A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

On spherical twisted conjugacy classes

Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let theta be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical theta-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. We generalize to good odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy classes.Comment: proof of Lemma 6.4 polished. The journal version is available at http://www.springerlink.com/content/k573l88256753640

Roots of the affine Cremona group

Let k[x_1,...,x_n] be the polynomial algebra in n variables and let A^n=Spec k[x_1,...,x_n]. In this note we show that the root vectors of the affine Cremona group Aut(A^n) with respect to the diagonal torus are exactly the locally nilpotent derivations x^a\times d/dx_i, where x^a is any monomial not depending on x_i. This answers a question due to Popov.Comment: 4 page

Schubert varieties and generalizations

This contribution reviews the main results on Schubert varieties and their generalizations It covers more or less the material of the lectures at the Seminar These were partly expository introducing material needed by other lecturers In particular Section reviews classical material used in several of the other contribution

A Difference Version of Nori's Theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).Comment: 29 page

A predictive phenomenological tool at small Bjorken-x

We present the results from global fits of inclusive DIS experimental data using the Balitsky-Kovchegov equation with running coupling.Comment: 5 pages, 2 figures, prepared for the Proceedings of 'Hot Quarks 2010

Formal properties in small codimension

In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces

Multivariate Poincar\'e series for algebras of SL2SL_2-invariants

Let \mathcal{C}_{\mathbi{d}}, \mathcal{I}_{\mathbi{d}}, \mathbi{d}{=}(d_1,d_2,..., d_n) be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,..., d_n.$ Formulas for computation of the multivariate Poincar\'e series \mathcal{P}(\mathcal{C}_{\mathbi{d}},z_1,z_2,...,z_n,t) and \mathcal{P}(\mathcal{I}_{\mathbi{d}},z_1,z_2,...,z_n) are found.Comment: 5 page

PDEPT: polymer-directed enzyme prodrug therapy

Polymer-directed enzyme prodrug therapy (PDEPT) is a novel two-step antitumour approach using a combination of a polymeric prodrug and polymer-enzyme conjugate to generate cytotoxic drug selectively at the tumour site. In this study the polymeric prodrug N-(2-hydroxypropyl) methacrylamide (HPMA) copolymer-Gly-Phe-Leu-Gly-doxorubicin conjugate PK1 (currently under Phase II clinical evaluation) was selected as the model prodrug, and HPMA copolymer-cathepsin B as a model for the activating enzyme conjugate. Following polymer conjugation (yield of 30–35%) HPMA copolymer-cathepsin B retained ~20–25% enzymatic activity in vitro. To investigate pharmacokinetics in vivo,125I-labelled HPMA copolymer-cathepsin B was administered intravenously (i.v.) to B16F10 tumour-bearing mice. HPMA copolymer-cathespin B exhibited a longer plasma half-life (free cathepsin B t1/2α= 2.8 h; bound cathepsin B t1/2α= 3.2 h) and a 4.2-fold increase in tumour accumulation compared to the free enzyme. When PK1 (10 mg kg−1dox-equiv.) was injected i.v. into C57 mice bearing subcutaneously (s.c.) palpable B16F10 tumours followed after 5 h by HPMA copolymer-cathepsin B there was a rapid increase in the rate of dox release within the tumour (3.6-fold increase in the AUC compared to that seen for PK1 alone). When PK1 and the PDEPT combination were used to treat established B16F10 melanoma tumour (single dose; 10 mg kg−1dox-equiv.), the antitumour activity (T/C%) seen for the combination PDEPT was 168% compared to 152% seen for PK1 alone, and 144% for free dox. Also, the PDEPT combination showed activity against a COR-L23 xenograft whereas PK1 did not. PDEPT has certain advantages compared to ADEPT and GDEPT. The relatively short plasma residence time of the polymeric prodrug allows subsequent administration of polymer-enzyme without fear of prodrug activation in the circulation and polymer-enzyme conjugates have reduced immunogenicity. This study proves the concept of PDEPT and further optimisation is warranted. © 2001 Cancer Research Campaign   http://www.bjcancer.co