Related papers: Fermat's and Catalan's equations over $M_2(\mathbb…

In this paper, the necessary and sufficient conditions in order that a smooth mapping F be a dependence of a complete solution of some second-order ordinary differential equation on Neumann conditions are deduced. These necessary and…

The existence of the meromorphic solutions to Fermat type delay-differential equation \begin{equation} f^n(z)+a(f^{(l)}(z+c))^m=p_1(z)e^{a_1z^k}+p_2(z)e^{a_2z^k}, \nonumber \end{equation} is derived by using Nevanlinna theory under certain…

Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof.…

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian…

The discrete Fourier transform matrix is one of the most important matrices in linear algebra, and submatrices of it arise in a variety of applications. Though the discrete Fourier transform matrix is unitary, its submatrices can be…

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…

We show that the Standard Model Yukawa matrices satisfy a set of simple yet nontrivial inequalities. The relations we derive are independent of the basis used to define the fermion fields, and, amongst other things, place strong constraints…

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial…

This paper is concerned with the positive definite solutions to the matrix equation $X+A^{\mathrm{H}}\bar{X}^{-1}A=I$ where $X$ is the unknown and $A$ is a given complex matrix. By introducing and studying a matrix operator on complex…

For a degree 2n finite sequence of real numbers $\beta \equiv \beta^{(2n)}= \{ \beta_{00},\beta_{10}, \beta_{01},\cdots, \beta_{2n,0}, \beta_{2n-1,1},\cdots, \beta_{1,2n-1},\beta_{0,2n} \}$ to have a representing measure $\mu $, it is…

We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2…

In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the…

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no…

In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=\gamma\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is…

We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b\in L_{d}$ and $c\in L_{q}$, $c\geq0$, $d>q\geq d/2$. We prove the solvability of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains,…

This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally…

In this paper, we derive a formula to compute the solution of the linear matrix equation $X=Af(X)B+C$ via finding any solution of a specific Stein matrix equation $\mathcal{X}=\mathcal{A} \mathcal{X} \mathcal{B}+\mathcal{C}$, where the…

In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0,…

The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \[d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad (x,y\in \R,\,k\in\{0,\ldots,n\}) \] is studied, where…

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=\mu\ge…