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.     

Periodic solutions for a neutral nonlinear dynamical equation on a time scale

Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel'ski? to show that the nonlinear neutral dynamic system with delay

     

Fluorescent DNA base replacements: Reporters and sensors for biological systems

We describe the design, synthesis, and properties of nucleoside monomers in which the DNA base is replaced by fluorescent hydrocarbons and heterocycles, and the assembly of these monomers into DNA-like molecules in which the all bases are fluorescent. As monomers, such molecules have useful applications as reporters in the DNA context. The use of fluorescent DNA bases, rather than more traditional fluorophores tethered to the DNA strand, gives a more predictable location and orientation, and yields a more direct response to changes that occur within the helix. In addition to uses as monomers, such compounds can be assembled into polychromophoric oligomers ("oligodeoxyfluorosides" or ODFs). ODFs are water soluble, discrete molecules and are easily arranged into specific sequences by use of a DNA synthesizer. They have displayed a number of properties not readily available in commercial fluorophores, including large Stokes shifts, tunable excitation and emission wavelengths, and sensing responses to physical changes or molecular species in solution. We describe an approach to assembling and screening large sets of oligofluorosides for rapid identification of molecules with desirable properties. Such compounds show promise for applications in biochemistry, biology, environmental and materials applications.     

Efficient electrocatalyst utilization: electrochemical deposition of Pt nanoparticles using nafion membrane as a template

We deposit Pt particles electrochemically on an electrode covered with a Nafion membrane. Platinum ions travel through the hydrophilic channels of the membrane, and platinum deposits are formed at the place where the channels make contact with the planar electrode. This procedure deposits the catalyst only at the end of the hydrophilic channels that cross the membrane; no catalyst is placed under the hydrophobic domains, where it would not be in contact with the electrolyte. By performing a series of cyclic voltammograms with this system, we show that deposition of the platinum through the membrane achieves better platinum utilization than deposition of platinum on the naked electrode followed by the placement of the membrane on top.