1,085 research outputs found
On Derived Equivalences of Categories of Sheaves Over Finite Posets
A finite poset X carries a natural structure of a topological space. Fix a
field k, and denote by D(X) the bounded derived category of sheaves of finite
dimensional k-vector spaces over X. Two posets X and Y are said to be derived
equivalent if D(X) and D(Y) are equivalent as triangulated categories.
We give explicit combinatorial properties of a poset which are invariant
under derived equivalence, among them are the number of points, the
Z-congruency class of the incidence matrix, and the Betti numbers.
Then we construct, for any closed subset Y of X, a strongly exceptional
collection in D(X) and use it to show an equivalence between D(X) and the
bounded derived category of a finite dimensional algebra A (depending on Y). We
give conditions on X and Y under which A becomes an incidence algebra of a
poset.
We deduce that a lexicographic sum of a collection of posets along a
bipartite graph is derived equivalent to the lexicographic sum of the same
collection along the opposite graph. This construction produces many new
derived equivalences of posets and generalizes other well known ones.
As a corollary we show that the derived equivalence class of an ordinal sum
of two posets does not depend on the order of summands. We give an example that
this is not true for three summands.Comment: 20 page
Trace Spaces: an Efficient New Technique for State-Space Reduction
State-space reduction techniques, used primarily in model-checkers, all rely
on the idea that some actions are independent, hence could be taken in any
(respective) order while put in parallel, without changing the semantics. It is
thus not necessary to consider all execution paths in the interleaving
semantics of a concurrent program, but rather some equivalence classes. The
purpose of this paper is to describe a new algorithm to compute such
equivalence classes, and a representative per class, which is based on ideas
originating in algebraic topology. We introduce a geometric semantics of
concurrent languages, where programs are interpreted as directed topological
spaces, and study its properties in order to devise an algorithm for computing
dihomotopy classes of execution paths. In particular, our algorithm is able to
compute a control-flow graph for concurrent programs, possibly containing
loops, which is "as reduced as possible" in the sense that it generates traces
modulo equivalence. A preliminary implementation was achieved, showing
promising results towards efficient methods to analyze concurrent programs,
with very promising results compared to partial-order reduction techniques
Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum
In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel-Mickey and the Chan-Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as \mtx{X^\adj X}, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties
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