An algorithm for solving global optimization problems with nonlinear constraints

In this paper we propose an algorithm using only the values of the objective function and constraints for solving one-dimensional global optimization problems where both the objective function and constraints are Lipschitzean and nonlinear. The constrained problem is reduced to an unconstrained one by the index scheme. To solve the reduced problem a new method with local tuning on the behavior of the objective function and constraints over different sectors of the search region is proposed. Sufficient conditions of global convergence are established. We also present results of some numerical experiments.     

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov-Schmidt reduction and the Crandall-Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.     

Three‐Component Gallium(III)‐Promoted Addition of Halide Anions and Acetylenes to Donor–Acceptor Cyclopropanes

A new strategy for the three‐component addition of halide anions and acetylenes to donor–acceptor cyclopropanes (DACs) is presented. This reaction, which occurs with high E selectivity, is promoted by gallium(III) salts and based on the 1,2‐zwitterionic reactivity of DACs. It opens up a new group of processes involving DACs. The reaction occurs readily with a broad range of substrates and is tolerant of various functional groups. This methodology makes it possible to assemble highly functionalized vinyl halides, which are very convenient building blocks in organic synthesis. A possible mechanism of this reaction and its stereochemical aspects are discussed in detail.     

Complex three-dimensional platinum-indium networks in the ternary indides Dy2Pt7In16 and Tb6Pt12In23

Well crystallized samples of Dy₂Pt₇In₁₆ and Tb₆Pt₁₂In₂₃ were synthesized by an indium flux technique. Arc-melted precursor alloys with the starting compositions ∼DyPt₃In₆ and ∼TbPtIn₄ were annealed with a slight excess of indium at 1200 K followed by slow cooling (5 K/h) to 870 K. Both indides were investigated by X-ray diffraction on powders and single crystals: Cmmm, a=1211.1(2), b=1997.8(3), c=439.50(6) pm, wR2=0.0518, 1138 F² values, 45 variable parameters for Dy₂Pt₇In₁₆ and C2/ma=2834.6(4), b=440.05(7), c=1477.1(3) pm, β=112.37(1)°, wR2=0.0753, 2543 F² values, 126 variable parameters for Tb₆Pt₁₂In₂₃. The platinum atoms in the terbium compound have a distorted trigonal prismatic coordination. In Dy₂Pt₇In₁₆, trigonal and square prismatic coordination occur. The shortest interatomic distances are observed for Pt-In followed by In-In contacts. Considering these strong interactions, both structures can be described by complex three-dimensional [Pt₇In₁₆] and [Pt₁₂In₂₃] networks. The networks leave distorted pentagonal channels in Dy₂Pt₇In₁₆, while pentagonal and hexagonal channels occur in Tb₆Pt₁₂In₂₃. The crystal chemistry and chemical bonding of the two indides are briefly discussed.     

Efficient stereoselective alkenylation through a homolytic domino reaction involving a 1,5 sulfur-to-carbon translocation

An effective method for cis-stereoselective attachment of functionalized alkenyl appendages to sp3 carbon is reported. This method is based on a free-radical process, involving a sequence of addition-elimination steps, resulting in alkenyl group transposition from divalent sulfur to a prochiral carbon radical. Absolute stereoselectivity is secured since the new carbon-carbon bond is formed in a ring-closure reaction leading to a bridged bicyclic carbon-centered radical intermediate. The latter undergoes beta-scission of the C-S bond, leaving the alkenyl side chain in its predetermined position while releasing a thiyl radical. This thiyl radical is trapped by tri-n-butylstyryltin, affording a (styrylsulfanyl)methyl side chain and a tri-n-butyltin radical that continues the chain. When 2-(alkenylsulfanyl)methyl-4-bromo(or iodo)pyrrolidines were used as starting materials 2,4-cis-disubstituted 4-alkenyl-2-(styrylsulfanyl)methylpyrrolidines were obtained as products (70-90% yield). Tri-n-butylstyryltin was used rather then the more common n-Bu3SnH as tin radicals sources because the latter led predominantly to bridged bicyclic 3-thia-6-azabicyclo[3.2.1]octanes (up to 77% yield). An additional advantage of using tri-n-butylstyryltin derives from the discovery that the resulting styrylsulfide functionality is an excellent synthetic equivalent to the formyl group. Thus, using a Pummerer-type oxidative desulfurization, 4-cis-alkenyl-proline aldehydes were obtained.     

Reactions of 4H-selenopyrans with halogens

The reactions of 2,4,4,6-tetraphenyl-4H-selenopyran with chlorine or bromine give the corresponding 1,1-dichloro- or 1,1-dibromo-4H-selenopyrans, whereas the reaction of 2,4,6-triphenyl-4H-selenopyran with bromine gives 2,4,6-triphenylselenopyrylium tribromide.     

New rare earth metal-rich indides RE14Ni3In3 (RE=Sc, Y, Gd-Tm, Lu)—synthesis and crystal chemistry

The rare earth-nickel-indides RE₁₄Ni₃In₃ (RE=Sc, Y, Gd-Tm, Lu) were synthesized from the elements by arc-melting and subsequent annealing. The compounds were investigated on the basis of X-ray powder and single crystal data: Lu₁₄Co₂In₃ type, P4₂/nmc, Z=4, a=888.1(1), c=2134.7(4), wR2=0.0653, 1381 F² values, 63 variables for Sc_13.89Ni_3.66In_2.45; a=961.2(1), c=2316.2(5), wR2=0.0633, 1741 F² values, 64 variables for Y_13.84Ni_3.19In_2.97; a=965.3(1), c=2330.5(5), wR2=0.0620, 1765 F² values, 63 variables for Gd₁₄Ni_3.29In_2.71; a=956.8(1), c=2298.4(5), wR2=0.0829, 1707 F² values, 64 variables for Tb_13.82Ni_3.36In_2.82; a=951.7(1), c=2289.0(5), wR2=0.0838, 1794 F² values, 64 variables for Dy_13.60Ni_3.34In_3.06; a=948.53(7), c=2270.6(1), wR2=0.1137, 1191 F² values, 64 variables for Ho_13.35Ni_3.17In_3.48; a=943.5(1), c=2269.1(5), wR2=0.0552, 1646 F² values, 64 variables for Er_13.53Ni_3.14In_3.33; a=938.42(7), c=2250.8(1), wR2=0.1051, 1611 F² values, 64 variables for Tm_13.47Ni_3.28In_3.25; a=937.3(1), c=2249.6(5), wR2=0.0692, 1604 F² values, 64 variables for Tm_13.80Ni_3.49In_2.71; and a=933.4(1), c=2263.0(5), wR2=0.0709, 1603 F² values, 64 variables for Lu_13.94Ni_3.07In_2.99. The RE₁₄Ni₃In₃ indides show significant Ni/In mixing on the 4c In1 site. Except the gadolinium compound, the RE₁₄Ni₃In₃ intermetallics also reveal RE/In mixing on the 4c RE1 site, leading to the refined compositions. Due to the high rare earth metal content, the seven crystallographically independent RE sites have between 9 and 10 nearest RE neighbors. The RE₁₄Ni₃In₃ structures can be described as a complex intergrowth of rare earth-based polyhedra. Both nickel sites have a distorted trigonal-prismatic rare earth coordination. An interesting feature is the In2-In2 dumb-bell at an In2-In2 distance of 304 pm (for Gd₁₄Ni_3.29In_2.71). The crystal chemical peculiarities of the RE₁₄Ni₃In₃ indides are briefly discussed.