13,616 research outputs found
Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines
Solving nesting problems or irregular strip packing problems is to position
polygons in a fixed width and unlimited length strip, obeying polygon integrity
containment constraints and non-overlapping constraints, in order to minimize
the used length of the strip. To ensure non-overlapping, we used separation
lines. A straight line is a separation line if given two polygons, all vertices
of one of the polygons are on one side of the line or on the line, and all
vertices of the other polygon are on the other side of the line or on the line.
Since we are considering free rotations of the polygons and separation lines,
the mathematical model of the studied problem is nonlinear. Therefore, we use
the nonlinear programming solver IPOPT (an algorithm of interior points type),
which is part of COIN-OR. Computational tests were run using established
benchmark instances and the results were compared with the ones obtained with
other methodologies in the literature that use free rotation
The Decomposition Theorem and the topology of algebraic maps
We give a motivated introduction to the theory of perverse sheaves,
culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and
Gabber. A goal of this survey is to show how the theory develops naturally from
classical constructions used in the study of topological properties of
algebraic varieties. While most proofs are omitted, we discuss several
approaches to the Decomposition Theorem, indicate some important applications
and examples.Comment: 117 pages. New title. Major structure changes. Final version of a
survey to appear in the Bulletin of the AM
f-cohomology and motives over number rings
This paper is concerned with an interpretation of f-cohomology, a
modification of motivic cohomology of motives over number fields, in terms of
motives over number rings. Under standard assumptions on mixed motives over
finite fields, number fields and number rings, we show that the two extant
definitions of f-cohomology of mixed motives over F--one via
ramification conditions on -adic realizations, another one via the
K-theory of proper regular models--both agree with motivic cohomology of
. Here is constructed by a limiting process in
terms of intermediate extension functors defined in analogy to
perverse sheaves.Comment: numbering has been updated to agree with the published versio
Drinfeld double of quantum groups, tilting modules and -modular data associated to complex reflection groups
Generalizing Lusztig's work, Malle has associated to some imprimitive complex
reflection group a set of "unipotent characters", which are in bijection of
the usual unipotent characters of the associated finite reductive group if
is a Weyl group. He also obtained a partition of these characters into families
and associated to each family a -modular datum. We construct a
categorification of some of these data, by studying the category of tilting
modules of the Drinfeld double of the quantum enveloping algebra of the Borel
of a simple complex Lie algebra.Comment: 42 pages, some comments added on a conjecture of Cunt
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