Shifting From Ziegler–Natta to Philips‐Type Catalyst? A Simple and Safe Access to Reduced Titanium Systems for Ethylene Polymerization

Silica‐supported titanium(IV) chloride is readily reduced by Mashima and co‐workers' reagent (1‐methyl‐3,6‐bis(trimethylsilyl)‐1,4‐cyclohexadiene) to afford materials active in ethylene polymerisation without need of aluminum alkyl cocatalyst.     

Long-term variability of global statistical properties of epileptic brain networks

We investigate the influence of various pathophysiologic and physiologic processes on global statistical properties of epileptic brain networks. We construct binary functional networks from long-term, multichannel electroencephalographic data recorded from 13 epilepsy patients, and the average shortest path length and the clustering coefficient serve as global statistical network characteristics. For time-resolved estimates of these characteristics we observe large fluctuations over time, however, with some periodic temporal structure. These fluctuations can--to a large extent--be attributed to daily rhythms while relevant aspects of the epileptic process contribute only marginally. Particularly, we could not observe clear cut changes in network states that can be regarded as predictive of an impending seizure. Our findings are of particular relevance for studies aiming at an improved understanding of the epileptic process with graph-theoretical approaches.     

On the Intersection Problem for Steiner Triple Systems of Different Orders

A Steiner triple system of order v, or STS(v), is a pair (V, ) with V a set of v points and a set of 3-subsets of V called blocks or triples, such that every pair of distinct elements of V occurs in exactly one triple. The intersection problem for STS is to determine the possible numbers of blocks common to two Steiner triple systems STS(u), (U, ), and STS(v), (V, ), with U⊆V. The case where U=V was solved by Lindner and Rosa in 1975. Here, we let U⊂V and completely solve this question for v−u=2,4 and for v≥2u−3. supported by NSERC research grant #OGP0170220. supported by NSERC postdoctoral fellowship. supported by NSERC research grant #OGP007621.     

Reverse lexicographic and lexicographic shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ_lex—an operation that transforms a monomial ideal of S = K[x_i: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence } are distinct. The limit ideal is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I. Research partially supported by NSF grants DMS 0070571 and DMS 0100141.     

Modularity of a certain Calabi-Yau threefold and combinatorial congruences

We prove a certain Calabi-Yau threefold is modular in the sense that the number of points on its reduction modulo p is expressed in terms of the pth coefficient of a weight 4 newform in S₄(Γ₀(6)). We also give a mod p² combinatorial expression for these coefficients. 2000 Mathematics Subject Classification: 11G25, 11G40     

On the R-Matrix Realization of Yangians and their Representations

We study the Yangians associated with the simple Lie algebras of type B, C or D. The algebra can be regarded as a quotient of the extended Yangian whose defining relations are written in an R-matrix form. In this paper we are concerned with the algebraic structure and representations of the algebra . We prove an analog of the Poincaré–Birkhoff–Witt theorem for and show that the Yangian can be realized as a subalgebra of . Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of which implies the corresponding theorem of Drinfeld for the Yangians . We also give explicit constructions for all fundamental representation of the Yangians. Communicated by Petr Kulish Dedicated to Daniel Arnaudon Submitted: November 22, 2005; Accepted: February 1, 2006     

Sugawara and Vertex Operator Constructions for Deformed Virasoro Algebras

From the defining exchange relations of the elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W_N algebras. The consistency conditions relating the parameters p, q, N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of available when N = 2. Communicated by Petr Kulish Submitted: January 13, 2006; Accepted: March 6, 2006 Dedicated to our friend Daniel Arnaudon     

Gold(I) and platinum(II) complexes with a new diphosphine ligand based on the cyclotriphosphazene platform

A new diphosphine ligand assembled on the cyclotriphosphazene platform forms linear chelate and dimetallic bridged complexes with Au(I) and a cis-chelate complex with Pt(II).     

Phenoxyl radicals: H-bonded and coordinated to Cu(II) and Zn(II)

Two pro-ligands ((R)LH) comprised of an o,p-di-tert-butyl-substituted phenol covalently bonded to a benzimidazole ((Bz)LH) or a 4,5-di-p-methoxyphenyl substituted imidazole ((PhOMe)LH), have been structurally characterised. Each possesses an intramolecular O-H[dot dot dot]N hydrogen bond between the phenolic O-H group and an imidazole nitrogen atom and (1)H NMR studies show that this bond is retained in solution. Each (R)LH undergoes an electrochemically reversible, one-electron, oxidation to form the [(R)LH] (+) radical cation that is considered to be stabilised by an intramolecular O...H-N hydrogen bond. The (R)LH pro-ligands react with M(BF(4))(2).H(2)O (M = Cu or Zn) in the presence of Et(3)N to form the corresponding [M((R)L)(2)] compound. [Cu((Bz)L)(2)] (), [Cu((PhOMe)L)(2)] (), [Zn((Bz)L)(2)] and [Zn((PhOMe)L)(2)] have been isolated and the structures of .4MeCN, .2MeOH, .2MeCN and .2MeCN determined by X-ray crystallography. In each compound the metal possesses an N(2)O(2)-coordination sphere: in .4MeCN and .2MeOH the {CuN(2)O(2)} centre has a distorted square planar geometry; in .2MeCN and .2MeCN the {ZnN(2)O(2)} centre has a distorted tetrahedral geometry. The X-band EPR spectra of both and , in CH(2)Cl(2)-DMF (9 : 1) solution at 77 K, are consistent with the presence of a Cu(ii) complex having the structure identified by X-ray crystallography. Electrochemical studies have shown that each undergo two, one-electron, oxidations; the potentials of these processes and the UV/vis and EPR properties of the products indicate that each oxidation is ligand-based. The first oxidation produces [M(II)((R)L)((R)L )](+), comprising a M(ii) centre bound to a phenoxide ((R)L) and a phenoxyl radical ((R)L ) ligand; these cations have been generated electrochemically and, for R = PhOMe, chemically by oxidation with Ag[BF(4)]. The second oxidation produces [M(II)((R)L )(2)](2+). The information obtained from these investigations shows that a suitable pro-ligand design allows a relatively inert phenoxyl radical to be generated, stabilised by either a hydrogen bond, as in [(R)LH] (+) (R = Bz or PhOMe), or by coordination to a metal, as in [M(II)((R)L)((R)L )](+) (M = Cu or Zn; R = Bz or PhOMe). Coordination to a metal is more effective than hydrogen bonding in stabilising a phenoxyl radical and Cu(ii) is slightly more effective than Zn(II) in this respect.     

Copper(I/II) complexes of a bis(tetrathiafulvalene)-2,2'-bipyridine: synthesis, characterization, magnetic and electrochemical properties

The coordination ability of the electroactive TTF-based chelating ligand 5,5'-bis(4,5-bis(thiomethyl)-4'-carbamoyltetrathiafulvalene)-2,2'-bipyridine (L) has been tested with Cu(I) and Cu(II) centres. [(L)2Cu(I)](PF6), [(L)2Cu(II)](OTf)2 and [(L)Cu(II)(DMF)3](OTf)2 have been synthesized. A single-crystal X-ray analysis was performed on [(L)Cu(II)(DMF)3](OTf)2, showing a distorted octahedral geometry around the Cu(II) centre, and the formation of dimeric units in the solid state through weak coordination in apical position of an amide oxygen atom from a neighbouring complex. Magnetic data show that the paramagnetic metallic centres are isolated, in agreement with the solid-state structure. Electrochemical measurements were performed on the three complexes and in all cases the Cu(I)/Cu(II) and TTF/TTF+*/TTF2+ redox processes were observed.