59,688 research outputs found
A Functor Converting Equivariant Homology to Homotopy
In this paper, we prove an equivariant version of the classical Dold-Thom
theorem. Associated to a finite group, a CW-complex on which this group acts
and a covariant coefficient system in the sense of Bredon, we functorially
construct a topological abelian group by the coend construction. Then we prove
that the homotopy groups of this topological abelian group are naturally
isomorphic to the Bredon equivariant homology of the CW-complex. At the end we
present several examples of this result.Comment: 11 pages. Major style change. The final published versio
Polynomial Optimization with Real Varieties
We consider the optimization problem of minimizing a polynomial f(x) subject
to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence
of sum of squares relaxations for finding the global minimum. Let K be the
feasible set. We prove the following results: i) If the real variety V_R(h) is
finite, then Lasserre's hierarchy has finite convergence, no matter the complex
variety V_C(h) is finite or not. This solves an open question in Laurent's
survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite
convergence of Lasserre's hierarchy is independent of the choice of defining
polynomials for the real variety V_R(h). iii) When K is finite, a refined
version of Lasserre's hierarchy (using the preordering of g) has finite
convergence.Comment: 12 page
Entropy degeneration of convex projective surfaces
We show that the volume entropy of the Hilbert metric on a closed convex
projective surface tends to zero as the corresponding Pick differential tends
to infinity. The proof is based on the theorem, due to Benoist and Hulin, that
the Hilbert metric and Blaschke metric are comparable.Comment: 5 page
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