A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

Rates of ligand exchange between >FeIII-OH2 functional groups on a nanometer-sized aqueous cluster and bulk solution

Variable-temperature 17O NMR experiments were conducted on the nanometer-sized Keplerate Mo72Fe30 cluster, with the stoichiometry [Mo72Fe30O252(CH3COO)12[Mo2O7(H2O)]2[H2Mo2O8(H2O)](H2O)91]. approximately 150H2O. This molecule contains on its surface 30 Fe(H2O) groups forming a well-defined icosidodecahedron, and we estimated the rates of exchange of the isolated >FeIII-OH2 waters with bulk aqueous solution. Both longitudinal and transverse 17O-relaxation times were measured, as well as chemical shifts, and these parameters were then fit to the Swift-Connick equations in order to obtain the rate parameters. Correspondingly, we estimate: k(ex)298 = 6.7(+/-0.8) x 106 s-1, which is about a factor of approximately 4 x 104 times larger than the corresponding rate coefficient for the Fe(OH2)63+ ion of k(ex)298 = 1.6 x 102 s-1 (Grant and Jordan, 1981; Inorg. Chem. 20, 55-60) and DeltaH and DeltaS are 26.3 +/- 0.6 kJ mol-1 and -26 +/- 0.9 J mol-1 K-1, respectively. High-pressure 17O NMR experiments were also conducted, but the cluster decomposed slightly under pressure, which precluded confident quantitative estimation of the DeltaV. However, the increase in the reduced transverse-relaxation time with pressure suggests a dissociative character, such as a D or Id mechanism. The enhanced reactivity of waters on the Mo72Fe30 cluster is associated with an increase in the FeIII-OH2 bond length in the solid state of approximately 0.1 A relative to the Fe(OH2)63+ ion, suggesting that a correlation exists between the FeIII-OH2 bond length and k(ex)298. Although there are only few high-spin Fe(III) complexes where both exchange rates and structural data are available, these few seem to support a general correlation.     

Monopropionate analogues of DOTA4- and DTPA5-: kinetics of formation and dissociation of their lanthanide(III) complexes

The replacement of an acetate function of the macrocyclic DOTA4-(DO3A-Nprop4-) or the acyclic DTPA5- in terminal position (DTTA-Nprop5-) has been recently shown to result in a significant increase of the water exchange rate on the Gd3+ complexes, which makes these chelates potential contrast agents for MRI applications. Here, two novel and straightforward synthetic routes to H4DO3A-Nprop are described. Protonation constants of DO3A-Nprop4- and stability constants with several alkaline earth and transition metal ions have been determined by potentiometry. For each metal, the thermodynamic stability constant is decreased in comparison to the DOTA chelates. The formation reaction of LnDO3A-Nprop- complexes (Ln=Ce, Gd and Yb) proceeds via the rapid formation of a diprotonated intermediate and its subsequent deprotonation and rearrangement in a slow, OH- catalyzed process. The stability of the LnH2DO3A-Nprop* intermediates is similar to those reported for the corresponding DOTA analogues. The rate constants of the OH- catalyzed deprotonation step increase with decreasing lanthanide ion size, and are slightly higher than for DOTA complexes. The kinetic inertness of GdDTTA-Nprop2- was characterized by the rates of its exchange reactions with Zn2+ and Eu3+. The rate of the reaction between GdDTTA-Nprop2- and Zn2+ increases with Zn2+ concentration, while it is independent of pH, implying that the exchange takes place predominantly via direct attack of the metal ion on the complex. In the Eu3+ exchange, the rate decreases with increasing concentration of the exchanging ion which is accounted for by the transitional formation of a dinuclear GdDTTA-NpropEu+ species. The kinetic inertness of the monopropionate GdDTTA-Nprop2- is decreased in comparison to GdDTPA2-: all rate constants, characterizing the dissociation reaction via either proton- or metal-catalyzed pathways being higher by 1-2 orders of magnitude. Similarly, a study of the acid-catalyzed dissociation of the macrocyclic CeDO3A-Nprop- showed a partial loss of the kinetic inertness with regard to the tetraacetate derivative CeDOTA-.     

Conservative methods for the Toda lattice equations

We are concerned with the numerical integration of the Todalattice equations by using different conservative methods. Numericalexperiments suggest that the global error for isospectral schemesdecreases exponentially with time but it is almost constantfor either symplectic or more general integrators. We providea theoretical explanation for these experimental findings.     

Optimal design accounting for uncertainty in loading amplitudes: A numerical investigation

A numerical investigation is performed addressing the optimal design of stiff structures accounting for uncertainty in loading amplitudes. A minimum volume problem is endowed with a stochastic compliance constraint handling normal distributions and solved adopting mathematical programming. The formulation, originally conceived for a single load case, is extended to handle multiple load cases. Numerical simulations are performed to test the proposed algorithms, pointing out features of the numerical procedures and peculiarities of the stochastic-based optimal solutions achieved for different values of the second-order moments. Comparisons with respect to conventional deterministic layouts are provided as well.     

Co2(CO)8-induced domino reactions of ethyl diazoacetate, carbon monoxide and ferrocenylimines leading to 2-(1-ferrocenyl-methylidene)-malonic acid derivatives

Novel 2-(1-ferrocenyl-methylidene)-malonic acid derivatives are obtained upon reacting ethyl diazoacetate, carbon monoxide and ferrocenylimines in the presence of Co₂(CO)₈ as catalyst under mild conditions. Presumably, the reaction involves three steps taking place in a domino fashion, (i) carbonylation of ethyl diazoacetate leading to a ketene derivative, (ii) [2+2] cycloaddition of the ketene with the ferrocenylimine present in the reaction mixture resulting in the formation of a β-lactam and (iii) N(1)-C(4) cleavage of the β-lactam ring. In most cases, 2-(1-ferrocenyl-methylidene)-malonic acid derivatives are obtained as a separable mixture of E- and Z-isomers in ratios depending on the structure of the imine component.     

Electrophoretic mobility and molecular distribution studies of poly(amidoamine) dendrimers of defined charges

Generation 5 ethylenediamine (EDA)-cored poly(amidoamine) (PAMAM) dendrimers (E5, E denotes the EDA core and 5 the generation number) with different degrees of acetylation and carboxylation were synthesized and used as a model system to investigate the effect of charge and the influence of dendrimer surface modifications on electrophoretic mobility (EM) and molecular distribution. The surface-modified dendrimers were characterized by size-exclusion chromatography, 1H NMR, MALDI-TOF-MS, PAGE, and CE. The focus of our study was to determine how EM changes as a function of particle charge and molecular mass, and how the molecular distribution changes due to surface modifications. We demonstrate that partially modified dendrimers have much broader migration peaks than those of fully surface functionalized or unmodified E5 dendrimers due to variations in the substitution of individual dendrimer surfaces. EM decreased nonlinearly with increases in surface acetylation for both PAMAM acetamides and PAMAM succinamic acids, indicating a complex migration activity in CE separations that is not solely due to charge/mass ratio changes. These studies provide new insights into dendrimer properties under an electric field, as well as into the characterization of dendrimer-based materials being developed for medical applications.     

Almost all triangle-free triple systems are tripartite

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite. Our proof uses the hypergraph regularity lemma of Frankl and R?dl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].