The Power of a Point for Some Real Algebraic Curves

According to various sources (e.g. [1, p. 102]), the terminology of the power of a point with respect to a circle is due to Steiner. His definition appears in most classical and contemporary geometry textbooks (to mention just a few references, see [2, 3, 4, 5]). The concept of the power of a point has been revisited not only in advanced Euclidean geometry, but also in computational geometry and other areas of mathematics.In the current literature there are two different definitions of the power of a point with respect to a circle, which we study in detail in section 2. In the first half of the twentieth century there have been published several attempts to generalise the concept of power of the point to real algebraic curves.</jats:p

Ternary Clifford Algebras

Ternary Clifford algebras are an essential ingredient in a cubic factorization of the Laplacian and using a ternary Clifford analysis build on such spaces one obtains a Dirac-type operator D such that D3 = Δ. This paper is a continuation of the work of the authors in describing properties of generalized ternary Clifford algebras. Here we explore a blade decomposition and symmetries of these algebras

Bicomplex Hyperfunctions

In this paper, we consider bicomplex holomorphic functions of several variables in BCn .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space Rn within the bicomplex space BCn , and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1

A Note on the Complex and Bicomplex Valued Neural Networks

In this paper we first write a proof of the perceptron convergence algorithm for the complex multivalued neural networks (CMVNNs). Our primary goal is to formulate and prove the perceptron convergence algorithm for the bicomplex multivalued neural networks (BMVNNs) and other important results in the theory of neural networks based on a bicomplex algebra

New Fueter Variables Associated to the Global Operator in the Quaternionic Case

The purpose of this paper is to develop a new theory of three non-commuting quaternionic variables and its related Schur analysis theory for a modified version of the quaternionic global operator

Discrete Wiener Algebra in the Bicomplex Setting, Spectral Factorization with Symmetry, and Superoscillations

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between bicomplex analysis and complex analysis with symmetry. We also write an application to superoscillations in this case

An Extension of the Complex–Real (C–R) Calculus to the Bicomplex Setting, with Applications

In this paper, we extend notions of complex ℂ−ℝ-calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms

The Infinite is the Chasm in Which Our Thoughts are Lost: Reflections on Sophie Germain\u27s Essays

Sophie Germain (1776–1831) is quite well-known to the mathematical community for her contributions to number theory [17] and elasticity theory (e.g., see [2, 5]). On the other hand, there have been few attempts to understand Sophie Germain as an intellectual of her time, as an independent thinker outside of academia, and as a female mathematician in France, facing the prejudice of the time of the First Empire and of the Bourbon Restoration, while pursuing her thoughts and interests and writing on them. Sophie Germain had to face a double challenge: the mathematical difficulty of the problems she approached and the socio-cultural context of her time, which never fully supported her interests, never appropriately rewarded her, and never allowed her to enjoy the recognition she deserved. In our attempt to understand the innermost Sophie Germain, we also try to grasp the place of her personality within her time and historical period. We will argue that she represents a unique case in both the history of mathematics and the context of Western European intellectuals at the beginning of the 19th century, deserving a further exploratory study of the connections of her work with the ideas of her time