78,344 research outputs found
Equivariant cobordism of schemes
We study the equivariant cobordism theory of schemes for action of linear
algebraic groups. We compare the equivariant cobordism theory for the action of
a linear algebraic groups with similar groups for the action of tori and deduce
some consequences for the cycle class map of the classifying space of an
algebraic groups.Comment: This revised version supercedes arxiv:1006:317
Adaptive control in rollforward recovery for extreme scale multigrid
With the increasing number of compute components, failures in future
exa-scale computer systems are expected to become more frequent. This motivates
the study of novel resilience techniques. Here, we extend a recently proposed
algorithm-based recovery method for multigrid iterations by introducing an
adaptive control. After a fault, the healthy part of the system continues the
iterative solution process, while the solution in the faulty domain is
re-constructed by an asynchronous on-line recovery. The computations in both
the faulty and healthy subdomains must be coordinated in a sensitive way, in
particular, both under and over-solving must be avoided. Both of these waste
computational resources and will therefore increase the overall
time-to-solution. To control the local recovery and guarantee an optimal
re-coupling, we introduce a stopping criterion based on a mathematical error
estimator. It involves hierarchical weighted sums of residuals within the
context of uniformly refined meshes and is well-suited in the context of
parallel high-performance computing. The re-coupling process is steered by
local contributions of the error estimator. We propose and compare two criteria
which differ in their weights. Failure scenarios when solving up to
unknowns on more than 245\,766 parallel processes will be
reported on a state-of-the-art peta-scale supercomputer demonstrating the
robustness of the method
Enumerative geometry for real varieties
We discuss the problem of whether a given problem in enumerative geometry can
have all of its solutions be real. In particular, we describe an approach to
problems of this type, and show how this can be used to show some enumerative
problems involving the Schubert calculus on Grassmannians may have all of their
solutions be real. We conclude by describing the work of Fulton and
Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane
conics such that each of the 3264 conics tangent to all five are real.Comment: Based upon the Author's talk at 1995 AMS Summer Research Institute in
Algebraic geometry. To appear in the Proceedings. 11 pages, extended version
with Postscript figures and appendix available at
http://www.msri.org/members/bio/sottile.html, or by request from Author
([email protected]
Some results on homoclinic and heteroclinic connections in planar systems
Consider a family of planar systems depending on two parameters and
having at most one limit cycle. Assume that the limit cycle disappears at some
homoclinic (or heteroclinic) connection when We present a method
that allows to obtain a sequence of explicit algebraic lower and upper bounds
for the bifurcation set The method is applied to two quadratic
families, one of them is the well-known Bogdanov-Takens system. One of the
results that we obtain for this system is the bifurcation curve for small
values of , given by . We obtain
the new three terms from purely algebraic calculations, without evaluating
Melnikov functions
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
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