Related papers: Bohr's Theorem for Monogenic Power Series
A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
In this work a class of massive scalar field theories with self-interactions described by a general potential is studied. Under the sole condition that the potential admits the Fourier representation, it is shown that such theories may be…
The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve in terms of coefficients of modular forms. We give another proof of this theorem which generalizes to higher dimensions.
A Lagrangian formulation is constructed for particle interpretations of quantum mechanics, a well-known example of such an interpretation being the Bohm model. The advantages of such a description are that the equations for particle motion,…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of…
In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.
We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables.…
It is shown that the measurement algebra of Schwinger, a characterization of the properties of Pauli measurements of the first and second kinds, forming the foundation of his formulation of quantum mechanics over the complex field, has a…
The classical Blasius--Chaplygin formula provides an elegant method for calculating the lift force on a two-dimensional body in steady, irrotational flow. The key ingredient is the definition of a complex-valued potential function…
In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the…
A derivation of the Bohm model, and some general comments about it, are given. A modification of the model which is formally local and Lorentz-invariant is introduced, and its properties studied for a simple experiment.
In the present paper, we give a brief review of $L^{1}$-convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.
Quaternionic analysis relies heavily on results on functions defined on domains in $\mathbb R^4$ (or $\mathbb R^3$) with values in $\mathbb H$. This theory is centered around the concept of $\psi-$hyperholomorphic functions i.e.,…
A strong consequence of quadratic forms becoming hyperbolic over the function field of a form is established. This result is invoked to obtain a new characterisation of hyperbolicity over function fields, and to recover a number of…
We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex to quaternions, the calculation of the real-valued parameters of the CKM matrix drastically changes. We aim to explain this…
A quaternionic partial differential equation is shown to be a generalisation of the Riccati ordinary differential equation and its relationship with the Schrodinger equation is established. Various approaches to the problem of finding…
In this paper, Ostrowski and Brauer type theorems are derived for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic…