This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy’s formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum ...

In this paper, we propose a new semi-parametric maximum likelihood (ML) estimate of a receiver operating characteristic (ROC) curve that satisfies the property of invariance of the ROC curve and is easy to compute. We show that our new estimator is sqrt[n]-consistent and has an asymptotically normal distribution. Our extensive simulation studies show that the proposed method is efficient and ro...

We introduce quaternionic Gabor filters for the classification of local image structure. These filters are constructed as windowed basis functions of the quaternionic Fourier transform. We show that – in contrast to the 2D complex Gabor filters – the quaternionic Gabor filters are intrinsically 2D filters. A generalized phase concept is introduced and compared to the classical one. It is shown ...

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic c...

The classical involute of a plane base curve intersects every tangent line at a right angle. This paper introduces a tanvolute, which intersects every tangent line at any given fixed angle. This minor change in the definition of a classical concept leads to a wealth of new examples and phenomena that go far beyond the original situation. Our treatment is based on two differential equations rela...

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

In this article, we examine the relationship between Darboux frames along parameter curves and frame of base curve ruled surface. We derive equations quaternionic shape operators, which can rotate tangent vectors around normal vector, find corresponding rotation matrices. Using these Gauss curvature mean explore how properties are found by use Frenet instead generator vectors. provide illustrat...

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 ...