Higher order bad loci

Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible, are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. Bad linear spaces are also investigated.     

Scattering of Magnetic Edge States

We consider a charged particle following the boundary of a two dimensional domain because a homogeneous magnetic field is applied. We develop the basic scattering theory for the corresponding quantum mechanical edge states. The scattering operator attains a limit for large magnetic fields which preserves Landau bands. We interpret the corresponding scattering phases in terms of classical trajectories. Communicated by Yosi Avron submitted 23/02/05, accepted 3/05/05     

One-pot palladium-catalyzed synthesis of 2,3-disubstituted bicyclo[2.2.1]heptanes and bicyclo[2.2.1]hept-5-enes

A new and highly stereoselective palladium-catalyzed synthesis is reported, based on two subsequent insertions of the bicyclo[2.2.1]heptene system into an aryl or vinylpalladium bond, formed in situ from aryl or vinyl bromides.     

Negative ion chemical ionization mass spectra of (η6-arene)Cr(CO)3

The negative ion chemical ionization mass spectra, with ammonia and methane as reagent gases, of the (η⁶-arene)Cr(CO)₃ complexes, where the arene is C₆H₅COCH₃, C₆H₅COC₂H₅, C₆H₅COC₃H₇, C₆H₅COC(CH₃)₃, 2-CH₃C₆H₄COC₃H₇, C₆H₅COOCH₃, C₆H₅CH₃, 1,3,5-(CH₃)₃C₆H₃ and C₆H₅CH₂COC₂H₅, are reported. Similar behaviour is observed with the two reagent gases, but ammonia shows a much higher abundance for the ions produced by reactions of [NH₂]^? with sample molecules. The compounds containing the C₆H₅CO group display an abundant [M]^{? ˙}, whereas the other compounds exhibit [M? H]^? as base peak, produced by ion/molecule reactions. A comparison of the negative ion chemical ionization mass spectra of the (η⁶-arene)Cr(CO)₃ complexes with those of the corresponding ligands shows the strong electron withdrawing power of the Cr(CO)₃ group in the gas phase.     

Transition metal-catalysed sequences of nucleophilic addition

Summary In presence of nickel or palladium catalysts, nucleophiles can attack vinylcyclopropanes with concomitant ring cleavage. With palladium catalysts in presence of appropriate substituents the terminal carbon atom of the resulting open chain is able to add to two molecules of a conjugated diene giving rise to long-chain unsaturated compounds.     

Stepwise retro 1,3-dipolar cycloaddition induced by electron. Impact on 3,5-diphenyl-1,2,4-oxadiazole

The retro 1,3-dipolar cycloaddition induced by electron impact on 3,5-diphenyl-1,2,4-oxadiazole is interpreted as a two-step process on the basis of the energetics and kinetics of the fragments [C₇H₅NO]⁺., [C₆H₅CN]⁺. and [C₆H₅CO]⁺.     

The Chemistry of Coumarin Derivatives. Part 5. Unusual course of the reaction of 4-hydroxycoumarin and aliphatic aldehydes

The reaction of 4-hydroxycoumarin and certain aliphatic aldehydes affords 1:1 or complex 2:2 adducts besides (or in place) of the expected 2:1 bis(coumarin) adducts. Reaction with heptanal, cyclohexanecarbaldehyde, and pivalaldehyde are reported as representative. The structure of the reaction products was established by spectroscopical techniques, including X-ray analysis, and their formation was mechanistically rationalized. Some of the 1:1 adducts are synthetically useful for the preparation of 3-alkyl-4-hydroxycoumarins.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

Crystal,molecular and electronic structure of 1,1-diaryl-2-halogenoethylenes

On the basis of semiempirical calculations, the conformation of the isolated molecule of 2-bromo-1,1-diphenylprop-1-ene corresponding to the minimum of total energy was determined. Assuming some information on the localization of the molecules in the crystal, also the conformation in a theoretical crystal was studied. The results have been compared with the crystal structure of 2-bromo-1,1-diphenylprop-1-ene, as obtained by X-ray diffraction. From the comparison a slight displacement of the C2 atom was suggested.

---

Zusammenfassung Die dem Minimum der Gesamtenergie entsprechende Konformation des isolierten Moleküls von 2-Brom-1,1-diphenylprop-1-en wurde durch halb-empirische Methoden bestimmt. Gewisse Ergebnisse über die Anordnung der Moleküle im Kristall machten es möglich, die Konformation in einem theoretischen Kristall zu untersuchen. Nach Vergleich der Resultate mit der durch Röntgen-Analyse bestimmten Kristallstruktur des 2-Brom-1,1-diphenylprop-1-en wurde eine kleine Verschiebung des C2-Atoms vorgeschlagen.

---

Résumé Par le moyen de calculs semiempiriques on a déterminé la conformation de la molécule isolée de 2-bromo-1, 1-diphénilprop-1-ène, correspondant au minimum d'énergie totale. On a aussi étudié la conformation dans le cristal théorique en se basant sur des informations concernant la localisation des molécules dans le cristal. On a comparé ces résultats avec ceux que nous avons obtenus par la détermination de la structure cristalline du 2-bromo-1, 1-diphénilprop-1-ène avec la diffraction des rayons X: cette comparaison nous a porté à déplacer un petit peu l'atome C 2.

---

---

This work was supported by a grant from the Italian Consiglio Nazionale delle Ricerche.     

A method for solving a conditional cauchy equation on abelian groups

Sunto In questa Nota è descritto wn metodo per la determinazione delle soluzioni della seguente equazione condizionale f(x+y) f(x) + f(y) + a implica f(x + y)=f(x) + f(y) dove f:GH, a H e G ed H sono gruppi abeliani qualsiasi. Sono poi determinate esplicitamente le soluzioni nei casi G=(r), G= e G=(p).

---

---

Lavoro svolto nell'ambito del G.N.A.F.A.-C.N.R.