Geometric properties of the double-point divisor

The locus of double points obtained by projecting a variety to a hypersurface in moves in a linear system which is shown to be ample if and only if is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined.

     

Galois Cohomology in Degree Three and Homogeneous Varieties

The central result of this paper is the following generalization of a result of the author on products of Severi–Brauer varieties. Let G be a semi-simple linear algebraic group over a field k. Let V be a generalized flag variety under G. Then there exist finite extensions k_i of k for 1im, elements _i in Br k_i and a natural exact sequence

群定律与群簇

Let ω1 and ω 2 be two homogeneous words and suppose that they respectively define the varieties Vω 1 and Vω2 of groups. Denote by θωi the standard exponent of ωi for i =1, 2, which was introduced in Ref. [1]. We obtain that if Vω1 lohtian Vω2, then ω1θ｜θ ω2.     

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q²) and H(4,q²) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q²). We also construct in H(3,q²), q=2^2h+1, h≥ 1, maximal partial spreads of size smaller than the size q²+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q²) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008     

Subdirect products of certain varieties of unary algebras

J. Płonka in [12] noted that one could expect that the regularization ℛ(K) of a variety K of unary algebras is a subdirect product of K and the variety D of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties K which are contained in the generalized variety TDir of the so-called trap-directable algebras. This work was supported by Grant 1227 of Ministry of Science and Technology, Republic of Serbia.     

$ {\cal L} $-subvarieties of the variety of idempotent semirings

We show that certain varieties of idempotent semirings are determined by some properties of Green's relations, provide equational bases for them and give conditions guaranteeing that some Green's relations are congruences. Received November 2, 1999; accepted in final form May 2, 2000.     

Spectral and spectral-polarization characteristics of potato leaves

The results of laboratory investigations of the spectral and spectral-polarization characteristics of radiation reflected from the leaves of potato(Solanum tuberosum) of different varieties are discussed. During the vegetation season of 1997, the angular dependence of the degree and azimuth of polarization of radiation reflected from potato leaves as well as the scattering indicatrices in the range 380–1080 nm were determined by a specially developed method with the use of a laboratory goniometric setup. The relationship between the spectral polarization characteristics of radiation and biological parameters of the potato has been obtained with the help of different methods of statistical analysis and explained on the basis of the known physical mechanisms. Translated from Zhurnal Prikladnoi Spektroskopii. Vol. 67, No. 4, pp. 524–529, July–August, 2000.     

Partial ovoids and partial spreads in hermitian polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3,q ²). For q = 2,3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3,9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].      

Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Some new relations on skew Schur function differences are established both combinatorially using Schützenberger’s jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive. For Manfred Schocker 1970–2006. S.J. van Willigenburg was supported in part by the National Sciences and Engineering Research Council of Canada.