We prove a general theorem that evaluates the Legendre symbol (A+B √ m p) under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

In this paper, we prove the generalized Hyers--Ulamstability of a quadratic and quartic functional equation inintuitionistic fuzzy Banach spaces.

We shall show that the three variable cubic inequality t(a + b + c) + (t − 2t)(ab + bc + ca) ≥ (2t − 1)(ab + bc + ca) + (3t − 6t + 3t − 6t + 3)abc holds for non-negative a, b, c, and for any real number t. We also show some similar three variable cyclic quartic inequalities.

Noncommutative quantum field theory of a complex scalar field is considered. There is a two-coupling noncommutative analogue of U(1)-invariant quartic interaction (φφ), namely Aφ ⋆ φ ⋆ φ ⋆ φ + Bφ ⋆ φ ⋆ φ ⋆ φ. For arbitrary values of A and B the model is nonrenormalizable. However, it is one-loop renormalizable in two special cases: B = 0 and A = B. Furthermore, in the case B = 0 the model does ...

The present article is concerned with the numerical solution of Benjamin-BonaMahony-Burgers (BBM-Burger) equation by quartic B-spline collocation method. The method is based on quartic B-spline basis functions for space integration, and Crank-Nicolson formulation for time integration. Numerical examples considered by different researchers are discussed to illustrate the efficiency, robustness a...

We investigate the parameter plane of the Newton’s method applied to the family of quartic polynomials pa,b(z) = z 4 +az + bz +az+ 1, where a and b are real parameters. We divide the parameter plane (a, b) ∈ R into twelve open and connected regions where p, p′ and p′′ have simple roots. In each of these regions we focus on the study of the Newton’s operator acting on the Riemann sphere.

In this paper, we completely solve the simultaneous Diophantine equations x − az = 1, y − bz = 1 provided the positive integers a and b satisfy b−a ∈ {1, 2, 4}. Further, we show that these equations possess at most one solutions in positive integers (x, y, z) if b − a is a prime or prime power, under mild conditions. Our approach is (relatively) elementary in nature and relies upon classical re...