A sufficient condition for a graph to be hamiltonian

Those non-hamiltonian graphsG withn vertices are characterized, which satisfy the Ore-type degree-conditiond(x)+d(y)n–2 for each pairx,yM of different nonadjacent vertices whereM consists of two vertices ofG. As an application a theorem on hamiltonian connectivity of graphs is given. Furthermore, a condition is presented which is sufficient for the existence of a covering of a graph by two disjoint paths with prescribed set of startpoints and prescribed set of endpoints. A class of graphs is described which have no covering of this kind.     

Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen

In [8] the author extended the concept of neighbouring functions (cp. [9]) to the case of several variables. Using these results it is shown that under some weak conditions a multiplicative functionf in two variables has a mean-value different from zero if and only if the two multiplicative functionsf ₁(n)=f(n, 1) andf ₂(n)=f(1,n) have mean-values different from zero. Applications to theorems ofDelange [3],Elliott [6] andDaboussi [1] are given.     

Approximations of inventory models

Summary A general procedure is presented for constructing approximations of discrete review single product dynamic inventory models. Bounds are derived for the approximations and compared with the ones ofHinderer [1978],Whitt [1978] for the approximation of a general dynamic program. Good order-policies are constructed. Finally, conditions are given under which the sequence of bounds associated with a sequence of approximating models converges to zero.

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Zusammenfassung Es wird ein allgemeines Verfahren zur Approximation periodisch inspizierter dynamischer Ein-Produkt-Lagerhaltungsmodelle betrachtet. Dazu werden Fehlerschranken gegeben. Diese werden verglichen mit denen vonHinderer [1978],Whitt [1978] für ein allgemeines dynamisches Entscheidungsmodell. Außerdem werden gute Bestellpolitiken konstruiert und hinreichende Bedingungen gegeben, unter denen die Fehlerschranken einer Folge von approximierenden Modellen gegen Null konvergiert.

     

Facile Synthesis and Palladium‐Catalyzed Cross‐Coupling Reactions of 2,3‐Bis(pinacolatoboryl)‐1,3‐butadiene

Treatment of 1,1‐bis(pinacolatoboryl)ethene with an excess of 1‐bromo‐1‐lithioethene gave 2,3‐bis(pinacolatoboryl)‐1,3‐butadiene in high yield. Palladium‐catalyzed cross‐coupling of the resulting diborylbutadiene with aryl iodides took place smoothly in the presence of a catalytic amount of Pd(OAc)₂/PPh₃ and aqueous KOH to give 2,3‐diaryl‐1,3‐butadienes in good yields. The coupling reaction with commercially available 4‐acetoxyphenylmethyl chloride under the same conditions followed by hydrolysis of the acetyl groups gave anolignan B in a one‐pot manner. A variety of [3]‐ to [6]dendralenes were synthesized by palladium‐catalyzed coupling of the diene or 1,1‐bis(pinacolato)borylethene with alkenyl or dienyl halides, respectively, in good yields.     

(Me2NH2)[(Ph3Sn)3(MoO4)2], ein Triorganozinnmolybdat mit Schichtstruktur

(Me₂NH₂)[(Ph₃Sn)₃(MoO₄)₂], a Triorganotin Molybdate with Layer Structure The reaction of [(Ph₃Sn)₂MoO₄] with (Me₂NH₂)Cl in an acetonitrile/water mixture leads to the formation of (Me₂NH₂)[(Ph₃Sn)₃(MoO₄)₂] ( 1 ). ( 1 ) crystallizes in the space group Pca2₁ with a = 1967.0(4), b = 1353.1(2) and c = 2176.6(5) pm. In the crystal structure of 1 Ph₃SnO₂ bipyramides and MoO₄ tetrahedra are linked by corner sharing to give a layer structure. Additionally the layers are connected by O···H···N hydrogen bridges between MoO₄ groups and [Me₂NH₂]⁺ ions to give a 3D network structure.     

Redox,transmetalation, and stacking properties of tetrathiafulvalene-2,3,6,7-tetrathiolate bridged tin,nickel, and palladium compounds

Here we report that capping the molecule TTFtt (TTFtt = tetrathiafulvalene-2,3,6,7-tetrathiolate) with dialkyl tin groups enables the isolation of a stable series of redox congeners and facile transmetalation to Ni and Pd. TTFtt has been proposed as an attractive building block for molecular materials for two decades as it combines the redox chemistry of TTF and dithiolene units. TTFttH₄, however, is inherently unstable and the incorporation of TTFtt units into complexes or materials typically proceeds through the in situ generation of the tetraanion TTFtt⁴⁻. Capping of TTFtt⁴⁻ with Bu₂Sn²⁺ units dramatically improves the stability of the TTFtt moiety and furthermore enables the isolation of a redox series where the TTF core carries the formal charges of 0, +1, and +2. All of these redox congeners show efficient and clean transmetalation to Ni and Pd resulting in an analogous series of bimetallic complexes capped by 1,2-bis(diphenylphosphino)ethane (dppe) ligands. Furthermore, by using the same transmetalation method, we synthesized analogous palladium complexes capped by 1,1′-bis(diphenylphosphino)ferrocene (dppf) which had been previously reported. All of these species have been thoroughly characterized through a systematic survey of chemical and electronic properties by techniques including cyclic voltammetry (CV), ultraviolet-visible-near infrared spectroscopy (UV-vis-NIR), electron paramagnetic resonance spectroscopy (EPR), nuclear magnetic resonance spectroscopy (NMR) and X-ray diffraction (XRD). These detailed synthetic and spectroscopic studies highlight important differences between the transmetalation strategy presented here and previously reported synthetic methods for the installation of TTFtt. In addition, the utility of this stabilization strategy can be illustrated by the observation of unusual TTF radical–radical packing in the solid state and dimerization in the solution state. Theoretical calculations based on variational 2-electron reduced density matrix methods have been used to investigate these unusual interactions and illustrate fundamentally different levels of covalency and overlap depending on the orientations of the TTF cores. Taken together, this work demonstrates that tin-capped TTFtt units are ideal reagents for the installation of redox-tunable TTFtt ligands enabling the generation of entirely new geometric and electronic structures.

Capping TTFtt enables facile transmetalation in three different oxidation states.     

Kinetics of the CH2Cl + CH3 and CHCl2 + CH3 radical-radical reactions

The CH2Cl + CH3 (1) and CHCl2 + CH3 (2) cross-radical reactions were studied by laser photolysis/photoionization mass spectroscopy. Overall rate constants were obtained in direct real-time experiments in the temperature region 301-800 K and bath gas (helium) density (6-12) x 10(16) atom cm(-3). The observed rate constant of reaction 1 can be represented by an Arrhenius expression k1 = 3.93 x 10(-11) exp(91 K/T) cm3 molecule(-1) s(-1) (+/-25%) or as an average temperature-independent value of k1= (4.8 +/- 0.7) x 10(-11) cm3 molecule(-1) s(-1). The rate constant of reaction 2 can be expressed as k2= 1.66 x 10(-11) exp(359 K/T) cm3 molecule(-1) s(-1) (+/-25%). C2H4 and C2H3Cl were detected as the primary products of reactions 1 and 2, respectively. The experimental values of the rate constant are in reasonable agreement with the prediction based on the "geometric mean rule." A separate experimental attempt to determine the rate constants of the high-temperature CH2Cl + O2 (10) and CHCl2 + O2 (11) reaction resulted in an upper limit of 1.2 x 10(-16) cm(3) molecule(-1) s(-1) for k10 and k11 at 800 K.     

Sequence selective recognition in the minor groove of dsDNA by pyrrole,imidazole-substituted bis-benzimidazole conjugates

A series of pyrrole, imidazole-substituted bis-benzimidazole conjugates, Py-Py-Im-gamma-biBenz, Py-Py-gamma-biBenz, Py-Im-gamma-biBenz, and Im-Py-gamma-biBenz (1-4), were prepared in an attempt to target dsDNA sequences possessing both A/T and G/C bps. The dsDNA interactions and sequence specificity of the conjugates have been characterized via spectrofluorometric titrations and thermal melting studies. All conjugates form 1:1 complexes with dsDNA at subnanomolar concentrations. The Im moiety selectively recognizes a G/C bp embedded in the A/T-rich binding site. This represents the first clear example of sequence selective recognition in a 1:1 motif.(1) The equilibrium association constant (K(1)) for complexation of a specific nine-bp dsDNA site, 5'-gcggTATGAAATTcgacg-3', by conjugate 1 is approximately 2.6 x 10(9) M(-1). Displacement of the G/C position or G/C-->A/T substitution within the nine-bp site decreases the K(1) by approximately 8-fold, whereas two continuous G/C bps decrease the K(1) by approximately 50-fold magnitude. The K(1) values for seven-bp dsDNA, 5'-gcggtaTGAAATTcgacg-3' and 5'-gcggtaCAAAATTcgacg-3', binding sites by conjugates Py-Im-gamma-biBenz (3) and Im-Py-gamma-biBenz (4) are approximately 2.3 x 10(9) and approximately 1.2 x 10(9) M(-1), respectively. However, the conjugates with no Im moiety, Py-Py-gamma-biBenz (2) and Py-Py-Py-gamma-biBenz (5 and 6), are specific for seven- to nine-bp A/T-rich sites and single A/T-->G/C bp substitution within the binding site decreases the K(1) values by 1-2 orders of magnitude.     

Regioselective Borylation of Porphyrins by C?H Bond Activation under Iridium Catalysis to Afford Useful Building Blocks for Porphyrin Assemblies

Highly regioselective and efficient borylation of a variety of porphyrins has been achieved by reaction with bis(pinacolato)diboron through C? H bond activation under iridium catalysis on the basis of the synthetic protocol developed by Miyaura, Hartwig, and Smith. A boryl group can be selectively introduced at sterically uncongested positions in the peripheral aryl groups of porphyrin substrates whose peripheral β‐positions are sterically hindered. Curiously, β substituents adjacent to the aryl group to be borylated have unexpectedly large effects on the regioselectivity, because the iridium catalyst can discriminate between subtle steric differences. Chemoselective borylation was also achieved for several functionalized porphyrins. This borylation protocol can be applied to various monomeric and oligomeric functional porphyrins, hence offering an efficient route to elaborate multiporphyrin‐based molecular constructs.