The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (a_ijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)²+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.     

Volume comparison and the σk-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.     

Novel antibiotics: second generation macrocyclic peptides designed to trap Holliday junctions

Described are the syntheses of 15 macrocyclic peptides designed to trap Holliday junctions (HJs) in bacteria during site-specific and homologous recombination. This leads to inhibiting bacterial growth. These second generation macrocycles were based on the C-2 symmetrical HJ. They were synthesized using a strategy that permits elucidation of the amino acid role in binding HJs. The syntheses of these macrocycles are an important step in the development of a new class of antibiotics.     

Regularity, partial elimination ideals and the canonical bundle

We present partial elimination ideals, which set-theoretically cut out the multiple point loci of a generic projection of a projective variety, as a way to bound the regularity of a variety in projective space. To do this, we utilize a combination of initial ideal methods and geometric methods. We first define partial elimination ideals and establish through initial ideal methods the way in which, for a given ideal, the regularity of the partial elimination ideals bounds the regularity of the given ideal. Then we explore the partial elimination ideals as a way to compute the canonical bundle of the generic projection of a variety and the canonical bundles of the multiple point loci of the projection, and we use Kodaira Vanishing to bound the regularity of the partial elimination ideals.

     

Surface tension and density measurements for indium and uranium using a sessile-drop apparatus with glow discharge cleaning

The sessile-drop method is used to measure the surface tension and density of liquid indium and uranium under high vacuum. Measurements are made over the temperature range 156–500°C for In and at the melting point for U. Surface oxides are efficiently removed with a glow discharge system. Drop profiles are captured by photograph and processed using nonlinear regression to yield the surface tension and density. In this regression procedure, normal distances from calculated profiles to data points are minimized. For indium, the density and surface tension measurements yield _mp = 7.05 × 10³kg/m³, d/dT = −0.776 kg/m³·°C, and γ_mp = 0.568 N/m, dγ/dT = −9.45 × 10⁻⁵ N/m·°C. The results for uranium at the melting point are _mp = 17.47 × 10³ kg/m³ and γ_mp = 1.653 N/m.     

Partition Lattice q-Analogs Related to q-Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F _q. Another member is constructed so as to mirror the partial order in the subspace lattice.     

On the stereochemistry of palmerolide A

Degradative studies of the anticancer macrolide palmerolide A have resulted in re-assignment of the C-7, C-10, and C-11 stereocenters.     

Orientation and relaxation of polymer–clay solutions studied by rheology and small-angle neutron scattering

The influence of shear on viscoelastic solutions of poly(ethylene oxide) (PEO) and clay [montmorillonite, i.e., Cloisite NA+ (CNA)] was investigated with rheology and small-angle neutron scattering (SANS). The steady-state viscosity and SANS were used to measure the shear-induced orientation and relaxation of the polymer and clay platelets. Anisotropic scattering patterns developed at much lower shear rates than in pure clay solutions. The scattering anisotropy saturated at low shear rates, and the CNA clay platelets aligned with the flow, with the surface normal parallel to the gradient direction. The cessation of shear led to partial and slow randomization of the CNA platelets, whereas extremely fast relaxation was observed for laponite (LRD) platelets. These PEO–CNA networklike solutions were compared with previously reported PEO–LRD networks, and the differences and similarities, with respect to the shear orientation, relaxation, and polymer–clay interactions, were examined. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3102–3112, 2004     

Irreducible Components of Fixed Point Subvarieties of Flag Varieties

Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P, and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T, and a Borel subgroup, B, of G, so that T?B∩L, B ? P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let ??_e be the subset of ?? consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let ??_ω be the B-orbit in ?? containing ^ωB. We consider the intersections ??_ω ∩ ??_e. The main result is that if dim ??_ω ∩ ??_e = dim ??_e, then ??_ω ∩ ??_e is an affine space. Thus, the irreducible components of ??_e are indexed by Weyl group elements. It is also shown that if G is of type A, then this set of Weyl group elements is a right cell in W.     

Orientational ordering of liquid crystals containing a difluoro-substituted phenyl ring

The orientational ordering of several liquid crystals containing a difluorosubstituted phenyl ring has been studied through the use of C-13 NMR. The fluorinated phenyl ring of these liquid crystals have C_s symmetry, so three order parameters are required to completely describe the ordering of this ring. All three of these order parameters have been calculated from carbon-fluorine dipolar coupling constants obtained from the carbon-fluorine splittings in the C-13 NMR spectra. Because of the complexity of the fluorine-coupled spectra, variable angle spinning (VAS) was used to resolve the carbon-fluorine splittings. In order to study the orientational ordering over wide ranges of temperature, we have developed an empirical correlation between the order parameter and the value of a carbon-fluorine dipolar coupling constant. This enabled us to study the change in the order parameter with temperature. The results of applying this method to several structurally similar liquid crystals containing the same type of difluorinated phenyl ring are presented. A comparison is made to a similar mono-fluorinated liquid crystal.