Related papers: On a problem of Peth\H{o}

Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left(…

We provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit…

Let $\alpha$ be a real number greater than $1$. We establish an effective lower bound for the distance between an integral power of $\alpha$ and its nearest integer.

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a…

In 2016 Izadi and Nabardi (b) showed (4-2-4) has infinitely many integer solutions. They used a specific congruent number elliptic curve.In 2019 Janfada and Nabardi,item C, showed that a necessary condition for n to have an integral…

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…

Let \Omega be a set of unsatisfiable clauses, an implicit resolution refutation of \Omega is a circuit \beta with a resolution proof {\alpha} of the statement "\beta describes a correct tree-like resolution refutation of \Omega". We show…

We consider the Dirichlet problem for the nonhomogeneous equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)$ in a bounded domain, where $p \neq q$, and $\alpha, \beta \in \mathbb{R}$ are parameters. We explore…

We prove that the Diophantine problem for spherical quadratic equations in free metabelian groups is solvable and, moreover, NP-complete

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}\{z\}$, and…

Let $\Omega\subset\r^n$ be a bounded mean convex domain. If $\alpha<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $\Omega$ for the $\alpha$-singular minimal surface equation with arbitrary…

The $(-\beta)$-integers are natural generalisations of the $\beta$-integers, and thus of the integers, for negative real bases. When $\beta$ is the analogue of a Parry number, we describe the structure of the set of $(-\beta)$-integers by a…

A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in…

We develop a geometric procedure for finding the Ap\'ery set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We…

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta>1$ for which a given point can be well approximated by the orbit of 1 is…

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad…

Let $f$ be a smooth plurisubharmonic function which solves $$ \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }\Omega\subset \mathbb C^n.$$ Suppose that the metric $\omega_{f}=\sqrt{-1}f_{i\bar j}dz_{i}\wedge d\bar z_{j}$ is complete and $f$…