Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

On decomposition of continuity

Summary The main purpose of this paper is to introduce the concepts of η-sets, ηζ-sets,η-continuity and ηζ-continuity and to obtain a decomposition of continuity.     

ZnTe6O13, a new ZnO–TeO2 phase

Our investigations into the ZnO–TeO₂ system have produced a new phase, zinc(II) hexatellurium(IV) tridecaoxide, ZnTe₆O₁₃, with trigonal (R) symmetry, synthesized by repeated heating and cooling to a maximum temperature of 1053 K. The asymmetric unit consists of a Zn atom coordinated in a distorted octahedral fashion by two unique tellurium(IV) oxide units that form trigonal–bipyramidal TeO₄ and TeO₃₊₁ corner‐ and edge‐shared polyhedra. Except for the Zn and an O atom, which occupy 6c positions, all atoms occupy 18f general positions.     

Pre—Separation Axioms in Fuzzifying Topology

1　 IntroductionYing[5,6 ] introduced and elementally developed so called fuzzifying topology with the semanticmethod of continuous valued L ogic.Shen[7] introduced and studied T0 -,T1-,T2 (Hausdorff) -,T3(regularity) -,T4 (normality) -separation axioms in fuzzifying topology.In [3 ]the concepts of thefamily of fuzzifying pre-open sets,fuzzifying pre-neighbourhood structure of a point and fuzzifyingpre-closure are introduced and studied.It is worth to mention that pre-separation axioms are …     

Synthesis and studies of antibacterial activity of pongaglabol

Pongaglabol [8-hydroxy-5-phenyl-furo[2,3-h]benzo(b)pyran-7-one] was synthesized and tested for antibacterial effects againstShigella dysenteriae, Salmonella typhi, Streptococcus β-haemolyticus andStaphylococcus aureus. The synthesized compounds were characterized using UV, IR and¹H NMR spectral data     

Influence of moisture on the viscoelastic relaxations in long aliphatic chain contained semiaromatic polyamide, (PA9‐T) fiber

The effect of moisture on the mechanical relaxation processes of semiaromatic semicrystalline polyamides synthesized by a long‐chain aliphatic diamine and terephthalic acid was investigated by dynamic viscoelastic analysis (DVA) and differential scanning calorimetry (DSC). Moreover, the implication of moisture with the amorphous and crystalline domains was also examined by temperature‐dependent wide‐angle X‐ray diffraction and Fourier transform infrared spectra. The characteristics of the relaxations such as α, β, γ, and the pronounced peak shoulder appeared at 25–100 °C in DVA tan δ curves were found to be strongly susceptible to the presence of moisture. With moisture evaporation, the peak shoulder of 25–100 °C and the β‐relaxation disappeared. The former is anticipated to originate from to the side group motion of hydrogen‐bonded water, whereas the later one is from the motions of the amide–water complex units. With the disappearance of the β‐relaxation, the γ‐relaxation appeared simultaneously in much lower temperatures and ultimately coupled with the δ‐relaxation. The γ‐relaxation is attributed to be associated with the molecular motion of the amide group and δ‐relaxation with for the motion of the methylene units. The existence of two types of water was identified in the polymer, namely, tightly bound and loosely bound. The tightly bound water is believed to be directly connected by hydrogen bonding with the strong polar groups and the loosely bound water weakly links with those connected water making hydrogen bridges. The moisture acts as a plasticizer in the polymer matrix, which causes quite a large depression in its glass transition temperature (T_g). WAXD and FTIR studies corroborated the existence of water solely in amorphous regions, i.e., no rapport of water with the crystalline parts. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2878–2891, 2003     

Heterocycles from carbohydrate precursors. Part XIX. The X-ray crystal structure determination of the reaction product of 4-(2-acetoxyethylidene)-4-hydroxy-2,3-dioxobutyro-1,4-lactone-2-(p-bromophenylhydrazone) with methylhydrazine

X-ray crystallographic data show that the product obtained in the reaction of 4-(2-acetoxy-ethylidene)-4-hydroxy-2,3-dioxobutyro-1,4-lactone-2-(p-bromophenylhydrazone) with methyl-hydrazine is the bicyclic compound 2,6-dimethyl-3,4-dioxo-2,3,4,6,7,8-hexahydropyridazino-[4,3-c]pyridazine 4-(p-bromophenylhydrazone) ( 10 ) and not as originally suggested 1-methyl-3-(1-methylpyrazolin-3-yl)-4,5-pyrazoledione 4-(p-bromophenylhydrazone) ( 8 ).     

Pyrrole studies part 41. Reactivity of 3,4-diformyl-2,5-dimethylpyrrole with diaminoalkanes. An unusual formation of 2-azafulvenes

3,4-Diformyl-2,5-dimethylpyrrole (1) reacts with ,ω-diamino-alkanes, NH₂(CH₂)_nNH₂t' to form either the potentially tautomeric 2:2 macrocyclic adduct (7a) (8), when N = 2, or the potentially tautomeric 1:1 bicyclic adduct (18) (19), when N = 4, 5, 6, and 12. ¹H and ¹³C N.m.r. spectral data indicate that the 2-azafulvene structures predominate for both types of cycloadducts. Only polymeric material was obtained when N = 3.