On the finiteness of optimal partitions

We prove that, under appropriate assumptions on the domain Ω and on the datumg, any optimal partition of Ω (minimizing the sum of the total perimeter and the approximation term is finite. Finiteness result for the problem of image segmentation in Artificial Vision can be deduced.

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Sunto Dimostriamo che, in opportune ipotesi sul dominio Ω e sul datog, ogni partizione ottimale di Ω (minimizzante il perimetro totale in Ω più il termine di approssimazione è finita. Se ne deducono risultati di finitezza per il problema della segmentazione di immagini in Visione Artificiale.

     

Two families of mixed finite elements for second order elliptic problems

Summary Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL ² () andH ^–5 () are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.     

Complete 1H and 13C NMR spectral assignment of cis- and trans- 3-(2-[2-(4-methylphenyl)ethenyl]phenyl])sydnones

1H and 13C NMR spectra of cis- and trans-3-(2-[2-(4-methylphenyl)ethenyl]phenyl])sydnones, the first stilbene-substituted mezoionic oxadiazolium rings, were fully assigned combining the information in various solvents, such as deuterated benzene, acetone and chloroform, using 2D NMR techniques.     

Regiocontrolled synthesis of enantiopure 3,3'-thiosubstituted biphenyls

Sulfenylation of 6,6'-dimethoxy-2,2'-dihydroxybiphenyl, used as a racemic mixture and single enantiomers, by phthalimidesulfenyl chloride afforded the corresponding 3,3'-N,N'-dithiophthalimide with complete regioselectivity. Simple manipulations of the latter compound allowed access to the corresponding bis-thiol or o-thioquinone as useful intermediates for the synthesis of new sulfur-containing open-chain and macrocyclic C(2) enantiopure ligands. The application of this methodology to the preparation of a biphenyl bearing two cysteine units as potential HIV-1 protease inhibitor is also described.     

Reactions of 1,4-diaza-3-methylbutadien-2-ylpalladium(II) derivatives with chloro-bridged rhodium(I) complexes

The reactions of the organometallic 1,4-diazabutadienes, RN=C(R′)C(Me)=NR″ [R = R″ = p-C₆H₄OMe, R′ = trans-PdCl(PPh₃)₂ (DAB); R = p-C₆H₄OMe, R″ = Me, R′ = trans-PdCl(PPh₃)₂ (DAB^I; R = R″ = p-C₆H₄OMe, R′ = Pd(dmtc)-(PPh₃), dmtc = dimethyldithiocarbamate (DAB^II); R = R″ = p-C₆H₄OMe, R′ = PdCl(diphos), diphos = 1,2-bis(diphenylphosphino)ethane (DAB^III)] with [RhCl(COD)]₂ (COD = 1,5-cyclooctadiene, Pd/Rh ratio = $\text{1}\text{2}$) depend on the nature of the ancillary ligands at the Pd atom in group R′. In the reactions with DAB and DAB^I transfer of one PPh₃ ligand from Pd to Rh occurs yielding [RhCl(COD)(PPh₃)] and the new binuclear complexes [Rh(COD) {RN=C(R?)-C(Me)=NR″}], in which the diazabutadiene moiety acts as a chelating bidentate ligand. Exchange of ligands between the two different metallic centers also occurs in the reaction with DAB^II. In this case, the migration of the bidentate dmtc anion yields [Rh(COD)Pdmtc] and [Rh(COD) {RN=C(R?)C(Me)=NR″}]. In contrast, the reaction with DAB^III leads to the ionic product [Rh(COD)- (DAB^III)][RhCl₂(COD)], with no transfer of ligands. The cationic complex [Rh(COD)(DAB^III)]⁺ can be isolated as the perchlorate salt from the same reaction (Pd/Rh ratio = 1/1) in the presence of an excess of NaClO₄. In all the binuclear complexes the coordinated 1,5-cyclooctadiene can be readily displaced by carbon monoxide to give the corresponding dicarbonyl derivatives. The reaction of [RhCl(CO)₂]₂ with DAB and/or DAB^I yields trinuclear complexes of the type [RhCl(CO)₂]₂(DAB), in which the diazabutadiene group acts as a bridging bidentate ligand. Some reactions of the organic diazabutadiene RN=C(Me)C(Me)=NR (R = p-C₆H₄OMe) are also reported for comparison.     

Intercalation of α,ω-alkyldiamines in layered α-zirconium phosphate and the inclusion behaviour of some of the intercalates obtained

The paper reports a study on the intercalation mechanism of NH₂(CH₂) _n NH₂ (withn=2, 4, 6, 8, 10) diamines in layered Zr(HOPO₃)₂·H₂O, performed by titrating the host with aqueous solutions of amines at 80°C. The intercalation reactions occur stepwise according to the ‘moving boundary’ model, with the formation of a number of intermediate intercalation compounds of formula Zr(HOPO₃)₂·xNH₂(CH₂) _n NH₂ (0<x<1) before obtaining the fully intercalated ones (x=1). For each diamine the batch titration curve and a diagram of the phases involved in the interaclation reaction are reported. Twenty-two intercalation compounds have been isolated and characterized by their composition, XRD patterns and thermal behaviour, and information on the disposition of the guests within the interlayer region have been derived. At full intercalation the diamines form a monolayer of extended molecules with their axis inclined at 58° to the plane of the sheet. The terminal amino groups are protonated by the —POH groups of the host, thus each diamine binds adjacent layers and, in a sense, transforms a layered structure into a framework structure that may have an accessible or potentially accessible porosity. The intercalation compound Zr(HOPO₃)₂·0.5NH₂(CH₂)₈NH₂ is indeed able to include polar molecules such as water and short chain alkanols.     

Charge-transfer interactions in face-to-face porphyrin-fullerene systems: solvent-dependent luminescence in the infrared spectral region

The cyclophane-type molecular dyads 1 x 2H and 1 x Zn, in which a doubly bridged porphyrin donor adopts a close, tangential orientation relative to the surface of a fullerene acceptor, were prepared by Bingel macrocylization. The porphyrin derivatives 2 x 2H and 2 x Zn with two appended, singly linked C60 moieties were also formed as side products. NMR investigations revealed that the latter compounds strongly prefer conformations with one of the carbon spheres nesting on the porphyrin surface, thereby taking a similar orientation to that of the fullerene moiety in the doubly bridged systems. Cyclic voltammetric measurements showed that the mutual electronic effects exerted by the fullerene on the porphyrin and vice versa are only small in all four dyads, despite the close proximity of the donor and acceptor components. The steady-state and time-resolved absorption and luminescence properties of 1 x Zn and 2 x Zn were investigated in toluene solution and it was shown that, upon light excitation, both the porphyrin- and the fullerene-centered excited states are deactivated to a lower-lying CT state, emitting in the IR spectral region (lambda max = 890 and 800 nm at 298 and 77 K, respectively). In the more polar solvent benzonitrile, this CT state is still detected but, owing to its very low energy (below 1.4 eV), is not luminescent and shorter-lived than in toluene. The remarkable observation of similar photophysical behavior of 1 x Zn and 2 x Zn suggests that a tight donor-acceptor distance cannot only be established in doubly bridged cyclophane-type structures but also in singly bridged dyads, by taking advantage of favourable fullerene-porphyrin ground-state interactions.     

Minimal boundaries enclosing a given volume

We prove some facts concerning surfaces of minimal area bounding regions of prescribed volume in ⁿ. The main result we prove is that the mean curvature of such a surface is constant, if possibly a discontinuous function of the enclosed volume. The boundary behaviour of the solutions is also discussed.