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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]

Abstract: Let λ be a complex number in the closed unit disc, and be a separable Hilbert space with the orthonormal basis, say, . A bounded operator T on is called a λ-Toeplitz operator if (where is the inner product on ). The function with for and for is, on the other hand, called the symbol of T. Let us denote T by . It can be verified directly that is an “eigenoperator” associated with the eigenvalue λ for the following map on : where S is the unilateral shift defined by , . In an earlier joint work, the author used a result of M.T. Jury regarding the Fredholm theory of a certain Toeplitz-composition -algebra to show that if φ is in the class and if has finite order, then the essential spectrum of is “rotationally invariant” with respect to λ, i.e., In this paper, we prove that the restriction for the symbol φ in the above result can be dropped entirely, and the equation actually holds for any φ in and any . It turns out that the key for removing the assumption on the smoothness of φ depends only on the definition of and some very elementary properties of S as a Fredholm operator. The applications of this phenomenon for include a generalization of A. Wintnerʼs result on the spectra of Toeplitz operators with bounded analytic symbols. [Copyright &y& Elsevier]

Abstract: Let be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space , let be the space of multipliers from to . Obviously, ideals and are quasi-Banach algebras and it is clear that ideal is a bimodule for . We study the set of all derivations from into . We show that any such derivation is automatically continuous and there exists an operator such that , moreover for some complex number , where is the modulus of concavity of the quasi-norm and is the identity operator on . In the special case, when is a symmetric Banach ideal of compact operators on our result yields the classical fact that any derivation on may be written as , where is some bounded operator on and . [Copyright &y& Elsevier]

Abstract: The purpose of this note is to report a curious relation between the Chebyshev polynomials of degree in a complex variable and the nested square roots of depth of the form being a complex number. Our approach leads us to generalize and recover, in an unified manner, the closed-form expressions recently given by Servi, corresponding to the case , and by Nyblom, corresponding to . [Copyright &y& Elsevier]

Abstract: Recently, A. Al-Salman (2011) gave a characterization of the boundedness of the parametric Marcinkiewicz integral when ρ is a positive real number. In this note, using a quite elementary method, we give a sufficient and necessary condition for the boundedness of when ρ is a complex number. [Copyright &y& Elsevier]

Abstract: Let k be a positive integer and be a family of meromorphic functions in a domain such that each has only zeros of multiplicity at least . If for each pair (f, g) in , and share a non-zero complex number a ignoring multiplicity, then is normal in D. [Copyright &y& Elsevier]

Abstract: This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems. [Copyright &y& Elsevier]

Abstract: Take positive integers . Let be a family of meromorphic functions in a domain such that each has only zeros of multiplicity at least k. If, for each pair in , and share a non-zero complex number a ignoring multiplicity, then is normal in D. [Copyright &y& Elsevier]

Abstract: In this paper, we consider some normality criterion concerning shared values. Let be a family of meromorphic functions defined in a domain . Let be positive integers and be a finite complex number. If and share a in for every pair of functions , then is normal in . [Copyright &y& Elsevier]

Abstract: Let be a polynomial of degree n and for any real or complex number α, let denote the polar derivative of the polynomial with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros inside or outside a circle. Our results shall generalize and sharpen some well-known polynomial inequalities. [Copyright &y& Elsevier]

Abstract: We prove q-Taylor series for Jackson q-difference operators. Absolute and uniform convergence to the original function are proved for analytic functions. We derive interpolation results for entire functions of q-exponential growth which is less than , , from its values at the nodes , a is a non-zero complex number with absolute and uniform convergence criteria. [Copyright &y& Elsevier]

Abstract: A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space if there exists a non-zero bounded linear operator X acting on such that . We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set . [Copyright &y& Elsevier]

Abstract: A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order n-dimensional tensors is a function of m and n. We denote it by and show that , , for and for . We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank. [Copyright &y& Elsevier]

Abstract: A Hilbert space operator is hereditarily normaloid (notation: ) if every part of T is normaloid. An operator is totally hereditarily normaloid (notation: ) if every invertible part of T is normaloid. We prove that THN-operators with Bishop''s property , also THN-contractions with a compact defect operator such that and non-zero isolated eigenvalues of T are normal, are not supercyclic. Take A and B in THN and let denote either of the elementary operators in : and , where and . We prove that if non-zero isolated eigenvalues of A and B are normal and , then is an isoloid operator such that the quasi-nilpotent part of equals for every complex number λ which is isolated in . If, additionally, has the single-valued extension property at all points not in the Weyl spectrum of , then , and the conjugate operator , satisfy Weyl''s theorem. [Copyright &y& Elsevier]

Abstract: A Banach space operator is said to be totally hereditarily normaloid, , if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an operator for some integer , , if the quasi-nilpotent part for every complex number λ. It is proved that if T is algebraically , or T is algebraically THN and is separable, then satisfies Weyl''s theorem for every function f analytic in an open neighborhood of , and satisfies a-Weyl''s theorem. If also has the single valued extension property, then satisfies a-Weyl''s theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of on which it is defined. [Copyright &y& Elsevier]

Abstract: We find the singular values and corresponding Schmidt pairs of a compact composition operator induced by , where , on the classical Hardy space. We do so by solving a functional equation that is a generalization of Schröder''s equation: find a function f, holomorphic on the open unit disc, and a complex number λ such that , where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator . [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]