We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For every suitable choice of a real subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case we get a volume Cauchy formula. In the Clifford algebra case, the choice of the ...

ABSTRACT. We extend a recent result on third and fourth-order Cauchy-Euler equations by establishing the Hyers-Ulam stability of higher-order linear non-homogeneous Cauchy-Euler dynamic equations on time scales. That is, if an approximate solution of a higher-order Cauchy-Euler equation exists, then there exists an exact solution to that dynamic equation that is close to the approximate one. We...

The second approach, called IFEP, to controlling search step size in FEP is to use both Cauchy mutation and Gaussian mutation. IFEP di ers from FEP slightly. Instead of using Cauchy mutation alone in FEP, IFEP generates two o spring from each parent, one by Cauchy mutation and the other by Gaussian mutation. The better one is then chosen as the o spring. The rest of the algorithm is exactly the...

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

In this chapter, the indirect Trefftz collocation method is applied to solve the inverse Cauchy problem of linear piezoelectricity. The static analysis of coupled fields in 2D piezoelectric solids is considered. The piezoelectric material is modelled as a homogenous, linear elastic, linear dielectric and transversal isotropic. The T-functions are obtained by the Stroh formalism. The inverse Cau...

We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of the pa...

The concept of I–convergence is an important generalization of statistical convergence which depends on the notion of an ideal I of subsets of the set N of positive integers. In this paper we introduce the ideas of I–Cauchy and I∗–Cauchy sequences in cone metric spaces and study their properties. We also investigate the relation between this new Cauchy type condition and the property of complet...

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a m...

Conventional quantum uncertainty relations (URs) contain dispersions of two observables. Generalized URs are known which contain three or more dispersions. They are derived here starting with suitable generalized Cauchy inequalities. It is shown what new information the generalized URs provide. Similar interpretation is given to generalized Cauchy inequalities. key words: quantum mechanics, unc...