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In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…
We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.
The general solution to the quantum master equation (and its $Sp(2)$ symmetric counterpart) is constructed explicitly in case of finite number of variables. It is shown that the finite-dimensional solution is physically trivial and thus can…
We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing…
In this paper we study the equation $Lu=f$, where $L$ is a $\C$-valued vector field in $\R^2$ with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers…
Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of…
In this paper, we study the algebraic form of the symmetric generalized Lam\'e equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes…
In this work, we show a convergence result for the discrete formulation of the generalised KPZ equation $\partial_t u = (\Delta u) + g(u)(\nabla u)^2 + k(\nabla u) + h(u) + f(u)\xi_t(x)$, where the $\xi$ is a real-valued random field,…
Relativistic field equations for a gas in special and general relativity are determined from the Boltzmann equation. The constitutive equations are obtained from the Chapman-Enskog methodology applied to a relativistic model equation…
We consider the equation $$ ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \subseteq F_q$. The question of solvability of such and…
We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific…
Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present an explicit formula for solving the d-bar-equation $f=\dbar{g}$ on the regular part of Y, where $f$ is a d-bar-closed $(0,1)$-form…
We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When…
In this paper we consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c$ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$.…
In the paper the conditions are obtained providing existence and uniqueness of the regular solution of the boundary problem for class of the second order homogeneous operator-differential equation with singular coefficients. High term of…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation \[(a_1X_1^m+\cdots + a_nX_n^m+a)^k=bX_1\cdots X_n \] over the finite field $\mathbb{F}_q$, where $m,k$ are positive integers, and $a,b,a_i\in…
After Abel Ruffini theorem and Galois Theory the search for a method or formula to solve quintic equation ends. This paper discuss about the radical solution of quintic equation using a method that could be proved in some simple steps. A…