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We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general G/P 's.
This book is a modern introduction to the theory of abelian varieties and theta functions. Here the Fourier transform techniques play a central role, appearing in several different contexts. In transcendental theory, the usual Fourier transform plays a major role in the representation theory of the Heisenberg group, the main building block for the theory of(More)
Forty-eight patients underwent electrical stimulation of the brain for treatment of chronic pain between 1978 and 1983. Average pain duration prior to treatment was 4.5 years. Before selection for this procedure patients underwent pain treatment in a multidisciplinary pain center, intensive psychological and psychiatric evaluation, and assessment of pain(More)
Finite arrangements of convex bodies were intensively investigated in the second half of the twentieth century. Connections were made to many other subjects, including crystallography, the local theory of Banach spaces, and combinatorial optimization. This book, the first one dedicated solely to the subject, provides an in-depth, state-of-the-art discussion(More)
The 'linear orbit' of a plane curve of degree d is its orbit in P d(d+3)/2 under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the(More)
The following elementary observation has proven useful in several enumerative geometry computations. Let X be any algebraic scheme over a eld, and let 2 K 0 (X) be an element in the Grothendieck group of vector bundles over X. Then has a well-deened rank rk , and Chern classes c k (). Also, as tensor product makes K 0 (X) a ring, we consider L], where L is(More)
About a decade ago physicists set off something of a " big bang " in the universe of algebraic geometry. A new approach to enumerative geometry, involving mirror symmetry, solved some old questions and raised many new ones. To mention just a few of the areas influenced by these developments, we have new approaches to the study of Calabi-Yau manifolds and(More)