Spatial chaotic structure of attractors of reaction-diffusion systems

The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.

     

Asymptotic behavior of eigenvalues and random updating schemes

For a stochastic matrix (Q _ij ^T ) _i,j=1 ^M withQ _ij ^T exp(–U(ij)/T) at the off-diagonal positions, we develop an algorithm to evaluate the asymptotic convergence rate of all eigenvalues ofQ _ij ^T asT 0 using Ventcel's optimal graphs. As an application we can compare the convergence rates of some random updating schemes used in image processing.This research was partially supported by the National Science Council, Taiwan and Air Force Office of Scientific Research Contract No. F49620 S5C 0144, and was completed while Tzuu-Shuh Chiang was visiting the Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA.     

The separation of tellurium and selenium by polyurethane foam sorbents

Tellurium and selenium can be sorbed from hydrochloric acid and hydrobromic acid solution by both polyether and polyester-based polyurethane foam. Although some acid is needed, the substitution of sodium chloride or sodium bromide increases the extraction significantly. Tellurium is extracted rapidly with > 99% sorbed in 2 min from 1.0/5.0M and 2.0/4.0M hydrochloric acid/sodium bromide. Selenium can also be sorbed quantitatively but much more slowly so that a separation is possible based on the relative rates of extraction. The capacity of polyether foam is 3% by weight of tellurium.     

Ditungsten Siloxide Hydrides, [(silox)(2)WH(n)()](2) (n = 1, 2; silox = (t)BuSiO), and Related Complexes

The addition of 4.0 equiv of Na(silox) to Na[W(2)Cl(7)(THF)(5)] afforded (silox)(2)ClW&tbd1;WCl(silox)(2) (1, 65%). Treatment of 1 with 2.0 equiv of MeMgBr in Et(2)O provided (silox)(2)MeW&tbd1;WMe(silox)(2) (2, 81%). In the presence of 1 atm of H(2), reduction of 1 with 2.0 equiv of Na/Hg in DME provided (silox)(2)HW&tbd1;WH(silox)(2) (3, 70%), characterized by a hydride resonance at delta 19.69 (J(WH) = 325 Hz, (1)H NMR). Exposure of 2 to 1 atm of H(2) yielded 3 and CH(4) via (silox)(2)HW&tbd1;WMe(silox)(2) (4); use of D(2) led to [(silox)(2)WD](2) (3-d(2)). Exposure of 3 to ethylene ( approximately 1 atm, 25 degrees C) in hexanes generated (silox)(2)EtW&tbd1;WEt(silox)(2) (5), but solutions of 5 reverted to 3 and free C(2)H(4) upon standing. NMR spectral data are consistent with a sterically locked, gauche, C(2) symmetry for 1-5. Thermolysis of 3 at 100 degrees C (4 h) resulted in partial conversion to (silox)(2)HW&tbd1;W(OSi(t)Bu(2)CMe(2)CH(2))(silox) (6a, approximately 60%) and free H(2), while extended thermolysis with degassing (5 d, 70 degrees C) produced a second cyclometalated rotational isomer, 6b (6a:6b approximately 3:1). When left at 25 degrees C (4 h) in sealed NMR tubes, 6 and free H(2) regenerated 3. Reduction of 1 with 2.0 equiv of Na/Hg in DME also afforded 6a (25%). When 3 was exposed to approximately 3 atm of H(2), equilibrium amounts of [(silox)(2)WH(2)](2) (7) were observed by (1)H NMR spectroscopy (3 + H(2) right harpoon over left harpoon 7; 25.9-88.7 degrees C, DeltaH = -9.6(4) kcal/mol, DeltaS = -21(2) eu). Benzene solutions of 3 and 1-3 atm of D(2) revealed incorporation of deuterium into the silox ligands, presumably via intermediate 6. In sealed tubes containing [(silox)(2)WCl](2) (1) and dihydrogen (1-3 atm), (1)H NMR spectral evidence for [(silox)(2)WCl](2)(&mgr;-H)(2) (8) was obtained, suggesting that formation of 3 from 1 proceeded via reduction of 8. Alternatively, 3 may be formed from direct reduction of 1 to give [(silox)(2)W](2) (9), followed by H(2) addition. Hydride chemical shifts for 7 are temperature dependent, varying from delta 1.39 (-70 degrees C, toluene-d(8)), to delta 3.68 (90 degrees C). (29)Si{(1)H} NMR spectra revealed a similar temperature dependence of the silox (delta 12.43, -60 degrees C, to delta 13.64, 45 degrees C) resonances. These effects may arise from thermal population of a low-lying, deltadelta, paramagnetic excited state of D(2)(d)() [(silox)(2)W](2)(&mgr;-H)(4) (DeltaE approximately 2.1 kcal/mol, chi(7a) approximately 0.03), an explanation favored over thermal equilibration with an energetically similar but structurally distinct isomer (e.g., [(silox)(2)WH(2)](2)(&mgr;-H)(2), DeltaG degrees approximately 0.69 kcal/mol, chi(7b) approximately 0.25) on the basis of spectral arguments. Extended Hückel and ab initio molecular orbital calculations on model complexes [(H(3)SiO)(2)W](2)(&mgr;-H)(4) (staggered bridged 7a', EHMO), [(H(3)SiO)(2)WH(2)](2) (all-terminal 7b', EHMO), [(H(3)SiO)(2)W](2) (9', EHMO), (HO)(4)W(2)(H(4)) (staggered-bridged 7", ab initio), and (HO)(4)W(2)(H(4)) (bent-terminal 7, ab initio) generally support the explanation of a thermally accessible excited state and assign 7 a geometry intermediate between the all-terminal and staggered-bridged forms.     

Oxidative addition of dialkylphosphites to complexes of iridium(I) and rhodium(I)

The PH bond of dialkylphosphites (dimethylphosphite, 5,5-dimethyl-1,3-dioxa-2-phosphorinane and 4,4,5,5-tetramethyl-1,3-dioxa-2-phospholane) oxidatively adds to irClL₂(L = PPh₃, AsPh₃) and IrCl(PMe₂Ph)₃ generated in situ to give six-coordinate hydrido(dialkylphosphonato)iridium(III) complexes, e.g. IrHClL₂[{(MeO)₂-PO}₂H] and IrHCl(PMe₂Ph)₃[PO(OMe)₂]. Addition of triphenylphosphine to a solution containing [IrCl(C₈H₁₄)₂]₂ and dimethylphosphite in a 1:2 mol ratio gives a five-coordinate hydrido (dimethylphosphonato)iridium(III) complex IrHCl(PPh₃)₂{PO(OMe)₂}, from which six-coordinate pyridine and acetylacetonato complexes IrHCl(PPh₃)₂(C₅H₅N){PO(OMe)₂} and IrH(PPh₃)₂(acac){PO(OMe)₂} can be obtained. The ligand arrangements in the various complexes are inferred from IR, ¹H and ³¹P NMR data.     

The Reaction of Triplet State Benzophenone with Acetylacetone

Irradiation of benzophenone in acetic acid containing acetylacetone resulted in regiospecific addition to form cis-2,2-dihpenyl-3-hydroxy-3-methyl-4-acetyloxetane and the rearrangement products therefrom. In the co-presence of copper ions, the regiospecificity is scrambled to give these products and a small amount of 1,1-diphenyl-1-buten-3-one, the secondary decomposition product of the other oxetane arising from the alternative orientation of the addition.