A new zero-finder for Tikhonov regularization

Tikhonov regularization with the regularization parameter determined by the discrepancy principle requires the computation of a zero of a rational function. We describe a cubically convergent zero-finder for this purpose. AMS subject classification (2000)  65F22, 65H05, 65R32     

Site-differentiated solid solution in (Na(1-x)Cu(x))2Ta4O11 and its electronic structure and optical properties

The (Na(1-x)Cu(x))(2)Ta(4)O(11) (0 ≤ x ≤ 0.78) solid-solution was synthesized within evacuated fused-silica vessels and characterized by powder X-ray diffraction techniques (space group: R3c (#167), Z = 6, a = 6.2061(2)-6.2131(2) ?, c = 36.712(1)-36.861(1) ?, for x = 0.37, 0.57, and 0.78). The structure consists of single layers of TaO(7) pentagonal bipyramids as well as layers of isolated TaO(6) octahedra surrounded by Na(+) and Cu(+) cations. Full-profile Rietveld refinements revealed a site-differentiated substitution of Na(+) cations located in the 12c (Wyckoff) crystallographic site for Cu(+) cations in the 18d crystallographic site. This site differentiation is driven by the linear coordination geometry afforded at the Cu(+) site compared to the distorted seven-coordinate geometry of the Na(+) site. Compositions more Cu-rich than x ~ 0.78, that is, closer to "Cu(2)Ta(4)O(11)", could not be synthesized owing to the destabilizing Na(+)/Cu(+) vacancies that increase with x up to the highest attainable value of ~26%. The UV-vis diffuse reflectance spectra show a significant red-shift of the bandgap size from ~4.0 eV to ~2.65 eV with increasing Cu(+) content across the series. Electronic structure calculations using the TB-LMTO-ASA approach show that the reduction in bandgap size arises from the introduction of Cu 3d(10) orbitals and the formation of a new higher-energy valence band. A direct bandgap transition emerges at k = Γ that is derived from the filled Cu 3d(10) and the empty Ta 5d(0) orbitals, including a small amount of mixing with the O 2p orbitals. The resulting conduction and valence band energies are determined to favorably bracket the redox potentials for water reduction and oxidation, meeting the thermodynamic requirement for photocatalytic water-splitting reactions.     

The ternary system cerium-rhodium-silicon

Phase relations have been established in the ternary system Ce-Rh-Si for the isothermal section at 800 °C based on X-ray powder diffraction and EPMA on about 80 alloys, which were prepared by arc melting under argon or by powder reaction sintering. From the 25 ternary compounds observed at 800 °C 13 phases have been reported earlier. Based on XPD Rietveld refinements the crystal structures for 9 new ternary phases were assigned to known structure types. Structural chemistry of these compounds follows the characteristics already outlined for their prototype structures: τ₇—Ce₃RhSi₃, (Ba₃Al₂Ge₂-type), τ₈—Ce₂Rh_3−xSi_3+x (Ce₂Rh_1.35Ge_4.65-type), τ₁₀—Ce₃Rh_4−xSi_4+x (U₃Ni₄Si₄-type), τ₁₁—CeRh₆Si₄ (LiCo₆P₄-type), τ₁₃—Ce₆Rh₃₀Si_19.3 (U₆Co₃₀Si₁₉-type), τ₁₈—Ce₄Rh₄Si₃ (Sm₄Pd₄Si₃-type), τ₂₁—CeRh₂Si (CeIr₂Si-type), τ₂₂—Ce₂Rh_3+xSi_1−x (Y₂Rh₃Ge-type) and τ₂₄—Ce₈(Rh_1−xSi_x)₂₄Si (Ce₈Pd₂₄Sb-type). For τ₂₅—Ce₄(Rh_1−xSi_x)₁₂Si a novel bcc structure was proposed from Rietveld analysis. Detailed crystal structure data were derived for τ₃—CeRhSi₂ (CeNiSi₂-type) and τ₆—Ce₂Rh₃Si₅ (U₂Co₃Si₅-type) by X-ray single crystal experiments, confirming the structure types. The crystal structures of τ₄—Ce₂₂Rh₂₂Si₅₆, τ₅—Ce₂₀Rh₂₇Si₅₃ and τ₂₃—Ce_33.3Rh_58.2−55.2Si_8.5−11.5 are unknown. High temperature compounds with compositions Ce₁₀Rh₅₁Si₃₃ (U₁₀Co₅₁Si₃₃-type) and CeRhSi (LaIrSi-type) have been observed in as-cast alloys but these phases do not participate in the phase equilibria at 800 °C.     

Ternary systems Sr-{Ni,Cu}-Si: Phase equilibria and crystal structure of ternary phases

Phase relations were established in the Sr-poor part of the ternary systems Sr-Ni-Si (900 °C) and Sr-Cu-Si (800 °C) by light optical microscopy, electron probe microanalysis and X-ray diffraction on as cast and annealed alloys. Two new ternary compounds SrNiSi₃ (BaNiSn₃-type) and SrNi_9−xSi_4+x (own-type) were found in the Sr-Ni-Si system along with previously reported Sr(Ni_xSi_1−x)₂ (AlB₂-type). The crystal structure of SrNi_9−xSi_4+x (own-type, x=2.7, a=0.78998(3), c=1.1337(2) nm; space group P4/nbm) was determined from X-ray single crystal counter to be a low symmetry derivative of the cubic, parent NaZn₁₃-type. At higher Si-content X-ray Rietveld refinements reveal the formation of a vacant site (□) corresponding to a formula SrNi_5.5Si_6.5□_1.0. Phase equilibria in the Sr-Cu-Si system are characterized by the compounds SrCu_2−xSi_2+x (ThCr₂Si₂-type), Sr(Cu_xSi_1−x)₂ (AlB₂-type), SrCu_9−xSi_4+x (0≤x≤1.0; CeNi_8.5Si_4.5-type) and SrCu_13−xSi_x (4≤x≤1.8; NaZn₁₃-type). The latter two structure types appear within a continuous solid solution. Neither a type-I nor a type-IX clathrate compound was encountered in the Sr-{Cu,Ni}-Si systems.Structural details are furthermore given for about 14 new ternary compounds from related alloy systems with Ba.     

X-ray structural study of intermetallic alloys RT2Si and RTSi2 (R=rare earth, T=noble metal)

Two series of intermetallic alloys, RT₂Si and RTSi₂, have been synthesized from stoichiometric compositions. The crystal structures of EuPt_1+xSi_2−x (CeNiSi₂-type), CeIr₂Si (new structure type), YbPd₂Si and YbPt₂Si (both YPd₂Si-type) have been elucidated from X-ray single crystal CCD data, which were confirmed by XPD experiments. The crystal structures of LaRh₂Si and LaIr₂Si (CeIr₂Si-type), {La,Ce,Pr,Nd}AgSi₂ (all TbFeSi₂-type), and EuPt₂Si (inverse CeNiSi₂-type) were characterized by XPD data. RT₂Si/RTSi₂ compounds were neither detected in as-cast alloys Sc₂₅Pt₅₀Si₂₅, Eu₂₅Os₂₅Si₅₀ and Eu₂₅Rh₂₅Si₅₀ nor after annealing at 900 °C. Instead, X-ray single crystal data prompted Eu₂Os₃Si₅ (Sc₂Fe₃Si₅-type) and EuRh_2+xSi_2−x (x=0.04, ThCr₂Si₂-type) as well as a new structure type for Sc₂Pt₃Si₂ (own type).     

Dimeric Capsules Formed by Tetra‐CMPO Derivatives of (Thia)Calix[4]arenes

Thiacalix[4]arene 2 , calix[4]arene 3 a and its tetraether fixed in the cone conformation 3 b form homo‐ and heterodimeric capsules in apolar solvents, which are held together by a seam of NH???O=P hydrogen bonds between carbamoylmethyl phospine oxide functions attached to their wide rim. Their internal volume of ～370 ų requires the inclusion of a suitable guest. Although neutral molecules such as adamantane (derivatives) or tetraethylammonium cations form kinetically stable complexes (¹H‐ and ³¹P‐time scale), the included solvent is rapidly exchanged. The internal mobility of the included tetraethylammonium cation is distinctly higher (ΔG=42.5 and 49.7 kJ mol^?1 for 3 a and 3 b ) than that for similar capsules of tetraurea calix[4]arenes 1 . Mixtures of 1 with 2 , 3 a , or 3 b contain only the two homodimers but the heterodimerization occurs with the tetraloop tetraurea 6 , which cannot form homodimers. Two dimers with cationic guests ( 2? (C₅H₅)₂Co⁺ ?2 and 3 a? Et₃NH⁺ ? H₂O ?3 a ) were confirmed by single‐crystal X‐ray analysis.     

Interaction of push–pull tert-enamines with phenylglyoxal

Abstract  

The reaction of push–pull enamines with 1,2-biselectrophilic phenylglyoxal was investigated. Phenylglyoxal was found to react depending on the structure of the push–pull enamine, affording either a hydroxyalkylation product at the methyl group or the cyclic product via participation of the methyl group and the β-carbon of the enamine.     

On (non-) monotonicity of cooperative solutions

Aggregate monotonicity of cooperative solutions is widely accepted as a desirable property, and examples where certain solution concepts (such as the nucleolus) violate this property are scarce and have no economic interpretation. We provide an example of a simple four-player game that points out at a class of economic contexts where aggregate monotonicity is not appealing.     

Effective learning in the presence of adaptive counterparts

Adaptive learning algorithms (ALAs) is an important class of agents that learn the utilities of their strategies jointly with the maintenance of the beliefs about their counterparts' future actions. In this paper, we propose an approach of learning in the presence of adaptive counterparts. Our Q-learning based algorithm, called Adaptive Dynamics Learner (ADL), assigns Q-values to the fixed-length interaction histories. This makes it capable of exploiting the strategy update dynamics of the adaptive learners. By so doing, ADL usually obtains higher utilities than those of equilibrium solutions. We tested our algorithm on a substantial representative set of the most known and demonstrative matrix games. We observed that ADL is highly effective in the presence of such ALAs as Adaptive Play Q-learning, Infinitesimal Gradient Ascent, Policy Hill-Climbing and Fictitious Play Q-learning. Further, in self-play ADL usually converges to a Pareto efficient average utility.     

Nonspecular total internal reflection of spatial solitons at the interface between highly birefringent media

We investigate total internal reflection of nonlocal spatial optical solitons at the interface between two differently oriented regions of a highly birefringent nematic liquid crystal. The solitons survive the interaction with an induced index mismatch and undergo nonspecular reflection, with an emerging angle differing appreciably from the incidence angle.