a normal $3$-pseudomanifold…

We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it…

We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…

In this paper, we want to give an exposition of our recent work on linear and nonlinear potential theory and their applications in conformal geometry. We use potential theory to study linear and quasilinear equations arising from conformal…

This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how…