First we introduce the concept of QD-hulls in arbitrary abelian groups. Then we use the concept to give a new characterization of purifiable torsion-free finite rank subgroups of an arbitrary abelian group. Finally we use it to formulate a splitting criterion for mixed groups of torsion-free finite rank.

In this article we give a decomposition for harmonic functions in an infinite network X which is similar to the Laurent decomposition of harmonic functions defined on an annulus in ℝⁿ, n ≥ 2. Also we give a decomposition for biharmonic functions on bihyperbolic infinite networks.

In this paper, we investigate the order of growth of solutions of the higher order linear differential equation

f^(k) + Σ^k-1_j=0 (h_je^P_j(z) + d_j) f^(j) = 0,

where P_j(z) (j = 0,1,…,k-1) are nonconstant polynomials such that deg P_j = n ≥ 1 and h_j(z), d_j(z) (j = 0,1,…,k-1) with h₀ ≢ 0 are meromorphic functions of finite order such that max {ρ (h_j),ρ(d_j): j = 0,1,…,k-1} < n. We prove that every meromorphic solution f ≢ 0 of the above equation is of infinite order. Then, we use the exponent of convergence of zeros or the exponent of convergence of poles of solutions to obtain an estimation of the hyper-order of solutions.

Given d ≥ 2 consider the family of monic polynomials P_c(z) = z^d + c, for c ∈ ℂ. Denote by J_c and HD(J_c) the Julia set of P_c and the Hausdorff dimension of J_c respectively, and let $\mathcal{M}$_d = {c|J_c is connected} be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c₀ ∈ ∂ $\mathcal{M}$_d: those for which the critical point is not recurrent by P_c0, 0 ∈ J_c0, and without parabolic cycles. We prove that if P_cn Ⅺ P_c0 algebraically, then for some C > 0, d_H(J_cn, J_c0) ≤ C|c_n - c₀|^1/d, where d_H denotes the Hausdorff distance. If, in addition, P_cn Ⅺ P_c0 preserving critical relations, then P_cn is semihyperbolic for all n ≫ 0, and HD(J_cn) Ⅺ HD(J_c0).

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-Σ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.

We show that the family of all operator monotone functions f on (-1,1) such that f(0) = 0 and f′(0) = 1 is a normal family and investigate some properties of odd operator monotone functions. We also characterize the odd operator monotone functions and even operator convex functions on (-1,1). As a consequence, we show that if f is an odd operator monotone function on (-1,1), then f is concave on (-1,0) and convex on (0,1).

In this paper we derive boundary integral identities for the bi-Laplacian eigenvalue problems under Dirichlet, Navier and simply-supported boundary conditions. By using these integral identities, we prove that the first eigenvalue of the eigenvalue problem under the simply-supported boundary conditions strictly increases with Poisson's ratio. In addition, we establish the boundary integral expressions for the strain energy calculation of the vibrating plates under the three types of boundary conditions.

On a Kähler manifold we consider trajectories under the influence of Kähler magnetic fields. They are smooth curves which are parameterized by their arclengths and whose velocities and normal vectors form complex lines. In this paper we study how trajectories spread, and give an estimate of norms of magnetic Jacobi fields from below and an estimate of area elements of trajectory-spheres.

Let X be an n-dimensional smooth projective variety defined over the field of complex numbers, let L be an ample and spanned line bundle on X. Then we classify (X,L) with b₂(X,L) = h²(X,ℂ)+1, where b₂(X,L) is the second sectional Betti number of (X,L).