On Semi- c -Periodic Functions

The main aim of this paper is to indicate that the notion of semi- c -periodicity is equivalent with the notion of c -periodicity, provided that c is a nonzero complex number whose absolute value is not equal to 1.

On the Skitovich–Darmois theorem for complex and quaternion random variables

We prove the following theorem. Let {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either {b\neq 0} or {b=0} and {a>0}. Let {\xi_{1}} and {\xi_{2}} be independent complex random variables. Assume that the linear forms {L_{1}=\xi_{1}+\xi_{2}} and {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent. Then {\xi_{j}} are degenerate random variables. (ii) Let {b=0} and {a<0}. Then there exist complex Gaussian random variables in the wide sense {\xi_{1}} and {\xi_{2}} such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms {L_{1}=\xi_{1}+\xi_{2}} and {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent.

Expressions for the g-Drazin inverse in a Banach algebra

We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a,b ? Ad. If ab = ?a?bab? for a nonzero complex number ?, then a + b ? Ad. The explicit representation of (a + b)d is presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.

Fixed points of differences of meromorphic functions

Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define $\Delta _{c}f=f(z+c)-f(z)$ Δ c f = f ( z + c ) − f ( z ) . The authors investigate the existence on the fixed points of $\Delta _{c}f$ Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of $f'$ f ′ . The existing theorem on the fixed points of $\Delta _{c}f$ Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

Extensions of modules over a class of Lie conformal algebras 𝒲(b)

Let [Formula: see text] be a class of free Lie conformal algebras of rank two with [Formula: see text]-basis [Formula: see text] and relations [Formula: see text] where [Formula: see text] is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras [Formula: see text].

Biderivations and bihomomorphisms in Banach algebras

In this paper, we solve the following bi-additive s-functional inequalities || f(x+y,z+w) + f(x+y,z-w)+f(x-y,z+w) + f (x-y,z-w)- 4f(x,z)||? ||s(2f(x+y,z-w)+ 2f(x-y,z + w)- 4f(x,z) + 4f(y,w)||(1) and ||2f(x+y,z-w) + 2f(x-y,z+w)-4f(x,z) + 4f(y,w)|| (2)? ||s(f(x+y,z+w)+ f(x+y,z-w) + f(x-y,z+w)+f(x-y,z-w)-4f(x,z))||, where s is a fixed nonzero complex number with |s| < 1. Moreover, we prove the Hyers-Ulam stability of biderivations and bihomomorphismsions in Banach algebras and unital C+-algebras, associated with the bi-additive s-functional inequalities (1) and (2).

Determinants of binomial-related circulant matrices

Due to their rich algebraic structures and wide applications, circulant matrices have been of interest and continuously studied. In this paper, n×n complex left and right circulant matrices whose first row consists of the coefficients in the expansion of (x + zy)n−1 are focused on, where z is a nonzero complex number and n is a positive integer. In the case where z ∈ {1, −1, i, −i}, explicit formulas for the determinants of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as the special case where z = 1. Finally, some remarks and open problems are discussed.

On k-circulant matrices with the Lucas numbers

Let k be a nonzero complex number. In this paper, we determine the eigenvalues of a k-circulant matrix whose first row is (L1,L2,..., Ln), where Ln is the nth Lucas number, and improve the result which can be obtained from the result of Theorem 7. [28]. The Euclidean norm of such matrix is obtained. Bounds for the spectral norm of a k-circulant matrix whose first row is (L-11, L-12,..., L-1n ) are also investigated. The obtained results are illustrated by examples.

On k-circulant matrices with arithmetic sequence

Let k be a nonzero complex number. In this paper we consider k-circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k = 1, the inverses of such (invertible) matrices are obtained (in a way different from the way presented in [1]), and the Moore-Penrose inverses of such (singular) matrices are derived.

ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.