Related papers: A Note on Quartic Equations with only Trivial Solu…
The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients $A_\epsilon$ specified by the small parameter $\epsilon>0$. We use an iteration method generated by an…
In this paper, by using the elliptic curves theory, we study the fourth power Diophantine equation ${ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }$, where $a_i$ and $n\geq3$ are fixed arbitrary integers. We solve the equation for…
We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A \subset Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
We give a method to solve generalized Fermat equations of type $x^4 + y^4 = q z^p$, for some prime values of $q$ and every prime $p$ bigger than 13. We illustrate the method by proving that there are no solutions for $q= 73, 89$ and 113.
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
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 a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results…
We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}),…
A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. We will show that, even if the lower order term is singular, it…
We study a canonical duality method to solve a mixed-integer nonconvex fourth-order polynomial minimization problem with fixed cost terms. This constrained nonconvex problem can be transformed into a continuous concave maximization dual…
This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a…
This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5 + mt^3 + nt^2 + pt +…
It is a generalization of Pell's equation $x^2-Dy^2=0$. Here, we show that: if our Diophantine equation has a particular integer solution and $ab$ is not a perfect square, then the equation has an infinite number of solutions; in this case…
A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new…
In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that…
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation…
Incremental gradient and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, without strong convexity, their convergence guarantees…
We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype…