Nonzero Complex Number Research Articles

On Repeated Values of the Riemann Zeta Function on the Critical Line

Let ζ(s) be the Riemann zeta function. In this paper, we study repeated values of ζ(s) on the critical line, and we give evidence to support our conjecture that for every nonzero complex number z, the equation ζ(1/2+it)=z has at most two solutions . We prove a number of related results, some of which are unconditional, and some of which depend on the truth of the Riemann hypothesis. We also propose some related conjectures that are implied by Montgomery's pair correlation conjecture.

On invertibility of combinations of k-potent operators

Some properties of combinations c1P1+c2P2-c3P1sP2k-1-s, where P1 and P2 are two different nonzero k-potents, c1, c2 and c3 are three nonzero complex numbers and positive integers k⩾2 and s⩽k-1, are obtained. Furthermore, the invertibility, the group invertibility and the k-potency of the linear combinations of k-potents are investigated, under certain commutativity properties imposed on them.

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let \({\mathcal{B}(X)}\) be the space of all bounded linear operators on X. For \({x \in X}\) and \({T \in \mathcal{B}(X)}\), let \({r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}}\) denote the local spectral radius of T at x. We prove that if \({\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)}\) is linear and surjective such that for every \({x \in X}\) we have rT(x) = 0 if and only if \({r_{\varphi(T)}(x) = 0}\), there exists then a nonzero complex number c such that \({\varphi(T) = cT}\) for all \({T \in \mathcal{B}(X) }\). We also prove that if Y is a complex Banach space and \({\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)}\) is linear and invertible for which there exists \({B \in \mathcal{B}(Y, X)}\) such that for \({y \in Y}\) we have rT(By) = 0 if and only if \({ r_{\varphi ( T) }(y)=0}\), then B is invertible and there exists a nonzero complex number c such that \({\varphi(T) =cB^{-1}TB}\) for all \({T \in \mathcal{B}(X)}\).

Entire Solutions of f(kz) = kf(z)f′(z)

A colleague asked whether x, sin x and sinh x are essentially the only solutions of the non-linear differential equation f(2x) = 2f(x)f′ (x) on [0, +∞) with boundary condition f′(0) = 1. Here we study the formal power series solutions of the more general equation f(kz) = kf(z)f′(z), where k is a non-zero complex number, with f(0) = 0, and we show that a transcendental, formal power series solution exists for only a countable set of k, and that each such solution is an entire function.

Tensor Product of Krammer’s Representations of the P<sub>3</sub>

We consider the complex specializations of Krammer’s representation of the pure braid group on three strings, namely K(q,t), where q and t are non-zero complex numbers. We then specialize the indeterminate t by one and replace by for simplicity. Then we present our main theorem that gives us sufficient conditions that guarantee the irreducibility of the tensor product of two irreducible complex specializations of Krammer’s repre- sentations .

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

Let T 1 and T 2 be two n × n tripotent matrices and c 1, c 2 two nonzero complex numbers. We mainly study the nonsingularity of combinations T = c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 of two tripotent matrices T 1 and T 2, and give some formulae for the inverse of c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 under some conditions. Some of these results are given in terms of group invertible matrices.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

The reduced Burau representation is a one-parameter representation of , the braid group on strings. Specializing the parameter to nonzero complex number gives a representation : , which is either irreducible or has an irreducible composition factor : . In our paper, we let , and we determine a sufficient condition for the irreducibility of the tensor product of irreducible Burau representations. This is a generalization of our previous work concerning the cases and .

Higher Dimensional Irreducible Representations of the Pure Braid Group

The reduced Gassner representation is a multi-parameter representation of $%P_{n},$ the pure braid group on n strings. Specializing the parameters $%t_{1},t_{2},...,t_{n}$ to nonzero complex numbers $x_{1},x_{2},...,x_{n}$gives a representation $G_{n}(x_{1},\ldots ,x_{n}):P_{n}\rightarrowGL(\mathbb{C}^{n-1})$ which is irreducible if and only if $x_{1}\ldotsx_{n}\neq 1$. In a previous work, we found a sufficient condition for theirreducibility of the tensor product of two irreducible Gassnerrepresentations. In our current work, we find a sufficient condition that guarantees the irreducibility of the tensor product of three Gassner representations. Next, a generalization of our result is given by considering the irreducibility of the tensor product of $k$ representations (\;$k \geq 3\;$).

Quasifinite representations of a class of Block type Lie algebras [formula omitted

Intrigued by a well-known theorem of Mathieu’s on Harish-Chandra modules over the Virasoro algebra, we show that any quasifinite irreducible module over a class of Block type Lie algebras B ( q ) is either a highest or lowest weight module, or else a uniformly bounded module, where the parameter q is a nonzero complex number. We also classify quasifinite irreducible highest weight B ( q ) -modules and irreducible B ( q ) -modules of the intermediate series. In particular, we obtain that an irreducible B ( q ) -module of the intermediate series may be a nontrivial extension of a V ir -module of the intermediate series if q is half of a negative integer, where V ir is a subalgebra of B ( q ) isomorphic to the Virasoro algebra.

Mellin transforms attached to certain automorphic integrals

Let k be any real number with k < 2 . We will consider complex-valued smooth functions f , f ˜ on H of period 1, having exponential decay at infinity (i.e. they are ≪ e − c y for y = ℑ ( z ) → ∞ with c > 0 ) and such that f | k W N = C f ˜ + q g . Here | k is an appropriately defined Petersson slash operator in weight k, C ∈ C ⁎ is a constant and q g ( z ) : = ∫ 0 i ∞ g ( τ ) ( τ − z ¯ ) − k d τ ¯ ( z ∈ H ) is a period integral attached to a holomorphic function g : H → C such that both g and g | 2 − k W N have period 1, have only positive terms in their Fourier expansions and the Fourier coefficients are of polynomial growth. An arbitrary power of a non-zero complex number is defined by means of the principal branch of the complex logarithm. Under the assumption that k < 1 , we will show that the Mellin transform M ( f , s ) ( σ ≫ 1 ) naturally attached to f has meromorphic continuation to C and we will establish an explicit formula for it (Section 2, Theorem 1). There are possible simple poles at the points s = − n where n = 0 , 1 , 2 , … and the residue at s = − n essentially is equal to the “ n-th period” ∫ 0 ∞ g ( i t ) t n d t of g. Moreover, there again is a functional equation relating M ( f , s ) and M ( f ˜ , k − s ) .

Uniqueness theorems of entire functions sharing a nonzero complex number with their difference operators

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (2009), 270–278) and deal with Question 1 in Liu and Yang (2009), where the entire functions are of finite orders.

Weak coherent states related to the multiplicative group

Generalized coherent states which are associated with the multiplicative group of non-zero complex numbers are introduced. They have the property of temporal stability, and they form a total set in some subspace of the full Hilbert space. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to regions where their Wigner functions take negative values. States with Bargmann functions sin (πAz) and exp (Bz)/[Γ(Az)]k are examples of the general formalism, and they are discussed in detail.

The spectrum of matrices depending on two idempotents

Let P and Q be two complex matrices satisfying P 2 = P and Q 2 = Q . For a , b nonzero complex numbers such that a P + b Q is diagonalizable, we relate the spectrum of a P + b Q to the spectra of P − Q , P Q , P Q P and P Q − Q P .

Normality criteria of meromorphic functions sharing one value

Let k be a positive integer and F be a family of meromorphic functions in a domain D ⊂ C such that each f ∈ F has only zeros of multiplicity at least k + 1 . If for each pair ( f, g) in F , f f ( k ) and g g ( k ) share a non-zero complex number a ignoring multiplicity, then F is normal in D.

Differential polynomials and shared values

Let f and g be non-constant meromorphic functions in C, a and b non-zero complex numbers and let n and k be natural numbers satisfying n ‚ 5k+17. We show that if the dierential polynomials f n + af (k) and g n + ag (k) share the value b CM, then f and g are either equal or at least closely related.

INVOLUTORY AND S+1-POTENCY OF LINEAR COMBINATIONS OF A TRIPOTENT MATRIX AND AN ARBITRARY MATRIX

Let and be nonzero complex matrices, denote a linear combination of the two matrices by , where , are nonzero complex numbers. In this paper, we research the problem of the linear combinations in the general case. We give a sufficient and necessary condition for A is an involutive matrix and s+1-potent matrix, respectively, where is a tripotent matrix, with . Then, using the results, we also give the sufficient and necessary conditions for the involutory of the linear combination A, where is a tripotent matrix, anti-idempotent matrix, and involutive matrix, respectively, and is a tripotent matrix, idempotent matrix, and involutive matrix, respectively, with .

Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n⩾m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.

Nonsingularity and group invertibility of linear combinations of two k-potent matrices

An n × n complex matrix A is said to be k-potent if A k = A. Let T 1 and T 2 be k-potent and c 1 and c 2 be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c 1 T 1 + c 2 T 2 of two k-potent matrices T 1 and T 2.

Theorems of W. W. Stothers and the Jacobian Conjecture in two variables

Differential equations of the form rp(z)q'(z) ― sp'(z)q(z) = γp(z) or rp(z)q'(z) ― sp'(z)q(z) = ―, where ― is a nonzero complex number and r, s are positive integers, have arisen in attempts to solve the two-variable Jacobian Conjecture. Solutions of such equations, in which p(z) and q(z) are monic complex polynomials of positive degrees r and s, give rise to extra-special pairs of polynomials in the sense of W. W. Stothers. Stothers showed that, modulo automorphisms of ℂ[z], there are only finitely many extra-special pairs of a given degree n. This implies that, modulo automorphisms of ℂ[z], there are only finitely many solutions of the above differential equations in which p(z) and q(z) are monic polynomials of given degrees r and s.