An alternative Cauchy functional equation on a semigroup

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation ${f(xy) - f(x) - f(y) \in \{ 0, 1\}}$ where ${f : S \to \mathbb{R}, S}$ is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions ${f : H \to \mathbb{R}}$ of the alternative functional equation on a semigroup ${H = \langle a, b| a^2 = a, b^2 = b \rangle }$ where the Cauchy equation is not stable.     

Simultaneous generalizations of the theorems of Ceva and Menelaus for field planes

In 1992, Klamkin and Liu proved a very general result in the Extended Euclidean Plane that contains the theorems of Ceva and Menelaus as special cases. In this article, we extend the Klamkin and Liu result to projective planes PG(2, 𝔽) where 𝔽 is a field.     

Connecting binuclear Pd(III) and mononuclear Pd(IV) chemistry by Pd-Pd bond cleavage

Oxidation of binuclear Pd(II) complexes with PhICl(2) or PhI(OAc)(2) has previously been shown to afford binuclear Pd(III) complexes featuring a Pd-Pd bond. In contrast, oxidation of binuclear Pd(II) complexes with electrophilic trifluoromethylating ("CF(3)(+)") reagents has been reported to afford mononuclear Pd(IV) complexes. Herein, we report experimental and computational studies of the oxidation of a binuclear Pd(II) complex with "CF(3)(+)" reagents. These studies suggest that a mononuclear Pd(IV) complex is generated by an oxidation-fragmentation sequence proceeding via fragmentation of an initially formed, formally binuclear Pd(III), intermediate. The observation that binuclear Pd(III) and mononuclear Pd(IV) complexes are accessible in the same reactions offers an opportunity for understanding the role of nuclearity in both oxidation and subsequent C-X bond-forming reactions.     

The Preparation of Luminescent,Mechanochromic Molecular Containers from Non-Emissive Components: The Box Cations, [Au6(Triphos)4Br]5+ and [Au6(Triphos)4Br2]4+

Two new molecular boxes, the mono-bromo box [Au₆(Triphos)₄Br](SbF₆)₅⋅6(CH₂Cl₂), 4 mB , and the dibromo box, [Au₆(Triphos)₄Br₂⋅H₂O](SbF₆)₄⋅4(CH₂Cl₂), 5 dB , have been prepared in crystalline form. Although constructed from non-luminescent components, both are strongly luminescent. Like its chloro counterpart, the mono-bromo box [Au₆(Triphos)₄Br](SbF₆)₅⋅6(CH₂Cl₂), 4 mB , is mechanochromic. Under grinding, it loses its luminescence. The bromo-bridged helicate, [(μ-Br){Au₃(Triphos)₂}₂](CF₃SO₃)₅⋅2(CH₂Cl₂), 3μ-H , with a cation that is isomeric with the box [Au₆(Triphos)₄Br]⁵⁺, has also been prepared and crystallographically characterized. Unlike its chloro analogue, [(μ-Br){Au₃(Triphos)₂}₂](CF₃SO₃)₅⋅2(CH₂Cl₂) is not luminescent. Thus, the cation produced upon grinding may be the cation present in the bromo-bridged helicate, [(μ-Br){Au₃(Triphos)₂}₂](CF₃SO₃)₅⋅2(CH₂Cl₂), 3μ-H . The dibromo box, [Au₆(Triphos)₄Br₂⋅H₂O](SbF₆)₄⋅4(CH₂Cl₂), 5 dB , is not significantly mechanochromic.     

Pulse labeling of dissolved (13) C-carbonate into tree xylem: developing a new method to determine the fate of recently fixed photosynthate

Stable carbon isotopes are often employed as tracers in plant and soil systems to study the fate and transformations of carbon as is it assimilated by the forest canopies and then translocated into the soil matrix and soil microorganisms. This experiment tested a new method of ¹³C‐labeling. We dissolved ¹³C‐carbonate into 12 mL of water and injected it into the xylem of a 6‐cm diameter tree. The isotopic composition of foliage, stem CO₂, and phloem contents were measured before the experiment and up to two weeks after the pulse label. Isotopic enrichments of 6.1‰ and 7.7‰ were observed in stem CO₂ and phloem contents, respectively. No enrichment in bulk foliage was observed. The pulse came through the phloem five days after the label was injected, consistent with expectations based on transport rates through the tree. The application of this xylem pulse‐labeling method may provide new insights into labile carbon sequestration in trees, perhaps even in much larger trees. Furthermore, the method could be applied under experimental treatments that would elucidate the mechanisms controlling the fate and transformation of recently fixed photosynthate in forests. Copyright © 2010 John Wiley & Sons, Ltd.