Related papers: Degree counting for Toda system with simple singul…

We explore the dynamics of FLRW cosmologies which consist of dark matter, radiation and dark energy with a quadratic equation of state. Standard cosmological singularities arise due to energy conditions which are violated by dark energy,…

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet…

An elementary, at the undergraduate level derivation is given of precise upper bounds of the number of various RNA secondary structures. The method works when the generating function has multiple singularities at the circle of convergence,…

Blowup equations and holomorphic anomaly equations are two universal yet completely different approaches to solve refined topological string theory on local Calabi-Yau threefolds corresponding to A- and B-model respectively. The former…

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case…

We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems…

We establish upper bounds on the blow up rate of the gradients of solutions of the Lam\'e system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system…

In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% \rho_{t}+k_{2}u\rho_{x}+(k_{1}+k_{2})\rho u_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\rho\rho_{x}=0. By the…

Interrelation between Thom's catastrophes and differential equations revisited. It is shown that versal deformations of critical points for singularities of A,D,E type are described by the systems of Hamilton-Jacobi type equations. For…

In this paper we consider the one dimensional spring-block model describing earthquake faulting. By using geometric singular perturbation theory and the blow-up method we provide a detailed description of the periodicity of the earthquake…

This paper studies a slow-fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for…

We classify all possible singularities in the electronic dispersion of two-dimensional systems that occur when the Fermi surface changes topology, using catastrophe theory. For systems with up to seven control parameters (i.e., pressure,…

This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated…

We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity…

We consider an area-minimizing integral current of dimension $m$ and codimension at least $2$ and fix an arbitrary interior singular point $q$ where at least one tangent cone is flat. For any vanishing sequence of scales around $q$ along…

A sharp signature of the black hole singularity in holography is a divergence in the boundary thermal two-point function at a specific point in the complex time plane. This divergence arises from a null geodesic that bounces off the black…

We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of…

We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first…

In this paper we consider the so-called Toda system of equations on a compact surface. In particular, we discuss the parity of the Leray-Schauder degree of that problem. Our main tool is a theorem of Krasnoselskii and Zabreiko on the degree…

This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $\Omega$. While, for a…