In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with three-dimensional Helmholtz-type equations is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L...

Introduction A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?”. Such a problem, called a stability problem of the functional equation, was formulated by Ulam [1] in 1940. In the next year, Hyers [2] gave a partial solution of Ulam’s probl...

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of the functional equation, was formulated by Ulam 1 in 1940. In the next year, Hyers 2 gave a partial solution of Ulam’s problem for the case of...

The refractive indices of E7 liquid-crystal mixture were measured at six visible and two infrared sl=1.55 and 10.6 mmd wavelengths at different temperatures, using Abbe and wedged cell refractometer methods, respectively. The experimental data of the visible wavelengths fit the extended Cauchy equations well. Using the extended Cauchy equations, we can extrapolate the refractive indices of E7 t...

We develop a numerical approach for computing the additive, multiplicative and compressive convolution operations from free probability theory. We utilize the regularity properties of free convolution to identify (pairs of) ‘admissible’ measures whose convolution results in a so-called ‘invertible measure’ which is either a smoothly-decaying measure supported on the entire real line (such as th...

This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over F2, originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determin...

Using fixed pointmethods, we investigate approximately higher ternary Jordan derivations in Banach ternaty algebras via the Cauchy functional equation$$f(lambda_{1}x+lambda_{2}y+lambda_3z)=lambda_1f(x)+lambda_2f(y)+lambda_3f(z)~.$$