Related papers: On $A^4 + hB^4 = C^4 + hD^4$

We provide infinitely many solutions of a Dirichlet problem on balls.

We are interested in the qualitative properties of positive entire solutions $u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation} \label{0.0} \Delta^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n \backslash…

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher…

In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4$: \begin{equation}\left\{ \begin{aligned} & -\Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in…

In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation &4(f(3x+y)+f(3x-y))=-12(f(x+y)+f(x-y)) &+12(f(2x+y)+f(2x-y))-8f(y)-192f(x)+f(2y)+30f(2x).

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a…

In this paper we prove the decidability of the HD0L ultimate periodicity problem.

In this survey, we studied the possibility of finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC)^{p-1}/4$ and its rational points computed using computational tools.

It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N.

We show that the number of integer solutions for a pair of bilinear equations in at least 2*6 variables has (up to logarithms) the expected upper bound unless there is a structural reason why it is not the case.

Let $f,g:{\Bbb R}\to{\Bbb R}$ be integrable functions, $f$ nowhere zero, and $\phi (u)=\int du/f(u)$ be invertible. An exact solution to the generalized nonhomogeneous inviscid Burgers' equation $u_t+g(u).u_x=f(u)$ is given, by quadratures.

In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic…

Considering the equation $$a^x+b^y = c^z,$$ Le gives an upper bound on the number of solutions $(x,y,z)$, and a bound on $z$. Here we give slight improvements to these results, also removing the restriction that $a$, $b$, $c$ be squarefree.

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of…

In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.

In this paper we prove that assuming Schanuel's conjecture, an exponential polynomial in one variable over the algebraic numbers has only finitely many algebraic solutions. This implies a positive answer to Shapiro's conjecture for…

Algebras simple with respect to an action of Sweedler's algebra $H_4$ deliver the easiest example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_4$-simple algebras and prove the…

After substantial progress over the last 15 years, the "algebraic CSP-dichotomy conjecture" reduces to the following: every local constraint satisfaction problem (CSP) associated with a finite idempotent algebra is tractable if and only if…

This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation…

During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously…