Related papers: A proof of Sylvester's theorem
We prove an abstract linking theorem that can be used to show existence of solutions to various types of variational elliptic equations, including Schr\"{o}dinger--Poisson--Slater type equations.
A proof of Sendov's conjecture is given.
We consider the eigenvalues and eigenvectors of an axisymmetric matrix$A$ with some special structures. We propose S-Oja-Brockett equation $\frac{dX}{dt}=AXB-XBX^TSAX,$ where $X(t) \in {\mathbb R}^{n \times m}$ with $m \leq n$, $S$ is a…
We study the rational solution of the Yang-Baxter equation with the supersymmetry algebra sl(2|1). The R-matrix acting in the tensor product of two arbitrary representations of the supersymmetry algebra can be represented as the product of…
In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the…
We give a rigorous proof for the linear stability of the Skyrmion. In addition, we provide new proofs for the existence of the Skyrmion and the GGMT bound.
We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of…
W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B.…
We prove the existence of axially symmetric solutions to the Vlasov--Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi's integral and the proof relies on an implicit…
We use the elimination theory to explicitly construct the (n-3)! order polynomial in one of the variables of the scattering equations. The answer can be given either in terms of a determinant of Sylvester type of dimension (n-3)! or a…
We find the all solutions to the $sl_q(2)$-invariant multi-parametric Yang-Baxter equations (YBE) at $q=i$ defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We are deriving the solutions in form of the linear…
Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $\pi: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is…
The Maxey--Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing…
We present a short and completely elementary proof for a double sum studied by Brent and Osburn in arXiv:1309.2795v2.
We provide in this article a new proof of the uniqueness of the flow solution to ordinary differential equations with $BV$ vector-fields that have divergence in $L^\infty$ (or in $L^1$) and that are nearly incompressible (see the text for…
We give a direct proof of the fact that elliptic solutions of the associative Yang-Baxter equation arise from appropriate spherical orders on an elliptic curve.
We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of…
We proved three theorems of $S$-version of the mulyiplicity one.
We prove the existence of positive solutions for a class of semipositone problem with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.
It is shown that all strongly symmetric elements are solutions of constant classical Yang-Baxter equation in Lie algebra, or of quantum Yang-Baxter equation in algebra. Otherwise, all solutions of constant classical Yang-Baxter equation…