Related papers: Singular solutions to the Seiberg-Witten and Freun…
We formulate Friedmann's equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $\beta$-times $t_\beta:=\int^t a^{-2\beta}$, where $a$ is the scale…
The purpose of these notes is to show that the methods introduced by Bauer and Furuta in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from Riemann…
It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the…
By extending a result of Kronheimer-Mrowka to the family setting, we prove a gluing formula for the family Seiberg-Witten invariant. This formula allows one to compute the invariant for a smooth family of 4-manifolds by cutting it open…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and…
Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional $\mathcal{N}=2$ SCFT. The Seiberg-Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the…
The bright soliton solutions and singular solutions are constructed for space-time fractional EW and modified EW equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and…
This paper focuses on investigation of the N-coupled Hirota equations arising in an optical fiber. Starting from analyzing the spectral problem, a kind of matrix Riemann-Hilbert problem is formulated strictly on the real axis. Then based on…
In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a…
The Seiberg--Witten map is a powerful tool in non-commutative field theory, and it has been recently obtained in the literature for gravity itself, to first order in non-commutativity. This paper, relying upon the pure-gravity form of the…
The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on…
In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations…
In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern \cite{Ch59}, we obtain the…
We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or generic versions of the key results of this approach for the case of positive energy and…
We prove a surgery formula for the ordinary Seiberg-Witten invariants, and surgery formulas for the families Seiberg-Witten invariants of families of $4$-manifolds obtained through fibrewise surgery. Our formula expresses the Seiberg-Witten…
By using the theory of analytic vectors and manifolds modelled on normed spaces, we provide a rigorous symplectic differential geometric approach to $t$-dependent Schr\"odinger equations on separable (possibly infinite-dimensional) Hilbert…
We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in $\mathbb{R}^3$ that satisfies an arbitrary elliptic Weingarten equation $W(\kappa_1,\kappa_2)=0$, and study the…
It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg-Witten (SW) monopole equations on Euclidean four-dimensional space R^4. We show that this is no longer true for the…
The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm]…