解拟变分不等式的超平面投影算法

拟变分不等式问题是变分不等式问题的一种推广,超平面投影算法是解变分不等式的一种重要方法.通过构造严格分离当前点与拟变分不等式解集的超平面,建立了解拟变分不等式的超平面投影算法.在一定的条件下,证明了该算法的全局收敛性.     

Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over _q. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over _q of the uniform matroid or, alternately, the number of _q-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.     

Tomographie Des Varietes Singulieres Et Theoremes De Lefschetz

Nous étudions l'homotopie d'une variétéquasi-projective dans un espace projectif complexe selon laméthode de Lefschetz, c'est-à-dire en considérantses sections par les hyperplans d'un pinceau (tomographie).En particulier, nous aboutissons à un théorèmedu type de Lefschetz qui généralise dans une certainedirection les meilleurs résultats connus dus àHamm, Lê, Goresky et MacPherson. Ce théorèmeest démontré par récurrence sur la dimensionde l'espace projectif ambiant à partir d'un théorèmesur les pinceaux d'axe générique qui constituele résultat principal de l'article. Ce dernier comparela topologie de la variété à celle de sasection par un hyperplan générique du pinceausur la base des comparaisons (section hyperplane générique– section par l'axe du pinceau) et (sections hyperplanesexceptionnelles – section par l'axe); l'incidence dessingularités est mesurée par un invariant appelé‘profondeur homotopique rectifiée globale’(analogue global de la notion de profondeur homotopique rectifiéede Grothendieck). We study the homotopy of a quasi-projectivevariety in a complex projective space following Lefschetz'smethod, that is, by considering its sections by the hyperplanesof a pencil (tomography). Specifically, we obtain a theoremof Lefschetz type which generalizes in a certain direction thebest-known results due to Hamm, Lê, Goresky and MacPherson.This theorem is proved by induction on the dimension of theambient projective space with the help of a theorem on pencilswith generic axis which is the main result of the paper. Thelatter compares the topology of the variety with that of itssection by a generic hyperplane of the pencil, on the basisof the following comparisons: section by a generic hyperplanewith section by the axis of the pencil; and sections by theexceptional hyperplanes with section by the axis. The effectof the singularities is measured by an invariant called ‘globalrectified homotopical depth’ (a global analogue of thenotion of rectified homotopical depth of Grothendieck). E-mail: eyral{at}cmi.univ-mrs.fr 2000 Mathematics Subject Classification:32S50, 14F35, 14F17.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

Varieties with a reducible hyperplane section whose two components are hypersurfaces

We classify smooth complex projective varieties of dimension admitting a divisor of the form among their hyperplane sections, both and of codimension in their respective linear spans. In this setting, one of the following holds: 1) is either the Veronese surface in or its general projection to , 2) and is contained in a quadric cone of rank or , 3) and .

     

The Valuations of the Near Octagon {\mathbb{H}}_4

Let , n  ≥ 2, be the near 2n-gon on the 2-factors of a complete graph with 2n + 2 vertices. In this paper, we classify the valuations of the near octagon . We use this classification to study isometric full embeddings of into DQ(8,2) and DH(7,4). We show that there is up to isomorphism a unique isometric full embedding of into each of these dual polar spaces. Further applications are expected in the classification of dense near polygons with lines of size 3.     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

Intrinsic normalizations of a hyperplane distribution on the Grassmann manifold. I

We consider intrinsic normalizations of a hyperplane distribution on the Grassmann manifold of ann-dimensional projective space. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 257–273, April–June, 1999. Translated by V. Mackevičius     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.