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

Author(s):

Kaijing Ling ◽

Lamei Yuan

Keyword(s):

Complex Number ◽

Conformal Algebras ◽

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].

Expressions for the g-Drazin inverse in a Banach algebra

Filomat ◽

10.2298/fil2011845c ◽

2020 ◽

Vol 34 (11) ◽

pp. 3845-3854

Author(s):

Huanyin Chen ◽

Marjan Sheibani

Keyword(s):

Banach Algebra ◽

Complex Number ◽

Block Matrix ◽

Explicit Representation ◽

Drazin Inverse ◽

Operator Matrices ◽

Nonzero Complex Number ◽

Generalized Drazin Inverse

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.

ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000125 ◽

2016 ◽

Vol 94 (2) ◽

pp. 245-253 ◽

Author(s):

TOUFIK ZAÏMI

Keyword(s):

Finite Number ◽

Algebraic Number ◽

Complex Number ◽

Convergent Sequence ◽

Pisot Number ◽

Pisot Numbers ◽

Limit Points ◽

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.

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788714000524 ◽

2014 ◽

Vol 98 (3) ◽

pp. 289-310 ◽

Author(s):

PETER BUNDSCHUH ◽

KEIJO VÄÄNÄNEN

Keyword(s):

Rational Function ◽

Complex Number ◽

Functional Equations ◽

Algebraic Independence ◽

Lucas Numbers ◽

Mahler Functions ◽

Independence Results ◽

Nonzero Complex Number ◽

Fibonacci And Lucas Numbers ◽

Reciprocal Sums

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

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

Georgian Mathematical Journal ◽

10.1515/gmj-2020-2076 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Gennadiy Feldman

Keyword(s):

Complex Number ◽

Random Variables ◽

Wide Sense ◽

Narrow Sense ◽

Gaussian Random Variables ◽

Linear Forms ◽

AbstractWe 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.

Fixed points of differences of meromorphic functions

Advances in Difference Equations ◽

10.1186/s13662-019-2386-8 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Zhaojun Wu ◽

Jia Wu

Keyword(s):

Meromorphic Function ◽

Fixed Points ◽

Finite Order ◽

Complex Number ◽

Meromorphic Functions ◽

Acta Math ◽

Transcendental Meromorphic Function ◽

Nonzero Complex Number ◽

Discrete Analogues

Abstract 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.

Linear mappings preserving square-zero matrices

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700015811 ◽

1993 ◽

Vol 48 (3) ◽

pp. 365-370 ◽

Cited By ~ 29

Author(s):

Peter Šemrl

Keyword(s):

Complex Number ◽

Linear Mapping ◽

Dimensional Case ◽

Complex Matrices ◽

Infinite Dimensional ◽

Let sln denote the set of all n × n complex matrices with trace zero. Suppose that ø: sln → sln is a bijective linear mapping preserving square-zero matrices. Then ø is either of the form ø(A) = cUAU-1 or ø(A) = cUAtU-1 where U is an invertible n × n matrix and c is a nonzero complex number. The same result holds if we assume that ø is a linear mapping preserving square-zero matrices in both directions. Applying this result we prove that a linear mapping ø defined on the algebra of all n × n matrices is an automorphism if and only if it preserves zero products in both directions and satisfies ø(I) = I. An extension of this last result to the infinite-dimensional case is considered.

Dynamics of a certain sequence of powers

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171200003136 ◽

2000 ◽

Vol 24 (4) ◽

pp. 283-288

Author(s):

Roman Sznajder ◽

Kanchan Basnyat

Keyword(s):

Functional Equation ◽

Complex Plane ◽

Complex Number ◽

For any nonzero complex numberzwe define a sequencea1(z)=z,a2(z)=za1(z),…,an+1(z)=zan(z),n∈ℕ. We attempt to describe the set of thesezfor which the sequence{an(z)}is convergent. While it is almost impossible to characterize this convergence set in the complex plane𝒞, we achieved it for positive reals. We also discussed some connection to the Euler's functional equation.

On k-circulant matrices with the Lucas numbers

Filomat ◽

10.2298/fil1811037r ◽

2018 ◽

Vol 32 (11) ◽

pp. 4037-4046

Author(s):

Biljana Radicic

Keyword(s):

Complex Number ◽

Circulant Matrix ◽

Spectral Norm ◽

Circulant Matrices ◽

Euclidean Norm ◽

Lucas Numbers ◽

Lucas Number ◽

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.

Biderivations and bihomomorphisms in Banach algebras

Filomat ◽

10.2298/fil1908317p ◽

2019 ◽

Vol 33 (8) ◽

pp. 2317-2328 ◽

Author(s):

Choonkil Park

Keyword(s):

Complex Number ◽

Banach Algebras ◽

Functional Inequalities ◽

Ulam Stability ◽

C 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).

On k-circulant matrices with arithmetic sequence

Filomat ◽

10.2298/fil1708517r ◽

2017 ◽

Vol 31 (8) ◽

pp. 2517-2525 ◽

Author(s):

Biljana Radicic

Keyword(s):

Complex Number ◽

Circulant Matrices ◽

Arithmetic Sequence ◽

Nonzero Complex Number ◽

Singular Matrices ◽

Invertible Matrices ◽

The Way

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.