An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over ...

A new and relatively simple version of the quaternion calculus is offered which is especially suitable for applications in molecular symmetry and structure. After introducing the real quaternion algebra and its classical matrix representation in the group SO(4) the relations with vectors in 3-space and the connection with the rotation group SO(3) through automorphism properties of the algebra a...

A split quaternion learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of three- and four-dimensional signals is proposed. The derivation takes into account the non-commutativity of the quaternion product, an aspect neglected in the derivation of the existing learning algorithms. It is shown that the additional information taken into acco...

The recently developed concepts of quaternion Fourier transform (QFT), quaternion convolution (QCV), and quaternion correlation, which are based on quaternion algebra, have been found to be useful for color image processing. However, the necessary computational algorithms and their complexity still need some attention. In this paper, we will develop efficient algorithms for QFT, QCV, and quater...

We study differential splitting fields of quaternion algebras with derivations. A algebra over a field k is always split by quadratic extension k. However, need not be any algebraic use solutions certain Riccati equations to provide bounds on the transcendence degree algebra.

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra. A quaternion algebra is a central simple algebra of dimension 4 over a field F . Generalizations of the notion of quaternion algebra to other commutative base rings R...

A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro’s “Pfister Factor Conjecture” is given for tensor products of at most five quaternion algebras. Throughout this paper, the characteristic of the base...

A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined “quadquaternion” signal and noise...