Related papers: Basis-free Solution to General Linear Quaternionic…
A generalized derivative nonlinear Schr\"odinger equation, \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x + (\gamma-1)(\gamma-2)|q|^4 q = 0 , is studied by means of Hirota's bilinear formalism. Soliton solutions are…
We construct the solution $\phi(t,{\bf x})$ of the quantum wave equation $\Box\phi + m^2\phi + \lambda:\!\!\phi^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!\phi^3(t,{\bf…
The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the…
Let $K_n$ be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer's parametric polynomial. We give all normal integral bases for $K_n$ only by the roots of the polynomial, which is a generalization of the work of Lehmer…
The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz, which provides some information about the polynomial map $P|_{\X_1\times...\times\X_n}$ when only…
While quantum computing provides an exponential advantage in solving system of linear equations, there is little work to solve system of nonlinear equations with quantum computing. We propose quantum Newton's method (QNM) for solving…
We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like…
An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative…
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion…
Using the complex Klein-Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and…
The analytic and formal solutions of certain family of $q$-difference-differential equations under the action of a complex perturbation parameter is considered. The previous study of the last two authors provides information in the case…
The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 +…
We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…
Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion…
In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations…
After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as…
One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it…
This study examines Quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be…
We clarify the linear algebra used in the quaternionic pluripotential theory so that proofs of several results there can be greatly simplified. In particular, we characterize and normalize real $2$-forms with respect to the quaternionic…
We argue that quaternions form a natural language for the description of quantum-mechanical wavefunctions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No…