An extension of Babbage’s criterion for primality

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n ? 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.     

Approximating the maximum 3-edge-colorable subgraph problem

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least of its edges, where γ_o(G) denotes the odd girth of G. In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13/15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6/7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound.For a fixed value of a parameter k≥1, the Maximum k-Edge-Colorable Subgraph Problem asks to k-edge-color the most of the edges of a simple graph received in input. The problem is known to be APX-hard for all k≥2. However, approximation algorithms with approximation ratios tending to 1 as k goes to infinity are also known. At present, the best known performance ratios for the cases k=2 and k=3 were 5/6 and 4/5, respectively. Since the proofs of our structural result are algorithmic, we obtain an improved approximation algorithm for the case k=3, achieving approximation ratio of 6/7. Better bounds, and allowing also for parallel edges, are obtained for graphs of higher odd girth (e.g., a bound of 13/15 when the input multigraph is restricted to be triangle-free, and of 19/21 when C₅’s are also banned).     

Topological Features of Massive Bosons on Two-Dimensional Einstein Space–Time

In this paper, we tackle the problem of constructing explicit examples of topological cocycles of Roberts’ net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two-dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces). The construction is carried out using only Cauchy data and related net of local algebras on the circle.     

Counteranion dependent protonation and aggregation of tetrakis(4-sulfonatophenyl)porphyrin in organic solvents

The tetrabutylammonium salt of 5,10,15,20-tetrakis(4-sulfonatophenyl)porphyrin (TPPS) is soluble in dichloromethane, and the general properties of this compound have been investigated as function of various added acids HX (X = Cl, Br, I, CF(3)COO, CF(3)SO(3), TFPB) through UV-vis absorption spectroscopy, steady state fluorescence emission, and resonance light-scattering (RLS) techniques. Upon addition of HX, the initial monomeric free base TPPS is readily converted in an aggregated species, whose spectroscopic features are independent of the nature of the counteranion X. All the spectroscopic evidence suggest a J-type arrangement of chromophores in this aggregate, involving strong hydrogen bonds, electrostatic, and dispersive interactions. In the specific case of chloride and bromide, in the presence of a TBAX excess, the addition of the corresponding acid leads to a monomeric ion-pair between the TBA cations and the diacid TPPS, whose central core is strongly interacting with the halide. On further increasing the acid concentration in these latter solutions, fully protonated species are formed that eventually start to aggregate.     

Platinum(II) complexes bearing 1,1'-bis(diphenylphosphino)ferrocene as building blocks for functionalized redox active porphyrins

Reactions of the cationic complex ions [PtMe(Me2SO)(PP)]+ (PP = dppf (1,1'-bis(diphenylphosphino)ferrocene) and dppe (1,2-bis(diphenylphosphino)ethane)) with 5,10,15,20-tetrakis(4-pyridyl)-21H,23H-porphyrin (TpyP) led to the formation of the symmetrical tetraplatinated porphyrin complexes, [PtMe(PP)]4TpyP.X4 (PP = dppf, X = CF3SO3-, 3, and PP = dppe, X = BF4-, 5) containing the organometallic fragment [PtMe(PP)]. The precursor sulfoxide platinum complexes [PtMe(Me2SO)(dppf)]CF3SO3, 2 and [PtMe(Me2SO)(dppe)]BF4, 4, were prepared by halide abstraction from [PtMeCl(dppf)], 1, and by controlled protonolysis of [PtMe2(dppe)] respectively, in the presence of a small amount of dimethyl sulfoxide. All these starting platinum(II) compounds, as well as the porphyrin derivatives 3 and 5, were fully characterized through elemental analysis, 1H NMR mono- and bidimensional, 31P[1H], 31P-1H HMBC, UV/Vis absorption and photophysical measurements. The X-ray crystal structure of complex 1 has been determined. In order to ascertain the electronic influence of ferrocene, the spectroscopic and redox properties of 3 were compared with those of TPyP and of the analogous 5. Cyclic voltammetry (CV), differential pulse voltammetry (DPV), 1H and 31P NMR data, and UV/Vis data, all suggest a certain degree of communication between the central porphyrin and the peripheral hetero-bimetallic fragments. In contrast, no detectable interaction among these peripheral groups seem to come into play. Unlikely from the porphyrin derivative 5, formation of well defined fluorescent mesoscopic ring structures was easily achieved by simple evaporation from diluted dichloromethane solutions of 3.     

Porphyrin deposition induced by UV irradiation

Simple irradiation by UV light of porphyrin TpyP and its zinc(II) and manganese(III) metal derivatives in chlorinated solvents leads to the deposition on silica of mesoscale aggregates. The process is due to the photochemical decomposition of the solvent with generation of HCl, which causes the precipitation of a protonated form. The morphology of the resulting objects is driven by the nature of the inserted metal ion.     

Regioselective one-pot synthesis of 9-alkyl-6-chloropyrido[3,2-e][1,2,4]triazolo[4,3-a]pyrazines. Reactivity of aliphatic and aromatic hydrazides

[reaction: see text] The one-pot synthesis of new 9-alkyl-6-chloropyrido[3,2-e][1,2,4]triazolo[4,3-a]pyrazines has been achieved. Hydrazides regioselectively reacted as nucleophiles with the 3-chloro substituent of 2,3-dichloropyrido[2,3-b]pyrazine. An intramolecular cyclization afforded the tricycle nonxanthine adenosine receptor antagonists.     

Synthesis of some 2,3-Dihydro-11bH-thiazolo[3,2-d][1,4]benzoxazepin-5-(6H)one derivatives. A new heterocyclic ring system

Condensation of (S)-penicillamine methyl or ethyl ester hydrochloride with salicylaldehyde and its C-5 derivatives, provided the diastereomeric thiazolidine derivatives 1 and 2 . The resulting amino function was acylated to afford the amides 3 and 4. Cyclization of the latter led to the 2,3-dihydro-11bH-thiazolo[3,2-d]-[1,4]benzoxazepin-5-(6H)ones 5 and 6. Conformational data for these heterocyclic compounds are discussed.     

Phosphorprotein intermediate in the Ca2+-dependent ATPase reaction of macrophage plasma membrane.

ATPase activity and phosphorylation by [gamma-32P] ATP of isolated plasma membrane of alveolar macorphages are stimulated in a parallel fashion by physiologic concentrations of Ca2+, with half-maximal activating effect of this ion at (3--7) X 10(-7) M. For various membrane preparations, a direct proportionality exists between Ca2+-dependent ATPase activity and amount of 32P incorporated. Labeling of membrane attains the steady-state level by 10 sec at 0 degrees C, and is rapidly reversed by adenosine diphosphate (ADP), K+ decreases the amount of membrane-bound 32P, mainly by enhancing the rate of dephosphorylation of the 32P-intermediate. Hydroxylamine causes a release of about 90% of 32P bound to the membrane, thus indicating that the 32P-intermediate contains an acyl-phosphate bond. When the labeled plasma membrane is solubilized and electrophoresed on acrylamide gels in the presence of sodium dodecyl sulphate, the radioactivity appears to be largely associated with a single protein fraction of 132,000 +/- 2,000 aarent molecular weight. These features of the macrophage Ca2+-ATPase suggest that the enzyme activity might be part of a surface-localized Ca1+-extrusion system, participating in the regulation of Ca2+-dependent activities of the macrophage.     

To what extent can cyclometalation promote associative or dissociative ligand substitution at platinum(II) complexes? A combined kinetic and theoretical approach

The ligand exchange rate constants for the reactions [Pt(bph)(SR2)2] + 2*SR2 --> [Pt(bph)(*SR2)2] + 2SR2 (bph = 2,2'-biphenyl dianion; R = Me and Et) and cis-[PtPh2(SMe2)2] + 2*SMe2 --> cis-[PtPh2(*SMe2)2] + 2SMe2 have been determined in CDCl3 as a function of ligand concentration and temperature, by 1H NMR isotopic labeling and magnetization transfer experiments. The rates of exchange show no dependence on ligand concentration and the kinetics are characterized by largely positive entropies of activation. The kinetics of displacement of the thioethers from [Pt(bph)(SR2)2] with the dinitrogen ligands 2,2'-bipyridine and 1,10-phenanthroline (N-N) to yield [Pt(bph)(N-N)], carried out in the presence of sufficient excess of thioether and N-N to ensure pseudo-first-order conditions, follow a nonlinear rate law k(obsd) = a[N-N]/(b[SR2] + [N-N]). The general pattern of behavior indicates that the rate-determining step for substitution is the dissociation of a thioether ligand and the formation of a three-coordinated [Pt(bph)(SR2)] intermediate. The value of the parameter a, which measures the rate of ligand dissociation, is constant and independent of the nature of N-N, and it is in reasonable agreement with the value of the rate of ligand exchange at the same temperature. Theoretical ab initio calculations were performed for both [Pt(bph)(SMe2)2] and cis-[PtPh2(SMe2)2], and for their three-coordinated derivatives upon the loss of one SMe2 ligand. The latter optimize in a T-shaped structure. Calculations were performed in the HF approximation (LANL2DZ basis set) and refined by introducing the correlation terms (Becke3LYP model). The activation enthalpies from the optimized vacuum-phase geometries are 52.3 and 72.2 kJ moll compared to the experimental values in CDCl3 solution, 80 +/- 1 and 93 +/- 1 kJ mol(-1) for [Pt(bph)(SMe2)2] and cis-[PtPh2(SMe2)2], respectively. The electrostatic potential maps of both parent compounds show a remarkable concentration of negative charge over the platinum atom which exerts a repulsion force on an axially incoming nucleophile. On the other hand, the strength of the organic carbanions trans to the leaving group and the stabilization of the T-shaped intermediate in the singlet ground state may also rationalize the preference for the dissociative mechanism. All of the kinetic and theoretical data support the latter hypothesis and indicate, in particular, that dissociation from the complex containing the planar 2,2'-biphenyl dianion is easier than from its analogue with single aryl ligands. Electron back-donation from filled d orbitals of the metal to empty pi* of the in-plane cyclometalated rings is weak or absent and is not operative in promoting an associative mode of activation.