Let p(z) be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the Voronoi property if it lies in the Voronoi cell of a root θ, V (θ), i.e. the set of points that are closer to θ than to the other roots. We prove at least one critical point has the Voronoi property and characterize the cases when both satisfy this property. It is known...

We prove that there are infinitely many inequivalent cubic binary forms F with content 1 for which the Thue equation F (x, y) = m has ≫ (logm) solutions in integers x and y for infinitely many integers m.

We shall prove that if F is a cubic binary form with integer coefficients and non-zero discriminant then there is a positive number c, which depends on F, such that the Thue equation F (x, y) = m has at least c(logm) solutions in integers x and y for infinitely many positive integers m.

We construct  a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result genera...

The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let G1, · be a group and let G2, ∗, d be a metric group with the metric d ·, · . Given > 0, does there exist a δ > 0, such that if a mapping h : G1 → G2 satisfies the inequality d h x · y , h x ∗ h y < δ for all x, y ∈ G1, then there exists a homomorphism ...

We prove generalized Hyres-Ulam-Rassias stability of the cubic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2(k − k)f(x) for all k ∈ N and the quartic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2k(k − 1)f(x)− 2(k − 1)f(y) for all k ∈ N in non-Archimedean normed spaces.

Abstract We have theoretically investigated the phenomenon of Eckhaus instability stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we deduced cubic cubic–quintic real Ginzburg–Landau equations ruling evolution pattern amplitude close to criticality. Starting ...

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of quadrangular discrete surfaces. The construction makes use of asymptotic coordi...

We derive equations of motion for the tachyon field living on an unstable non-BPS D-brane in the level truncated open cubic superstring field theory in the first non-trivial approximation. We construct a special time dependent solution to this equation which describes the rolling tachyon. It starts from the perturbative vacuum and approaches one of stable vacua in infinite time. We investigate ...

We study the field isomorphism problem of cubic generic polynomial X + sX + s over the field of rational numbers with the specialization of the parameter s to nonzero rational integers m via primitive solutions to the family of cubic Thue equations x − 2mxy − 9mxy −m(2m+ 27)y = λ where λ is a divisor of m(4m+ 27).