78,344 research outputs found

    Equivariant cobordism of schemes

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    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

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    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 6.9â‹…10116.9\cdot10^{11} 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

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    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

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    Consider a family of planar systems depending on two parameters (n,b)(n,b) and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when Φ(n,b)=0.\Phi(n,b)=0. We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set Φ(n,b)=0.{\Phi(n,b)=0}. 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 nn, given by b=57n1/2+72/2401n−30024/45294865n3/2−2352961656/11108339166925n2+O(n5/2)b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2}). We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions

    Recognizing Graph Theoretic Properties with Polynomial Ideals

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    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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