Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Mean growth and geometric zero distribution of solutions of linear differential equations

The aim of this paper is to consider certain conditions on the coefficient A of the differential equation f″ + Af = 0 in the unit disc which place all normal solutions f in the union of Hardy spaces or result in the zero-sequence of each non-trivial solution being uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.     

A study of neutrinoless double-beta decay in the framework of QRPA

A study concerning the valence space dependence of neutrinoless doubleβ ⁻ decay observables is performed for several nuclei in the framework of QRPA using the multiple commutator method (MCM). Calculations and comparison to earlier results show that at least for the ground state to ground state transition the dependence is much weaker than in the two neutrino doubleβ ⁻ decay. It is still a difficult task to extract the parameters of the neutrinoless doubleβ ⁻ decay since the half-life expression contains products of the parameters 〈m _ν 〉 〈η〉 and 〈λ〉. Presented by M. Aunola at the Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997.     

A new maximum likelihood approach with asymmetric residual distribution for multicomponent mass spectra analysis

This paper proposes a new maximum likelihood approach for the deconvolution of identity and quantity of individual compounds based on the multicomponent mass spectra measured by mass spectrometry (MS). Mixture analysis of multicomponent mass spectra is, typically, based on a linear multicomponent mass spectrum model, where the compounds of the measured spectra to be solved are explicitly stated and assumed to be known. In many cases, however, the measured spectrum may contain unknown compounds that are not explicitly stated in the model and a commonly used least square (LS) solution fails. Moreover, a standard improvement over the LS method in these cases, namely the M-estimation (ME) approach, also suffers from this same problem. Our method overcomes the limitations of the LS and ME methods by modeling the effect of the unknown compound(s) to the residual of the linear model. The experimental results presented show that this new approach can separate more robustly the complex multicomponent mass spectra into their individual constituents compared to the LS and ME methods.     

Synthesis, characterization, reactivity and theoretical studies of ruthenium carbonyl complexes containing ortho-substituted triphenyl phosphanes

A series of ruthenium o-phosphane complexes was synthesized and characterized. The reactivity of the prepared complexes was studied by using them as catalysts for the hydroformylation of 1-hexene. The activities depended on the binding mode of the phosphane and on the strength of the ruthenium-phosphane interaction. Strongly coordinated chelating [2-(dimethylamino)phenyl]-(diphenyl) phosphane and [2-(methylthio)phenyl]-(diphenyl) phosphane showed poor activity, while weakly chelated [2-(methoxy)phenyl]-(diphenyl) phosphane and non-chelating phosphanes such as [2-(methyl)phenyl]-(diphenyl) phosphane or [2-(ethyl)phenyl]-(diphenyl) phosphane led to higher activities.     

Solid-contact ion-selective electrodes for aromatic cations based on pi-coordinating soft carriers

Ion-selective electrodes (ISEs) based on pi-coordinating carriers were prepared and investigated as potentiometric sensors for aromatic cations, using N-methylpyridinium as a model aromatic cation. Derivatives of tetraphenylborate were studied as charged carriers in plasticized poly(vinyl chloride) membranes. Furthermore, neutral compounds containing pi-coordinating anthryl groups were studied as neutral carriers. Bis(2-ethylhexyl)sebacate (DOS) and 2-nitrophenyl octyl ether (o-NPOE) were used as non-polar and polar plasticizer, respectively. ISEs were constructed by using poly(3,4-ethylenedioxythiophene) (PEDOT) as solid-contact material. Conventional ISEs with internal filling solution were used for comparison. The potentiometric responses of the ISEs were investigated using N-methylpyridinium as primary ion. The results show that the selectivity of the ISEs is influenced significantly by both the plasticizer and the charged carriers, while the neutral carriers studied have only a minor influence on the selectivity. The role of cation-pi interactions between aromatic cations and the membrane components is discussed.     

A novel heterobidentate chiral phosphine and its coordination chemistry in transition metal clusters

The optically active ligand R,R-PHAZAN (1,3-bis[(1R)-1-Phenylethyl]-2-(2-thienyl)-1,3,2-diazaphospholane) has been prepared and the products resulting from the reactions with Rh₆(CO)₁₅NCMe, H₃RhOs₃(CO)₁₂, and H₄Ru₄(CO)₁₂ have been investigated by X-ray crystallography and a variety of multinuclear NMR methods. X-ray studies show that PHAZAN can behave as a bidentate ligand in Rh₆(CO)₁₄(μ₂,κ²-R,R-PHAZAN) (with coordination through P and S) or a monodentate ligand (through P coordination) in H₄Ru₄(CO)₁₁(κ¹-R,R-PHAZAN) and NMR studies show that these structures are retained in solution. In Rh₆(CO)₁₄(μ₂,κ²-R,R-PHAZAN), edge-bridging coordination of PHAZAN results in the formation of an additional two novel chiral centres and these are observed in solution. Reaction of PHAZAN with H₃RhOs₃(CO)₁₂ results in cleavage of the thienyl group and formation of the phosphido cluster, H₂RhOs₃(CO)₁₁(μ₂-PNN), (PNN = 1,3-bis-(1-phenylethyl)-[1,3,2]diazaphospholidine-2-yl). A variety of NMR measurements show that the hydride site-occupancies in the solid state are retained in solution and there is evidence for interaction of an ortho-phenyl hydrogen and a hydride through “dihydrogen” bonding.     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)