Nonzero Complex Number Research Articles

Low Phase-Rank Approximation

In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.

On Complex Differential‐Difference Equations of Malmquist Type

In this paper, we shall give some properties of complex differential‐difference equation of the form where and are non‐negative integers, ci(i = 1,2, …, t) are distinct, nonzero complex numbers, αμ(z)(μ = 1,2, …, n) and βν(z)(ν = 1,2, …, m) are small functions relative to f(z), and R(z, f(z)) is a rational function in f(z) with coefficients, which are small functions of f(z). We also consider related complex functional equations in the paper. These results are improvements of some several previous results.

ENTIRE FUNCTIONS THAT SHARE A SET WITH THEIR DIFFERENCES

<abstract> In this paper, we study the uniqueness of entire functions concerning deficient value and exponent of convergence, and have mainly proved the following theorem: Let $S=\{1, \omega , \omega ^2, \cdots , \omega ^{n-1}\}$, where $\omega ^n=1$, $n\ge 1$ is an integer, let $k$ be a positive integer, and let $f$ be a nonconstant entire function such that $\lambda(f)&lt;\rho(f)&lt;\infty$. If $f(z)$ and $\Delta _{\eta }^kf(z)$ share $S$ IM, where $\eta $ is a nonzero complex number, then $f(z)=e^{az+b}$, where $a(\neq0)$ and $b$ are two finite complex numbers. The results obtained in this paper improve some results due to Li ([<xref ref-type="bibr" rid="b15">15</xref>]). </abstract>

Notes on the quadraticity of linear combinations of a cubic matrix and a quadratic matrix that commute

Let $A_{1}$ and $A_{2}$ be an $\{\alpha_{1}, \beta_{1}, \gamma_{1}\}$-cubic matrix and an $\{\alpha_{2}, \beta_{2}\}$-quadratic matrix, respectively, with $\alpha_{1} \neq \beta_{1}$, $\beta_{1} \neq \gamma_{1}$, $\alpha_{1} \neq \gamma_{1}$ and $\alpha_{2}\neq \beta_{2}$. In this work, we characterize all situations in which the linear combination $A_{3}=a_{1}A_{1}+a_{2}A_{2}$ with the assumption $A_{1}A_{2}=A_{2}A_{1}$ is an $\{\alpha_{3}, \beta_{3}\}$-quadratic matrix, where $a_{1}$ and $a_{2}$ are unknown nonzero complex numbers.

On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix

We characterize the involutiveness of the linear combinations of the form $a{\mathbf{A}} + b{\mathbf{B}}$ when $a,b$ are nonzero complex numbers, ${\mathbf{A}}$ is a quadratic $n \times n$ nonzero matrix and ${\mathbf{B}}$ is an arbitrary $n \times n$ nonzero matrix, under certain properties imposed on $\mathbf{A}$ and $\mathbf{B}$.

Formula omitted]-free modules over the Block algebra [formula omitted

In this paper, we construct and study a new class of modules over the Block algebra B(q), where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U(h)-free B(q)-modules of rank-1.

Unicity on Entire Function Concerning Its Differential-Difference Operators

In this paper, we study the uniqueness of entire function and its differential-difference operators. We prove the following result: let f be a transcendental entire function of finite order, let \(\eta \) be a non-zero complex number, \(n\ge 1, k\ge 0\) two integers and let a and b be two distinct finite complex numbers. If f and \((\Delta _{\eta }^{n}f)^{(k)}\) share a CM and share b IM, then \(f\equiv (\Delta _{\eta }^{n}f)^{(k)}\).

On the logarithm of the Riemann zeta-function near the nontrivial zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|\zeta(\rho+z)|)$ and $(\arg\zeta(\rho+z)).$ Here $\rho=\frac12+i\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\gamma \leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|\zeta^\prime(\rho)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.

Structural properties of multiple zeta values

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built from an arbitrary sequence of nonzero complex numbers. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

On k-circulant matrices involving the Pell–Lucas (and the modified Pell) numbers

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is $$(Q_{1},Q_{2},\ldots ,Q_{n})$$ , where $$Q_{n}$$ is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 2013)) is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(Q_{1}^{-1},Q_{2}^{-1},\ldots ,Q_{n}^{-1})$$ are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is $$(q_{1},q_{2},\ldots ,q_{n})$$ , where $$q_{n}$$ is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(q_{1}^{-1},q_{2}^{-1},\ldots ,q_{n}^{-1})$$ are given

Completeness conditions of systems of Bessel functions in weighted L2-spaces interms of entire functions

Let $J_{\nu}$ be the Bessel function of the first kind of index $\nu\ge 1/2$, $p\in\mathbb R$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system $\big\{x^{-p-1}\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$ in the weighted space $L^2((0;1);x^{2p} dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$.

Uniqueness of Derivatives and Shifts of Meromorphic Functions

Recently, some uniqueness theorems about meromorphic functions f(z) concerning their derivatives $$f'(z)$$ and shifts $$f(z+c)$$ with three CM sharing values have been obtained. In this paper, we continue to study this topic. We consider not only high order derivatives instead of just $$f'(z)$$ , but also IM sharing value instead of CM sharing value. In fact, we mainly prove that for a non-constant meromorphic function f(z) of hyper order strictly less than 1, if $$f^{(k)}(z)$$ and $$f(z+c)$$ share $$0,\infty $$ CM and 1 IM, then $$f^{(k)}(z)\equiv f(z+c)$$ , where c is a non-zero finite complex number. Our main theorem generalizes and greatly improves the related result due to Qi–Li–Yang. In addition, we give some discussion of this issue and obtain a uniqueness theorem concerning defective values in Sect. 3.

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 ofc-periodicity, provided thatcis a nonzero complex number whose absolute value is not equal to 1.

Meromorphic Functions Sharing four Values with their Difference Operators

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

AbstractIn 1996, aq-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrevet al. [J. Phys.A29(1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category$\mathcal{O}$for the quantum Schrödinger algebra$U_q(\mathfrak{s})$, whereqis a nonzero complex number which is not a root of unity. If the central charge$\dot z\neq 0$, using the module$B_{\dot z}$over the quantum Weyl algebra$H_q$, we show that there is an equivalence between the full subcategory$\mathcal{O}[\dot Z]$consisting of modules with the central charge$\dot z$and the BGG category$\mathcal{O}^{(\mathfrak{sl}_2)}$for the quantum group$U_q(\mathfrak{sl}_2)$. In the case that$\dot z = 0$, we study the subcategory$\mathcal{A}$consisting of finite dimensional$U_q(\mathfrak{s})$-modules of type 1 with zero action ofZ. We directly construct an equivalence functor from$\mathcal{A}$to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional$U_q(\mathfrak{s})$-modules is wild.

Meromorphic solutions of higher order non-homogeneous linear difference equations

In this paper, we investigate the growth of meromorphic solutions of nonhomogeneous linear difference equation A_n(z)f(z + c_n) + · · · + A_1(z)f(z + c_1) + A_0(z)f(z) = A_{n+1}(z), where A_{n+1 (z), · · · , A0 (z) are (entire) or meromorphic functions and c_j (1, · · · , n) are non-zero distinct complex numbers. Under some conditions on the (lower) order and the (lower) type of the coefficients, we obtain estimates on the lower bound of the order of meromorphic solutions of the above equation. We extend early results due to Luo and Zheng.

Structure of a class of Lie conformal algebras of Block type

Let p be a nonzero complex number. Recently, a class of infinite rank Lie conformal algebras was introduced. In this article, we study structure theory of this class of Lie conformal algebras. Specifically, we completely determine conformal derivations, conformal biderivations, and certain second cohomologies of Communicated by K. C. Misra

On nonsingularity and group invertibility of combinations of two group invertible matrices

Let A and B be two group invertible matrices, we study the rank, the nonsingularity and the group invertibility of A-B, AA#-BB#, c1A + c2B, c1A + c2B + c3AA#B where c1,c2 are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c1A + c2B + c3and c1A + c2B + c3+ c4BA to be nonsingular and group invertible are presented, which generalized some related results of Ben?tez, Liu, Koliha and Zuo [4, 17, 19, 25].

Linear factorization of hypercyclic functions for differential operators

On the Fréchet space of entire functions H(C), we show that every nonscalar continuous linear operator L:H(C)→H(C) which commutes with differentiation has a hypercyclic vector f(z) in the form of the infinite product of linear polynomials:f(z)=∏j=1∞(1−zaj), where each aj is a nonzero complex number.

On the distribution of powers of a Gaussian Pisot number

A Gaussian Pisot number is an algebraic integer with modulus greater than one whose other conjugates, over the quadratic field Q(−1), are of modulus less than one. Given a nonreal algebraic number α, with modulus greater than one, we prove that there is a nonzero complex number λ such that the series ∑n∈N(Re(λαn)2+Im(λαn)2) converges (resp. such that the sequence ({Re(λαn)},{Im(λαn)})n∈N has a unique limit point), if and only if α is a Gaussian Pisot number (resp. α is a Gaussian Pisot number satisfying P(1)≥2 or deg(P)=2, where P is the minimal polynomial of α over Q).