On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015     

In search for mixed transition metal/lanthanide single-molecule magnets: Synthetic routes to NiII/TbIII and NiII/DyIII clusters featuring a 2-pyridyl oximate ligand

Employing the mononuclear complex [Ni{(py)C(Me)NO}₂{(py)C(Me)NOH}] (1) as ‘ligand’ [(py)C(Me)NOH = methyl 2-pyridyl ketone oxime], the use of the ‘metal complexes as ligands’ approach has led to the synthesis of the mixed Ni^II/Ln^III complexes [NiTb{(py)C(Me)NO}₂(NO₃)₃{(py)C(Me)NOH}] (2), [Ni₂Ln₂{(py)C(Me)NO}₆(NO₃)₄] (Ln = Dy, 3; Ln = Tb, 4) and [Ni₂Tb{(py)C(Me)NO}₆](NO₃) (5). The structures of 2, 3, and 5, and the magnetic properties of 2 and 5 are briefly discussed.     

Triangular Ni2Ln and Ni2Y complexes derived from di-2-pyridyl ketone: Synthesis, structures and magnetic properties

The synthetic investigation of the Ni^II/M(NO₃)₃·6H₂O/di-2-pyridyl ketone [(py)₂CO] tertiary reaction system in EtOH has yielded triangular Ni₂M cationic complexes (M = lanthanide, Y). The reaction between Ln(NO₃)₃·6H₂O, Ni(ClO₄)₂·6H₂O, (py)₂CO and base (1:3:3:3) in EtOH under gentle heating gave the isostructural complexes [Ni₂Ln{(py)₂C(OEt)(O)}₃{(py)₂C(OH)(O)}(NO₃)(H₂O)](ClO₄)₂ (Ln = Gd, 2; Ln = Tb, 3) in high yields. The ligands (py)₂C(OEt)(O)⁻ and (py)₂C(OH)(O)⁻ are the monoanions of the hemiketal and gem-diol derivatives of (py)₂CO, respectively, formed in situ in the presence of the metal ions. The cations of 2 and 3 consist of one 8-coordinate Ln^III and two distorted octahedral Ni^II atoms in an essentially isosceles, triangular arrangement capped by a central μ₃-Ο⁻ atom of the unique 3.3011 (Harris notation) (py)₂C(OH)(O)⁻ ligand. Each metal-metal edge is bridged by the deprotonated O atom of one 2.2011 (py)₂C(OEt)(O)⁻ ligand. The isostructural complexes [Ni₂M{(py)₂C(OEt)(O)}₄(NO₃)(H₂O)]₂[M(NO₃)₅](ClO₄)₂ (M = Y, 4 ; M = Tb, 5 ; M = Dy, 6) were prepared by the 1:1 reaction of the mononuclear “metalloligand” [Ni(O₂CMe){(py)₂CO}{(py)₂C(OH)₂}](ClO₄) (1) and M(NO₃)₃·6H₂O in EtOH under mild heating in moderate to good yields. The structures of the dications of 4-6 are similar to those in 2 and 3, the only difference being the replacement of the unique 3.3011 (py)₂C(OH)(O)⁻ ligand of the latter by one 3.3011 (py)₂C(OEt)(O)⁻ group in the former. The Y^III, Tb^III and Dy^III atoms in [M(NO₃)₅]²⁻ are coordinated by five bidentate chelating nitrato groups. Characteristic IR bands of the complexes are discussed in terms of the known structures and the coordination modes of the ligands. Variable temperature, solid-state direct current magnetic susceptibility and magnetization studies were carried out on dried samples of 2-4. The data indicate ferromagnetic Ni?Ni and Ni?Gd exchange interactions, and an S_T = 11/2 ground state for 2. Complex 3 is characterized by a high-spin ground state while the ferromagnetic Ni?Ni interaction for 2 is independently supported by the study of 4. No out-of-phase, alternating current susceptibility signals have been detected for 3 that would be indicative of SMM behavior.     

Synthesis and properties of oligonucleotides that contain a triazole‐linked nucleic acid dimer

New chemically modified oligonucleotides at the site of the backbone are needed to improve the properties of oligonucleotides. A practical synthesis for a triazole‐linked nucleoside dimer based on a PNA‐like structure has been developed. This involves synthesizing two uracil‐based monomers that contain either an azide or an alkyne functionality, followed by copper‐catalyzed 1,3‐dipolar cycloaddition. This dimer was incorporated within an oligonucleotide via phosphoramidite chemistry and UV‐monitored thermal denaturation data illustrates slight destabilization relative to its target complementary sequence. This chemically modified dimer will allow for a future investigation of its properties within DNA and RNA‐based applications. J. Heterocyclic Chem., (2011).     

A high-nuclearity 3d/4f metal oxime cluster: an unusual Ni(8)Dy(8) "core-shell" complex from the use of 2-pyridinealdoxime

The initial employment of 2-pyridinealdoxime in 3d/4f chemistry has led to a Ni(II)(8)Dy(III)(8) cluster with an unprecedented metal topology; the compound has an unusual structure, is the highest-nuclearity metal oxime cluster to date, and exhibits slow magnetization relaxation.     

Topology and statistics in zero dimensions

It has been suggested that space-time may be intrinsically not continuous, but discrete. Here we review some topological notions of discrete manifolds, in particular ones made out of finite number of points, and discuss the possibilities for statistics in such spaces.     

Innovation diffusion modeling: the deterministic, stochastic and chaotic case

The field of innovation diffusion modeling showed a tremendous growth process during the last decades. Numerous qualitative and quantitative studies have been presented followed by significant applications in various scientific fields. This review paper explores the main quantitative developments on innovation diffusion and the gradual progress from the original deterministic models to their stochastic and chaotic alternatives. Related applications are presented.     

A mononuclear complex and a cubane cluster from the initial use of 2-(hydroxymethyl)pyridine in nickel(II) carboxylate chemistry

The reactions of 2-(hydromethyl)pyridine, hmpH, with Ni(O₂CMe)₂·4H₂O in H₂O, in the absence of counterions, have been investigated. The synthetic study has led to the two new complexes [Ni(O₂CMe)₂(hmpH)₂] (1) and [Ni₄(O₂CMe)₄(hmp)₄(H₂O)₂] (2). Complex 1 can also be transformed into 2 by reacting with an excess of NaOH in H₂O. The structures of 1 and 2·2.25H₂O·0.5(1,4-dioxane) have been solved by single-crystal, X-ray crystallography. The octahedral Ni^II center in centrosymmetric 1 is coordinated by two 1.10 (Harris notation) MeCO₂⁻ groups and two N,O-chelating (1.11) hpmH ligands. The tetranuclear cluster molecule of 2·2.25H₂O·0.5(1,4-dioxane) possesses a distorted cubane {Ni₄(μ₃-OR′)₄}⁴⁺ core [R′ = (2-pyridyl)CH₂–] with the Ni^II ions and the oxygen atoms from the 3.31 hmp⁻ ligands occupying alternate vertices of the cube. Two 2.11 MeCO₂⁻ groups cap two opposite faces of the cube, while two 1.10 MeCO₂⁻ ions and two aqua ligands complete the octahedral coordination sphere of the metal centers. Characteristic IR bands for the two complexes are discussed in terms of the nature of bonding and the structures of the two complexes. The variable-temperature magnetic properties of 2 have been modeled with two J values, and reveal antiferromagnetic exchange interactions between the four Ni^II ions to give a diamagnetic ground state.     

The first cobalt metallacrowns: preparation and characterization of mixed-valence cobalt(II/III), inverse 12-metallacrown-4 complexes

Aerobic reactions of Co(O(2)CMe)(2).4H(2)O with di-2-pyridyl ketone oxime (Hpko) in the presence of counterions (ClO(4)(-), PF(6-)) give the tetranuclear, mixed-valence cobalt(II/III) clusters [Co(II)(2)Co(III)(2)(OR)(2)(O(2)CMe)(2)(pko)(4)S(2)]X(2) [R = H, S = MeOH, X = ClO(4) (1); R = Me, S = EtOH, X = PF(6) (2)] depending on the solvent mixture. Complexes 1 and 2 are the first Co members in the family of metallacrowns adopting the extremely rare inverse 12-metallacrown-4 motif.