A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions

Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λ ∈ V ? ? ⁿ . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions \({\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}\) at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form \(\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} \), z ∈ ?, where P _k and Q _k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P _k and Q _k .     

Gelfand-Kirillov dimension of algebras with locally nilpotent derivations

Let R be a finitely generated algebra over a field of characteristic 0 with a locally nilpotent derivation δ ≠ 0. We show that if {ie313-1}, where the invariants {ie313-2} are prime and satisfy a polynomial identity, then {ie313-3}. Furthermore, when R is a domain, the same conclusion holds without the assumption that R is finitely generated. This enables us to obtain a result on skew polynomial rings. These results extend work of Bell and Smoktunowicz on domains with GK dimension in the interval [2, 3).     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

Ojective ideals in modular lattices

The concept of an extending ideal in a modular lattice is introduced. A translation of module-theoretical concept of ojectivity (i.e. generalized relative injectivity) in the context of the lattice of ideals of a modular lattice is introduced. In a modular lattice satisfying a certain condition, a characterization is given for direct summands of an extending ideal to be mutually ojective. We define exchangeable decomposition and internal exchange property of an ideal in a modular lattice. It is shown that a finite decomposition of an extending ideal is exchangeable if and only if its summands are mutually ojective.     

Toeplitz Operators with Uniformly Continuous Symbols

Let T _f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi: 10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on C ⁿ and the Bergman metric on \({\Omega}\), respectively, the operator T _f is bounded if and only if f is bounded. Moreover, T _f is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).     

Charge carrier decay and diffusion in organic–inorganic CH3NH3PbI3–xClx perovskite based solar cell

In recent years, organic–inorganic lead halides attracted widespread interest, mainly due to their impressive photoconversion properties and low‐cost solution processing. In this study, we employed small amplitude transient photovoltage and photocurrent spectroscopy to investigate charge transport and recombination properties of perovskite CH₃NH₃PbI_3–xCl_x solar cell under realistic light harvesting conditions (<1 sun). Cell structure resembles outlay commonly found in organic photovoltaics, with perovskite absorber being sandwiched between two thin layers of organic polymers. Tested device displayed high power conversion efficiency (10.3%), good fill factor and negligible hysteresis effect. Fundamental device parameters were characterized at various open‐circuit voltages (V_oc) by examination of small voltage and current perturbations created by the low intensity pulsed laser excitations. The obtained results exhibit long charge carrier lifetimes and fast charge transport over the full range of applied optical bias, as well as remarkable diffusion lengths exceeding 1 μm. (© 2015 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)