Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Graphs and Hermitian matrices: Exact interlacing

We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest eigenvalue of a graph.     

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

     

On the Existence and Construction of Tight Fourth Order Quadrature Rules for the Sphere

We study the so-called tight quadrature rules for polynomials of degree 4 on the unit sphere S ^D-1 and present precise formulae for the first 6 components of the nodes in terms of the parameter u := . In particular, we reobtain the well-known necessary condition for the existence of such rules saying that u has to be an odd integer and we sharpen it under an additional assumption.As a constructive application, two explicit tight fourth order quadrature rules for the case D = 7 are given.     

Engineering bands of extended electronic states in a class of topologically disordered and quasiperiodic lattices

We show that a discrete tight-binding model representing either a random or a quasiperiodic array of bonds can have the entire energy spectrum or a substantial part of it absolutely continuous, populated by extended eigenfunctions only, when atomic sites are coupled to the lattice locally, or non-locally from one side. The event can be fine-tuned by controlling only the host–adatom coupling in one case, while in two other cases cited here an additional external magnetic field is necessary. The delocalization of electronic states for the group of systems presented here is sensitive to a subtle correlation between the numerical values of the Hamiltonian parameters – a fact that is not common in the conventional cases of Anderson localization. Our results are analytically exact, and supported by numerical evaluation of the density of states and electronic transmission coefficient.     

Trimming of metric spaces and the tight span

We use the trimming transformations to study the tight span of a metric space.     

High current dosimetry techniques

A review of techniques for electrical measurement of high current ion beams is presented. A computer analysis of the secondary-particle current-collection schemes is made for dosimetry-cup configurations commonly used on ion implanters with electrostatic scanning. The design of a dosimetry cup for a high current implanter with mechanical scanning is discussed. Also beam monitoring techniques for uniformity measurements are reviewed.     

Detection of cellular damage by hydrogen peroxide using SV40-T2 cells on shear horizontal surface acoustic wave (SH-SAW) sensor

The rat lung epithelial cell line SV40-T2 was used to develop a cellular biosensing system to assay for environmental toxicants. The novel approach on which this system is based involves direct attachment of cultured rat or human cells onto a cell-adhesive matrix on the device through which shear horizontal surface acoustic waves (SH-SAW) are transmitted using 50 MHz SAW resonator. This novel design enables sensitive monitoring of changes of the electrophysical characteristics of cells, such as their conductivity and relative permittivity. A time-dependent change of phase of SAW and change of insertion loss (change of amplitude) were observed when the cells were treated with 0.5 or 1.0 mM H₂O₂. The change of insertion loss was biphasic, with an early phase (1–3 h) and a late phase (3–6 h). The late phase coincided with the destruction of cell–cell tight junctions detected by measurement of the transepithelial electrical resistance and paracellular permeability; in contrast, the early phase coincided with the destruction of intracellular actin filaments by H₂O₂. The early-phase effect of H₂O₂ on phase shift may be attributable to the change of intracellular permittivity by a change of cellular polarity. Immunofluorescence microscopy showed the disappearance of zonula occludens protein 1 from the region of cell–cell contact. These results suggest the correlation between the change of insertion loss as an SAW parameter and the destruction of tight junctions of the cells on the SH-SAW device in the late phase.     

A Note on “Lot-sizing with fixed charges on stocks: The convex hull”

We update and complete the proof of Proposition 7 in Van Vyve and Ortega (2004) [1], which states that the projection of a facility location reformulation of an uncapacitated lot-sizing problem with fixed charges on stocks (ULSW) to the original space is equivalent to that of the tight shortest path reformulation of ULSW. Their proof is interesting and consists of two cases, only first of which is analyzed in detail. We show that the second case exhibits several challenges not present in the first one and necessitates an updated proof.