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Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion.

For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion.

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann differential equations for generalized complex differentiable functions are derived. For random versions of the considered new type of generalized complex numbers, moments are introduced and uniform distributions on discs with respect to functionals of the considered type are analyzed. Moreover, generalized uniform distributions on corresponding circles are studied and a connection with generalized circle numbers, which are natural relatives of π, is established. Finally, random generalized complex numbers are considered which are star-shaped distributed.

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Just as Gauss’s interpretation of complex numbers as points in a number plane in the form of a suitably formulated axiom found its way into the vector representation of Fourier transforms, this is the case with Euler’s formula and Riemann’s Zeta function considered here. The description of the connection between variables through complex numbers as it is given in Euler’s formula and emphasized by Riemann is reflected here with great flexibility in the introduction of non-classically generalized complex numbers and the vector representation of the generalized Zeta function based on them. For describing such dependencies of two variables with the help of generalized complex numbers, we introduce manifolds underlying certain Lie groups as level sets of norms, antinorms or semi-antinorms. No undefined or “imaginary” quantities are used for this. In contrast to the approach of Hamilton and his numerous successors, the vector-valued vector product of non-classically generalized complex numbers is commutative, and the whole number system satisfies a weak distributivity property as considered by Hankel, but not the strong one.

We first shortly review, in part throwing a new light on, basics of ball numbers for balls having a positively homogeneous Minkowski functional and turn over then to a new particular class of ball numbers of balls having a Minkowski functional being homogeneous with respect to multiplication with a specific diagonal matrix. Applications to crystal breeding, temperature expansion and normalizing density generating functions in big data analysis are indicated and a challenging problem from the inhomogeneity program is stated.

Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R3. Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric analogy. Because a change in the radius now leads to a change in the shape of the sphere, a three-dimensional dynamic complex structure is created. Statements about invariant probability densities, generalized uniform distributions on generalized spheres, geometric measure representations, and dynamic ball numbers associated with this structure are also presented.

For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss-Laplace law. The latter will be considered here as an element of the newly-introduced family of (p,q) -spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on (p,q) -circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box-Muller simulation method. In passing, we prove an extension of the sector and circle number functions.

This paper aims to introduce a mathematical-philosophical type of question from the fascinating world of generalized circle numbers to the widest possible readership. We start with recalling well-known (in part from school) properties of the polygonal approximation of the common circle when approximating the famous circle number π by convergent sequences of upper and lower bounds being based upon the lengths of polygons. Next, we shortly refer to some results from the literature where suitably defined generalized circle numbers of l p - and l p , q -circles, π p and π p , q , respectively, are considered and turn afterwards over to the approximation of an l p -circle by a family of l p , q -circles with q converging to p, q → p . Then we ask whether or not there holds the continuity property π p , q → π p as q → p . The answer to this question leads us to the answer of the question stated in the paper’s title. Presenting here for illustration true paintings instead of strong technical or mathematical drawings intends both to stimulate opening heart and senses of the reader for recognizing generalized circles in his real life and to suggest the philosophical challenge of the consequences coming out from the demonstrated non-continuity property.