AN EXTRAGRADIENT METHOD FOR RELAXED COCOERCIVE VARIATIONAL INEQUALITY AND EQUILIBRIUM PROBLEMS

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of solutions of the variational inequality prob- lem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao[17], Takahashi[12] and many others.     

Hydrogen bonding in labdane diterpenoids, labda-7,12(E),14-triene-17-oic acid and labda-12(Z),14,17-triene-18-oic acid

Two labdane diterpenoids, labda-7,12(E),14-triene-17-oic acid (1) and labda-12(Z),14,17-triene-18-oic acid (2), C₂₀H₃₀O₂, have been isolated from Croton oblongifolius. Both 1 and 2 crystallized in the monoclinic system, space group C2, with cell dimensions of a = 21.912(1) Å, b = 7.4002(4) Å, c = 11.5079(7) Å, = 101.999(1)^o and a = 21.308(2) Å, b = 11.9067(9) Å, c = 7.5606(6) Å, = 100.763(1)^o, respectively. Compound 1, a rare example of carboxylic group bound to a cyclohexene ring, forms an infinite intermolecular hydrogen-bonded polymer [O1 O2(–x + 1/2, y + 1/2, –z + 2) 2.697(2) Å], whereas molecules of compound 2 are linked to form an asymmetric hydrogen-bonded dimer [O1 O2(–x, y, –z) 2.657(3) Å].     

Synthesis of tripodal aza crown ether calix[4]arenes and their supramolecular chemistry with transition-, alkali metal ions and anions

Tripodal aza crown ether calix[4]arenes, 5a, 5b, 6a and 6b, have been synthesized. The structure of protonated 5a was elucidated by X-ray crystallography to be a self-threaded rotaxane. Complexation studies of 5a and 5b towards anions using Na⁺ as countercation were carried out by ¹H NMR titration in dimethylsulfoxide-d₆ and the mixture of chloroform-d and methanol-d₄, respectively. Ligands 5a and 5b were able to form 1:1 complexes with Br⁻, I⁻ and NO₃⁻ and the complexation stability varied as follows: NO₃⁻>I⁻>Br⁻. The effect of countercation on anion complexation was also investigated. The results showed that the association constants of 5a towards Br⁻ in the presence of various cations varied as K⁺>Bu₄N⁺>Na⁺. The enhancement in anion complexation ability of 5a may result from the rearrangement of the tripodal ammonium unit in the presence of K⁺. The neutral forms, 6a and 6b, were able to form complexes with transition metal ions such as Co²⁺, Ni²⁺, Cu²⁺ and Zn²⁺. The stability of the complexes followed the sequence: Ni²⁺2+Cu²⁺Zn²⁺. Compounds 6a and 6b may, therefore, potentially be used as either transition metal ion or anion receptors that can be controlled by pH of the solution.     

(Triphenylarsine)iodinemonobromine: a charge‐transfer adduct in which arsenic selectively bonds to iodine

The title molecule, (iodobromo)triphenylarsenic(III), [As(BrI)(C₆H₅)₃], has a dart shape, with the phenyl rings arranged in a propeller conformation [I—As—C—C 61.3 (4), 43.8 (4) and 54.1 (4)°]. There is no indication that the halogen atoms have mixed site occupancies. Packing forces displace the I atom away from one phenyl ring [I—As—C 117.3 (2)°] towards the other two [I—As—C 109.8 (2) and 108.3 (2)°] and produce an even more pronounced leaning of the terminal bromine [As—I—Br 174.78 (2)°].     

Independence of continued fractions in the field of Laurent series

Criteria for independence, both algebraic and linear, are derived for continued fraction expansions of elements in the field of Laurent series. These criteria are then applied to examples involving elements recently discovered to have explicit series and continued fraction expansions. Communicated by Attila Pethő     

Two types of explicit continued fractions

Summary Two types of explicit continued fractions are presented. The continued fractions of the first type include those discovered by Shallit in 1979 and 1982, which were later generalized by Pethő. They are further extended here using Peth\H o's method. The continued fractions of the second type include those whose partial denominators form an arithmetic progression as expounded by Lehmer in 1973. We give here another derivation based on a modification of Komatsu's method and derive its generalization. Similar results are also established for continued fractions in the field of formal series over a finite base field.     

A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.     

Exogenous Spermidine Alleviates UV-Induced Growth Inhibition of Synechocystis sp. PCC 6803 via Reduction of Hydrogen Peroxide and Malonaldehyde Levels

Effects of exogenously added spermidine (Spd) to UV-treated Synechocystis sp. PCC 6803 cultures on their growth, intracellular pigments, hydrogen peroxide (H₂O₂), malonaldehyde (MDA) contents, and antioxidant enzymes were investigated. Growth inhibition of cells subjected to 1-h UV-A, UV-B, and UV-C irradiation was abolished in culture added with 0.5 mM Spd. Both chlorophyll a and carotenoid contents were decreased under UV radiations in cells grown in BG₁₁ medium. However, the contents of these two pigments were slightly increased under UV radiations in Spd-supplemented cells with the consequence of enhanced oxygen evolution. Intracellular levels of H₂O₂ and MDA generated during 1-h UV irradiation were decreased when the culture medium contained 0.5 mM Spd. The antioxidative enzymes, catalase, and superoxide dismutase had a little or no response towards Spd supplementation under UV irradiation except for some increase in superoxide dismutase activity under UV-C. Total intracellular polyamines were decreased during Spd supplementation under UV stress; however, the cells showed a drastic increase in the amount of Put under this condition. Altogether, exogenous Spd is likely a potential compound that enables Synechocystis cells to cope with UV stress.