Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

Split Fibonacci Quaternions

Starting from ideas given by Horadam in [5] , in this paper, we will define the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion. We used the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations between the split Fibonacci, split Lucas and the split generalized Fibonacci quaternions. Moreover, we give Binet formulas and Cassini identities for these quaternions.     

Involutions of Complexified Quaternions and Split Quaternions

An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.     

Commutative Quaternion Matrices

In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their fundamental matrices. After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of commutative quaternion matrices and give some of their properties.     

On Complex Split Quaternion Matrices

In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 × 4 complex matrix representation. On the other hand, we give answers to the following two basic questions “If AB =  I, is it true that BA =  I for complex split quaternion matrices?” and “How can the inverse of a complex split quaternion matrix be found?”. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.     

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.     

Canal Surfaces with Quaternions

Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.     

On a new generalization of Fibonacci quaternions

In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k -Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Binet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative.     

On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions

In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra.     

The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

Coefficient rings of numerical semigroup algebras

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.

     

Biquaternion (Complexified Quaternion) Roots of −1

The roots of −1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are derived. There are trivial solutions (the complex operator, and any unit pure real quaternion), and non-trivial solutions consisting of complex numbers with perpendicular pure quaternion real and imaginary parts. The moduli of the two perpendicular pure quaternions are expressible by a single parameter by using a hyperbolic trigonometric identity.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.     

Spline Split Quaternion Interpolation in Minkowski Space

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).     

Compact totally disconnected k-divisible semigroups

One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups     

Dual Quaternion Involutions and Anti-Involutions

An involution or anti-involution is a self-inverse linear mapping. In this paper, we present involutions and anti-involutions of dual quaternions. In order to do this, quaternion conjugate, dual conjugate and total conjugate are defined for a dual quaternion and these conjugates are used in some transformations in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion and dual quaternion involutions and anti-involutions are given.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

A Parametrization of the Irreducible Representations of a Compact Inverse Semigroup

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.     

On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R _q about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.