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.     

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.     

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     

On the Synthesis of 3-Acetyl-1-aryl-1,4,5,6-tetrahydro-1 H-1,2,4-triazepin-7-ones by Reaction of Nitrilimines with 3-Aminopropanoic Acid

Nitrilimines were prepared from N-arylhydrazono chlorides and reacted with -alanine yielding the corresponding amidrazones, which were treated with 1,1-carbonyldiimidazole in THF affording the hitherto unknown 3-acetyl-1-aryl-1,4,5,6-tetrahydro-1,2,4-triazepin-7-ones.     

An alternative synthesis of 2-(N-arylhydrazono)-1-benzothiophen-3-ones

In the presence of triethylamine, 2-mercaptobenzoic acid readily adds onto acylhydrazonoyl chlorides (1a-c) (precursors of the reactive nitrile imine 1,3-dipolar species) to afford good yields of the corresponding 2-[(2-oxo-1-arylhydrazonopropan-1-yl)mercapto]benzoic acids (2a-c). The latter acyclic adducts, in THF in the presence of 1,1'-carbonyldiimidazole, undergo intramolecular cyclization involving the activated carboxy and the enol functionality to deliver the respective 2-(N-arylhydrazono)-3-oxobenzothiophenes (3a-c). In the solid state, the latter compounds adopt the (Z)-geometry around the C=N double bond as evidenced by single crystal X-ray structure determination for 3b.     

Highly Efficient and Practical Syntheses of Lavendamycin Methyl Ester and Related Novel Quinolindiones

The novel 7-(N-formyl-, 7-(N-acetyl-, and 7-(N-isobutyrylamino)-2-methylquinoline-5,8-diones were synthesized in excellent overall yields in three steps via the nitration of the commercially available 8-hydroxy-2-methylquinoline followed by a reduction-acylation step and then oxidation. Acid hydrolysis of 7-(N-acetylamino)-2-methylquinoline-5,8-dione (14a) afforded the novel 7-aminoquinoline-5,8-dione 7 in excellent yields. Due to our efficient preparation of dione 14a, we now report a short and practical method for the total synthesis of the potent antitumor agent lavendamycin methyl ester (1b) with an excellent overall yield.     

On the Synthesis of Some 2-Arylhydrazono-3-oxothieno[2,3-<Emphasis Type="Italic">b</Emphasis>]pyridines by Reaction of Nitrilimines with 2-Mercaptonicotinic Acid

Summary. Nitrilimines prepared from N-arylhydrazono chlorides reacted with 2-mercaptonicotinic acid yielding the corresponding addition products, 2-[(2-oxo-1-arylhydrazonopropan-1-yl)mercapto] nicotinic acids, which were treated with 1,1-carbonyldiimidazole in THF affording by cyclocondensation the corresponding hitherto unknown 2-arylhydrazono-3-oxothieno[2,3-b]pyridines.     

Instability of nonlinear bound states

We establish a sharp instability theorem for the bound states of lowest energy of the nonlinear Klein-Gordon equation,u _tt–u+f(u)=0, and the nonlinear Schrödinger equation, –iu _t–u+f(u)=0.Supported in part by NSF Grants MCS 81-21487 and MCS 82-01599     

Determination of enzyme kinetic parameters of cyclic CMP-specific phosphodiesterase by quantitative fast atom bombardment tandem mass spectrometry

The determination of cytidine 3′,5′-cyclic monophosphate-specific phosphodiesterase activity by means of fast-atom bombardment (FAB) mass Spectrometry with mass-analysed ion kinetic energy (MIKE) spectrum scanning is described. Initial efforts to determine the activity of the enzyme by this method were unsuccessful owing to the obfuscation of sample-related peaks by peaks emanating from the incubation buffer and cation adducts; dilution of buffer and a desalting procedure overcame these difficulties. In the resulting positive-ion FAB mass spectra, characteristic peaks of the enzyme substrate and product could be readily identified and the protonated molecular ions selected for MIKE scanning. By spiking enzyme incubates with known amounts of substrate and product, and measuring peak heights in the MIKE spectra of both spiked and unspiked samples, the substrate/product ratio at the end of a series of phosphodiesterase incubations was determined. From the data obtained, the K_m and V_max of the phosphodiesterase were calculated as 6.08 mM and 11 μmol min^?1 mg^?1, respectively, showing good agreement with the analogous values of 8.06 mM and 5.8 μmol^?1 min^?1 mg^?1 obtained by radioactive assay.     

Poly(4-vinylpyridine)：As a green, efficient and commercial available basic catalyst for the synthesis of chromene derivatives

Poly(4-vinylpyridine) is reported as a green,commercial available and efficient basic recyclable catalyst for the synthesis of chromene derivatives.This catalyst can be easily recovered by simple filtration and recycled up to 5 consecutive runs without any loss of its efficiency.