Nonzero Complex Number Research Articles

Inertia of complex unit gain graphs

Let T={z∈C:|z|=1} be a subgroup of the multiplicative group of all nonzero complex numbers C×. A T-gain graph is a triple Φ=(G,T,φ) consisting of a graph G=(V,E), the circle group T and a gain function φ:E→→T such that φ(eij)=φ(eji)−1=φ(eji)¯. The adjacency matrix A(Φ) of the T-gain graph Φ=(G,φ) of order n is an n × n complex matrix (aij), whereaij={φ(eij),ifviisadjacenttovj,0,otherwise.Evidently this matrix is Hermitian. The inertia of Φ is defined to be the triple In(Φ)=(i+(Φ),i−(Φ),i0(Φ)), where i+(Φ),i−(Φ),i0(Φ) are numbers of the positive, negative and zero eigenvalues of A(Φ) including multiplicities, respectively. In this paper we investigate some properties of inertia of T-gain graph.

On k-circulant matrices (with geometric sequence)

Let k be a nonzero complex number. In this paper we show how the inverse of a nonsingular k-circulant matrix can be obtained. The method is used to determine the inverse of a nonsingular k-circulant matrix with geometric sequence. If k = 1, then we get the result presented in the paper A.C.F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Res. 1(4) (2012), 593– 603. Also, we derive the Moore-Penrose inverse of a singular k-circulant matrix with geometric sequence. At the end of the paper, we illustrate the obtained results by examples.

Nonlinear maps preserving Jordan triple η-⁎-products

Let η≠−1 be a non-zero complex number, and let ϕ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections, satisfying ϕ(I)=I and preserving the Jordan triple η-⁎-product. It is showed that ϕ is a linear ⁎-isomorphism if η is not real and ϕ is the sum of a linear ⁎-isomorphism and a conjugate linear ⁎-isomorphism if η is real.

On the Irreducibility of Artin's Group of Graphs

We consider the graph $E_{3,1}$ with three generators $\sigma_1, \sigma_2, \delta$, where $\sigma_1$ has an edge with each of $\;\sigma_2$ and $\;\delta$. We then define the Artin group of the graph $E_{3,1}$ and consider its reduced Perron representation of degree three. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation.<br />

Non-Existence of Common Hypercyclic Entire Functions for Certain Families of Translation Operators

Let $${\mathcal {H}}({\mathbb {C}})$$ be the set of entire functions endowed with the topology of local uniform convergence. Fix a sequence of non-zero complex numbers $$({\lambda }_n)$$ with $$\liminf _{n}\frac{|{\lambda }_{n+1}|}{|{\lambda }_n|}>2.$$ We prove that there exists no entire function $$f$$ such that for every $$b\in \mathbb {C}\setminus \{ 0\}$$ the set $$\{ f(z+{\lambda }_nb): n=1,2,\ldots \}$$ is dense in $${\mathcal {H}}({\mathbb {C}}).$$ This, on one hand gives a negative answer to Costakis (Approximation by translates of entire functions, Complex and harmonic analysis. Destech Publ., Inc., Lancaster, pp 213–219 2007, Question 2) and on the other hand shows that certain results from Tsirivas (Existence of common hypercyclic vectors for translation operators, 2014; Common hypercyclic functions for translation operators with large gaps, 2014) are sharp.

Generalized Lacunary Statistical Difference Sequence Spaces of Fractional Order

We generalize the lacunary statistical convergence by introducing the generalized difference operatorΔναof fractional order, whereαis a proper fraction andν=(νk)is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator and investigate the topological structures of related sequence spaces. Furthermore, we introduce some properties of the strongly Cesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relations of these spaces. We also determine the relationship between lacunary statistical and strong Cesaro difference sequence spaces of fractional order.

Efficient realization of nonzero spectra by polynomial matrices

A theorem of Boyle and Handelman gives necessary and sufficient conditions for an n-tuple of nonzero complex numbers to be the nonzero spectrum of some matrix with nonnegative entries, but is not constructive and puts no bound on the necessary dimension of the matrix. Working with polynomial matrices, we constructively reprove this theorem in a special case, with a bound on the size of the polynomial matrix required to realize a given polynomial.

Associating vertex algebras with the unitary Lie algebra

In this paper, we associate vertex algebras and their two different kinds of module categories with the unitary Lie algebra uˆN(CΓ˜) for N≥2 being a positive integer and Γ˜={qn|n∈Z}, where the nonzero complex number q is not a root of unity. It is proved that for any complex number ℓ, the category of restricted uˆN(CΓ˜)-modules of level ℓ is canonically isomorphic to the category of quasi modules for certain vertex algebra. And we also prove that the category of restricted uˆN(CΓ˜)-modules of level ℓ is isomorphic to the category of Γ-equivariant ϕ-coordinated quasi modules for the same vertex algebra, where Γ is an automorphism group of this vertex algebra.

Meromorphic solutions of a type of system of complex differential-difference equations

Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form (*){∑j=1nαj(z)f1(λj1)(z+cj)=R2(z,f2(z)),∑j=1nβj(z)f2λj2(z+cj)=R1(z,f1(z)). where λij (j = 1,2,···, n; i = 1,2) are finite non-negative integers, and cj (j = 1,2,·,n) are distinct, nonzero complex numbers, αj(z), βj(z) (j = 1,2,·,n) are small functions relative to fi(z) (i = 1,2) respectively, Ri(z,f(z)) (i = 1,2) are rational in fi(z) (i = 1,2) with coefficients which are small functions of fi(z)(i = 1,2) respectively.

Twisted Γ-Lie algebras and their vertex operator representations

Let Γ be a generic subgroup of the multiplicative group C⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of ZN and an arbitrary finite order isometry of ZN preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As an application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras.

Multiplicative Subgroups of ℂ that Contain Regular Jordan Curves

SummaryIn this paper, we characterize subgroups of the multiplicative group ℂ× of nonzero complex numbers that contain Jordan curves of area zero.

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

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.

Approximations of Analytic Functions via Generalized Power Product Expansions

Let f(x)=1+∑n=1∞anxn be a formal power series. For a fixed set of nonzero complex numbers {rk}k=1∞ we convert f(x) into the formal product ∏k=1∞(1+gkxk)rk, namely the Generalized Power Product Expansion. We provide estimates on the domain of absolute convergence of the infinite product when f(x)=1+∑n=1∞anxn is absolutely convergent. This makes it possible to use the truncated Generalized Power Product Expansions ∏n=1M(1+gkxk)rk as approximations to the analytic function f(x). The results are made possible by certain intriguing algebraic properties characteristic of the expansions for the case of rk≥1. An asymptotic formula for the gk associated with the majorizing power series is provided. A combinatorial interpretation of the Generalized Power Product Expansion with {rk}k=1∞ being integers is also given.

On the quadraticity of linear combinations of quadratic matrices

Let and be nonzero quadratic matrices and let and be nonzero complex numbers. Necessary and sufficient conditions for the quadraticity of the linear combinations of the form are obtained. Our main results contain many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices. Finally, some numerical examples are given to exemplify the main results.

A-Points of the Riemann Zeta-Function on the Critical Line

We investigate the proportion of the nontrivial roots of the equation $\zeta (s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to $\zeta (s)=a$ lie on the line $\Re s=1/2$ for any nonzero complex number $a$.

Tuba’s Representation of the Pure Braid Group on Three Strands

We consider Tuba’s representation of the pure braid group, P3, defined by the map ψ : P3 −→ GL (V ), where V is an algebraically closed field. We then specialize the indeterminates used in defining the representation to nonzero complex numbers. Our objective is to find necessary and sufficient conditions that guarantee the irreducibility of Tuba’s representations of the pure braid group P3 with dimensions d = 2 and d = 3.

Irreducible quasifinite modules over a class of Lie algebras of block type

For any nonzero complex number $q$, there is a Lie algebra of Block type, denoted by $\mathcal{B}(q)$. In this paper, a complete classification of irreducible quasifinite modules is given. More precisely, an irreducible quasifinite module is a highest weight or lowest weight module, or a module of intermediate series. As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct product of the Virasoro algebra and a module of intermediate series, is also obtained. Our method is conceptional, instead of computational.

Normal families of meromorphic functions concerning shared values

In this article, we prove the following normality criterion: let n ≥ 2, m, k be three positive integers, and a be a non-zero complex number. Let ℱ be a family of meromorphic functions in a domain D such that each f ∈ ℱ has only zeros of multiplicity at least max{k, 2}. If for each pair of f and g in ℱ, f m (f (k)) n and g m (g (k)) n share the value a IM, then ℱ is normal in D. This extends Hu and Meng's result.

No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$

Consider the family $f_{\lambda, \gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}$ where $\lambda$ and $\gamma$ are non-zero complex numbers. It contains the sine family $\lambda \sin z$ and is a natural extension of the sine family. We give a direct proof of that the escaping set $I_{\lambda, \gamma}$ of $f_{\lambda, \gamma}$ supports no $f_{\lambda,\gamma}$-invariant line fields.

A remark on the uniqueness of the Dirichlet series with a Riemann-type function equation

We show that if for a nonzero complex number c the inverse images L1−1(c) and L2−1(c) of two functions in the extended Selberg class are the same, then L1(s) and L2(s) must be identical.