Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let C$C$ be a closed convex subset of a real Hilbert space H$H$ and assume that T$T$ is an asymptotically κ$\kappa$-strict pseudo-contraction on C$C$ with a fixed point, for some 0≤κ<1$0\le \kappa <1$. Given an initial guess _x0∈C$x_0\in C$ and given also a real sequence {_αn}$\left\{{\alpha }_n\right\}$ in (0, 1), the modified Mann’s algorithm generates a sequence {_xn}$\left\{x_n\right\}$ via the formula: _xn+1=_αnxn+(1−_αn)Tⁿ_xn$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n$, n≥0$n\ge 0$. It is proved that if the control sequence {_αn}$\left\{{\alpha }_n\right\}$ is chosen so that κ+δ<_αn<1−δ$\kappa +\delta <{\alpha }_n<1-\delta$ for some δ∈(0,1)$\delta \in (0,1)$, then {_xn}$\left\{x_n\right\}$ converges weakly to a fixed point of T$T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.     

Synthesis,crystal structure and magnetism of a three-dimensional manganese(II) complex with carbamyldicyanomethanide anion (<Emphasis Type="Italic">cda</Emphasis>) as a bridging ligand

A three-dimensional complex {[Mn(cda)₂]·2MeOH}_n has been synthesized and its crystal structure determined by X-ray crystallography. In the complex each Mn ion is located in a distorted octahedral environment with two oxygen atoms O(1D) and O(1E) from two amide groups, and four nitrogen atoms N(2), N(2A), N(3B) and N(3C) from four nitrile groups of four cda anions, respectively; each cda anion as a ₃-bridging ligand provides two nitrile nitrogen atoms, and an amide oxygen atoms to coordinate three Mn ions. Thus, a three-dimensional network consisting of Mn ions and cda bridging ligands is constructed with solvent MeOH molecules trapped in the cavities. The variable-temperature magnetic susceptibility of the complex was measured in the 5–300K range. The magnetic coupling parameter is consistent with an antiferromagnetic exchange and generates the antiferromagnetic coupling parameter, 2J=–0.2cm^–1.     

刺乌头碱氢溴酸盐的合成及晶体结构

报道了以双溴代烷烃和刺乌头碱合成刺乌头碱氢溴酸盐的方法. 用元素分析、红外光谱、高分辨质谱和核磁共振进行了表征. 并用X射线单晶衍射确定了标题化合物的绝对构型. 晶体结构表明, 该化合物通过分子间氢键形成了网状类似超分子结构. 晶体属于单斜晶系, P2₁空间群, 晶胞参数: a＝1.0619(2) nm, b＝1.2196(3) nm, c＝1.2282(2) nm, β＝90.87(1)°, V＝1.59037(54) nm³, Z＝2, D_m＝1.428 g/cm^－3, F(000)＝720.0, µ＝1.349 mm^－1. 环 A, B, C, D, E和F分别呈船式、椅式、信封式、船式、船式和信封式. 其绝对构型被确定为1S,4S,5S,7S,8S,9S,10S,11S,13R,14S,16S,17R.     

Study on the non-isothermal kinetics of decomposition of 4Na<Subscript>2</Subscript>SO<Subscript>4</Subscript>·2H<Subscript>2</Subscript>O<Subscript>2</Subscript>·NaCl

The non-isothermal decomposition kinetics of 4Na₂SO₄·2H₂O₂·NaCl have been investigated by simultaneous TG-DSC in nitrogen atmosphere and in air. The decomposition processes undergo a single step reaction. The multivariate nonlinear regression technique is used to distinguish kinetic model of 4Na₂SO₄·2H₂O₂·NaCl. Results indicate that the reaction type Cn can well describe the decomposition process, the decomposition mechanism is n-dimensional autocatalysis. The kinetic parameters, n, A and E are obtained via multivariate nonlinear regression. The n ^th-order with autocatalysis model is used to simulate the thermal decomposition of 4Na₂SO₄·2H₂O₂·NaCl under isothermal conditions at various temperatures. The flow rate of gas has little effect on the decomposition of 4Na₂SO₄·2H₂O₂·NaCl.     

Perturbation techniques for nonexpansive mappings with applications

Perturbation techniques for nonexpansive mappings are studied. An iterative algorithm involving perturbed mappings in a Banach space is proposed and proved to be strongly convergent to a fixed point of the original mapping. These techniques are applied to solve the split feasibility problem and the multiple-sets split feasibility problem, and to find zeros of accretive operators.     

Explicit Hierarchical Fixed Point Approach to Variational Inequalities

An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances of parameter selections. Applications in hierarchical minimization problems are also included.     

Compound Poisson Law for Hitting Times to Periodic Orbits in Two-Dimensional Hyperbolic Systems

We show that a compound Poisson distribution holds for scaled exceedances of observables \(\phi \) uniquely maximized at a periodic point \(\zeta \) in a variety of two-dimensional hyperbolic dynamical systems with singularities \((M,T,\mu )\), including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form \(\phi (z)=-\ln d(z,\zeta )\) where d is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index \(\theta \). We calculate \(\theta \) in terms of the derivative of the map T. Furthermore if we define \(M_n=\max \{\phi ,\ldots ,\phi \circ T^n\}\) and \(u_n (\tau )\) by \(\lim _{n\rightarrow \infty } n\mu (\phi >u_n (\tau ) )=\tau \) the maximal process satisfies an extreme value law of form \(\mu (M_n \le u_n)=e^{-\theta \tau }\). These results generalize to a broader class of functions maximized at \(\zeta \), though the formulas regarding the parameters in the distribution need to be modified.