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We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series. [ABSTRACT FROM AUTHOR]

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed. [ABSTRACT FROM AUTHOR]

We deal with the following functional equation: f ( x y ) + λ f ( x ) f ( y ) = ϕ ( x , y ) , where λ is a complex number and f and ϕ are complex mappings defined on a semigroup. Moreover, we assume an addition formula of trigonometric type for ϕ . Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003. [ABSTRACT FROM AUTHOR]

Let J be the set of all infinite complex continued fractions with partial numerators equal to one and partial denominators b 1 , b 2 , … , where each b k is a complex number of the form m + n i with m being a positive integer and n being any integer. In this paper we give an algorithm to determine lower bounds for the Hausdorff dimension of J . We show that the Hausdorff dimension of J is greater than 1.825, which is the best known lower bound, to the best of our knowledge. The ideas used to obtain this improved bound is different from those used by other authors and could be applied to estimate Hausdorff dimensions of other sets as well. [ABSTRACT FROM AUTHOR]

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P ( z ) = ∑ j = 0 n a j z j is called conjugate reciprocal if a n − j = a ‾ j , a j ∈ C for each j = 0 , 1 , … , n . Let ∂ D be the unit circle of the complex plane. We prove that there is an absolute constant ε > 0 such that max z ∈ ∂ D ⁡ | f ( z ) | ≥ ( 1 + ε ) 4 / 3 m 1 / 2 , for every conjugate reciprocal unimodular polynomial of degree m . We also prove that there is an absolute constant ε > 0 such that M q ( f ′ ) ≤ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 1 ≤ q < 2 , and M q ( f ′ ) ≥ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 2 < q , for every conjugate reciprocal unimodular polynomial of degree m , where M q ( g ) = ( 1 2 π ∫ 0 2 π | g ( e i t ) | q d t ) 1 / q , q > 0 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This study considers three stochastic transforms for jump diffusion processes: the stochastic Laplace transform, stochastic Fourier transform, and stochastic wavelet transform. First, we introduce the stochastic Laplace transform for processes adapted by Brownian filtration as a solution for complex number valued backward stochastic differential equations (BSDEs). This transform can be explained well by the Girsanov theorem, which also allows us to define the stochastic Laplace transform for jump diffusion processes directly. Based on this perspective, we give natural definitions of the stochastic Fourier transform and stochastic wavelet transform. The advantages of these stochastic transforms are all related to the uniqueness of the processes. Compared with the classical transforms, the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes, while they also agree with the corresponding parameters in the classical transforms, which can represent the frequency property of the processes. In addition, these three stochastic transforms can also be regarded as the solutions of related BSDEs with jumps. [ABSTRACT FROM AUTHOR]

Abstract: We say that a complex number λ is an extended eigenvalue of a bounded linear operator T on a Hilbert space if there exists a nonzero bounded linear operator X acting on , called extended eigenvector associated to λ, and satisfying the equation . In this paper we describe the sets of extended eigenvalues and extended eigenvectors for the product of a positive and a self-adjoint operator which are both injective. We also treat the case of normal operators. [Copyright &y& Elsevier]

Let X be a complex Banach space and denote by B (X) the algebra of all linear bounded operators on X. For T ∈ B (X) , denote by r (T) its spectral radius. We prove that if φ : B (X) → B (X) is a linear and surjective map such that i) given any A , B ∈ B (X) , there exists a positive real number M A , B such that r (φ (A) + λ φ (B)) ≤ M A , B r (A + λ B) for all complex numbers λ satisfying | λ | ≤ 1 ; ii) r (φ (T)) = 0 implies r (T) = 0 , then there exist a linear functional f : B (X) → C , a nonzero complex number c and either a bounded invertible linear map U : X → X such that φ (T) = c U T U − 1 + f (T) I (T ∈ B (X)) , or a bounded invertible linear map U : X ⁎ → X such that φ (T) = c U T ⁎ U − 1 + f (T) I (T ∈ B (X)). Also, the linear functional f satisfies the property that given any A , B ∈ B (X) there exists N A , B ≥ 0 such that | f (A) + λ f (B) | ≤ N A , B r (A + λ B) for all complex numbers λ satisfying | λ | ≤ 1. [ABSTRACT FROM AUTHOR]

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 − z a j ) , where each a j is a nonzero complex number. [ABSTRACT FROM AUTHOR]