Optimization over analytic functions whose founrier coefficients are constrained

This article gives necessary and sufficient conditions for local solutions to several very general constrained optimization problems over spaces of analytic functions.The results presented here have many applications, a particular instance of which is the sup-norm approximation of functions continuous on the unit circle in the complex plane by functions continuous on the circle and analytic on the open disk and whose Fourier coefficients satisfy prescribed linear relations.Also, the results in this article generalize Nevanlinna-Pick and Caratheodory-Fejer Interpolation results to allow values of arbitrary derivatives of functions to be assigned or merely bounded. Classically, NP and CF solve only problems with consecutive derivatives specified.In engineering, constraints on the Fourier coefficients of a frequency response function correspond to constraints on its time domain behavior. Indeed the central problems of control theory involve both time and frequency domain constraints. That is precisely what the results in this paper handle.Supported in part by the AFOSR and the NSF     

A note about bistellar operations on PL-manifolds with boundary

In 1990, U. Pachner proved that simplicial triangulations of the same PL-manifold (with boundary) are always connected by a finite sequence of transformations belonging to two different groups:shelling operations (and their inverses), which work mostly with the boundary triangulations, andbistellar operations, which affect only the interior of the triangulations.The purpose of this note is to prove that, in case of simplicial triangulations coinciding on the boundary, bistellar operations are sufficient to solve the homeomorphism problem.Work performed under the auspicies of the GNSAGA of the CNR (National Research Council of Italy) and financially supported by MURST of Italy (project Geometria Reale e Complessa).     

Sesquilinear-orthogonally quadratic mappings

Summary The conditional Jordan-von Neumann functional equation for a mappingG: (X, +, ) (Y, +), that is,G(x + y) + G(x–y) = 2G(x) + 2G(y) for allx, y X withx y, was first studied by Vajzovi in 1966. He gave the general form of the continuous scalar valued solutions of (*) on a Hibert space with its natural orthogonality. Later his result was generalized toA-orthogo-nalities on a Hilbert space, which satisfyx ^A y Ax, y = 0 whereA is a selfadjoint operator. In particular, Drljevi in 1986 determined the continuous scalar valued solutions and recently Fochi showed that theA-orthogonally quadratic functionals are exactly the quadratic ones.Here we further generalize their results to a symmetric orthogonality induced by a sesquilinear form on a vector space and for arbitrary mappings with values in an abelian group. The main result states that such a mapping can satisfy (*) only if it is quadratic. In the proof extensive use is made of the theory of sesquilinear-orthogonally additive mappings as developed in an earlier paper of ours.The above mentioned results are valid only for the cases of dimension 3 and a 2-dimensional counter example is presented. Finally, an interesting concept of orthogonality is suggested for possible future investigation.     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

Coverage dependence of field emission flicker noise due to lithium adsorbed on the W(112) surface

Spectral density functions of field emission current fluctuations were investigated by a probe hole technique for lithium adsorbates on the tungsten (112) surface plane in the sub- to double monolayer range. The dependence of the spectral density on the lithium surface concentration obtained slightly above room temperature, shows maxima and minima. A correspondence is found between the extrema of the coverage dependent spectral density, features of the work function, and the film structures occurring with different adparticle densities. Commensurate lithium structures are clearly manifest in the noise power.     

Novel addition in trifluoromethylation of [70]fullerene

Pyrolytic trifluoromethylation of [70]fullerene with CF₃CO₂Ag at 300 °C results in the addition of up to 12 CF₃ groups to the fullerene cage. Forty-six C₇₀(CF₃)_n derivatives (numbers in parentheses) were separated by two-stage high pressure liquid chromatography (HPLC) as follows: n = 2(2), 4(16), 6(9), 8(14) 10(5), some being characterised by NMR. The range of derivatives is much greater than for other [70]fullerene reactions, and as with [60]fullerene trifluoromethylation, no single derivative is dominant, indicating that kinetic stability mainly controls product formation. NMR spectra show most derivatives to be unsymmetrical, with combinations of quartets and septets (overlapping quartets) due to contiguous (‘linear’) addend arrays, having significantly different coupling constants of the ‘terminal’ quartets of between 9.1 and 17.7 Hz. These differences, together with those observed previously in trifluoromethylation of [60]fullerene are consistent with addition across both 6:6- and 5:6-ring junctions. Of the two C₇₀(CF₃)₂ isomers, one has either C_s or C₂ symmetry, the other has C₁ symmetry, whilst the C₇₀(CF₃)₄ derivatives fall into four categories: (i) symmetrical compounds (one gives only two singlets in the NMR); (ii) unsymmetrical compounds that show a ‘linear’ coupling sequence; (iii) unsymmetrical compounds having a remote pairs of adjacent groups; (iv) compounds having a coupled array of three CF₃ groups, together with a remote group suggesting sterically-driven migration. The first evaluation of differential NMR couplings across 6:6- and 5:6-bonds in a fullerene has been made using C₆₀F₆ as a model.     

碳笼化合物的高效液相色谱研究

采用化学键合固定相（ＤＮＡＰ－ＳＰ）及二元混合流动相（５０％苯－正已烷），对碳笼化合物Ｃ６０与Ｃ７０的高效液相色谱（ＨＰＬＣ）分析及分离进行了研究。比较了多种流动相对Ｃ６０、Ｃ７０分离效果的影响，苯／正已烷混合体系是最好的流动相。     

Darstellung,Eigenschaften und elektronische Raman-Spektren von Bis(chloro)phthalocyaninatoferrat(III), -ruthenat(III) und -osmat(III)

Preparation, Properties and Electronic Raman Spectra of Bis(chloro)-phthalocyaninatoferrate(III), -ruthenate(III) and -osmate(III) Bis(chloro)phthalocyaninatometalates of Fe^III, Ru^III and Os^III [MCl₂Pc(2-)]^?, with an electronic low spin ground state are formed by the reaction of [FeClPc(2-)] resp. H[MX₂Pc(2?)] (M = Ru, Os; X = Cl, I) with excess chloride in weakly coordinating solvents (DMF, THF) and are isolated as (n-Bu₄N) salts. The asym. M? Cl stretch (ν_as(MCl)) is observed in the f.i.r. at 288 cm^?1 (Fe), 295 cm^?1 (Ru), 298 cm^?1 (Os), ν_as(MN) at 330 cm^?1 (Fe), 327 cm^?1 (Ru), and 317 cm^?1 (Os); only ν_s(OsCl) at 311 cm^?1 is resonance Raman (r.r.) enhanced with blue excitation. The m.i.r. and FT-Raman spectra are typical for hexacoordinated phthalocyanines of tervalent metal ions. The UV-vis spectra show besides the characteristic π-π* transitions (B, Q, N, L band) of the Pc ligand a number of extra bands at 12–15 kK and 18–24 kK due to trip-doublet and (Pc→M)CT transitions. The effect of metal substitution is discussed. The r.r. spectra obtained by excitation between the B and Q band (λ₀ = 476.5 nm) are dominated by the intraconfigurational transition Γ₇ Γ ₈ arrising from the spin-orbit splitting of the electronic ground state for Fe^III at 536 cm^?1, for Ru^III at 961 cm^?1 and Os^III at 3 028 cm^?1. Thus the spin-orbit coupling constant increases very greatly down the iron group: Fe^III (357 cm^?1)< Ru^III (641 cm^?1)< Os^III (2 019 cm^?1). The Γ₇ Γ ₈-transition is followed by a very pronounced vibrational finestructure being composed in the r.r. spectra by the coupling with ν_s(MCl), δ(MClN) and the most intense fundamental vibrations of the Pc ligand. In absorption only vibronically induced transitions are observed for the Ru and Os complex at 1 700-2800 rsp. 3100-5800 em^?1 instead of the 0-0 phonon transitions. The most intense lines are attributed to combinations of the intense odd vibrational mo-des at ≈ 740 and 1120 cm^?1 with ν₅(MCI), δ(MClN).     

Crosslinking of alkylpolysiloxane filsm on various types of glass surfaces including fused silica using γ-radiation of a60cobalt-source. Comparison to crosslinking by thermal peroxid treatment

Summary Crosslinking of alkylpolysiloxane stationary phases, especially in thick film capillary columns, is useful for the homogenous coating of fused silica and pretreated alkaliglass surfaces. The films of the stationary liquid are immobilized against solvent rinsing using CH₂Cl₂, pentane, and acetone, and maintain homogenity even at high temperature. Various doses of -radiation from a⁶⁰Co. source were used for the crosslinking instead of the thermal peroxid treatment recently described by other authors. The effect of the -radiation crosslinking-procedure was investigated in comparison to the peroxid method in regard of: decrease of stationary phase content by solvent rinsing, separation efficiency, tailing behaviour, and bleeding of the columns obtained. Similar results as with the cumylperoxid-treatment were achieved using the -radiation-method. By -radiation no polar functional groups or moleculs are introduced into the stationary phase, however, as with the peroxid method. Less than 20% of the various stationary liquids are usually removed from the columns by solvent rising after crosslinking using both methods depending on the doses of radiation and the cumylperoxid concentration applied respectively.Dedicated to Dr. L. S. Ettre on the occasion of his 60th birthday.     

Molecular electrochemistry of hydrofullerenes C70H36—46

Electrochemistry of a mixture of hydrofullerenes C₇₀H_36—46 composed of C₇₀H₃₆, C₇₀H₃₈, C₇₀H₄₄, and C₇₀H₄₆ (50, 20, 14, and 15%, respectively) was studied by cyclic voltammetry in THF and CH₂Cl₂ in the –43—–13 °C temperature range. Two cathodic peaks, namely, one-electron reversible (E° = –3.16 V (Fc^0/+), Fc is ferrocene) and irreversible (E _p = –3.37 V (Fc^0/+)) were observed for this mixture in THF. The irreversible broad oxidation peak (E _p = 1.22 V (Fc^0/+)) was observed in CH₂Cl₂. The reversible reduction peak (E° = –3.16 V) and irreversible oxidation peak (E _p = 1.22 V) were attributed to the most stable hydrofullerene C₇₀H₃₆. The irreversible reduction (E _p = –3.37 V) and oxidation (E _p = 1.22 V) peaks were attributed to hydrofullerenes C₇₀H_44—46 with a higher degree of hydrogenation. The values of an electrochemical gap, which is an analog of the energy gap (HOMO—LUMO), are 4.38 and 4.59 V for C₇₀H₃₆ and C₇₀H_44—46, respectively, and indicate that these hydrofullerenes are sufficiently hard molecules with low reactivity in redox reactions.