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In this letter we present the first multiparticle solutions to Einstein's field equations in the presence of matter. These solutions are iteratively obtained via the perturbiner method, which can circumvent gravity's infinite number of…

In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both…

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on…

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence)…

This paper is an expansion of my lecture for David Epstein's birthday, which traced a logical progression from ideas of Euclid on subdividing polygons to some recent research on invariants of hyperbolic 3-manifolds. This `logical…

We study a generalization of the Fuchsian triangle groups to the hyperbolic 3-space, namely, the groups generated by half-turns in three hyperbolic lines. The role of the hyperbolic triangles is now played by the right-angled hexagons. This…

We apply discrete algebraic Morse theory to calculate the Anick resolution of the group algebra of the group $G_3^2$. As a corollary, we evaluate Hochschild cohomologies of $G_3^2$ with coefficients in all 1-dimensional bimodules. Almost…

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun")…

We present an alternative proof of the Alexander-Hirschowitz Theorem in dimension 3 using degenerations of toric varieties.

We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

We develop the theory of Riemann-Hilbert problems necessary for the results in part one of this series of papers. In particular, we obtain solutions for a family of non-linear Riemann-Hilbert problems through classical contraction…

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same…

In this article, we first consider solutions to a semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in \Omega,\\ u=0,\ \ &x\in\partial…

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…

In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact…

This paper aims to expand on the open case $k=n$ regarding Proposition 3.6[1] and hopefully foster curiosity for its resolution.

This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repov\v{s}, Yamabe-type…

We modify a three-field formulation of the Poisson problem with Nitsche approach for approximating Dirichlet boundary conditions. Nitsche approach allows us to weakly impose Dirichlet boundary condition but still preserves the optimal…

We present novel approaches for solving Milne's equation, which was introduced in 1930 as an efficient numerical scheme for the Schr\"odinger equation. Milne's equation appears in a wide class of physical problems, ranging from astrophysics…