Gradient estimates and blow-up analysis for stationary harmonic maps

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary maps are m \leq n - 4 rectifiable. Both of these results follow from a general analysis on the defect measures and energy concentration sets associated with a weakly converging sequence of stationary harmonic maps.Comment: 45 pages, published versio

Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows

We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke’s flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold

Numerical algorithm and complexity analysis for diagonalization of multivariate homogeneous polynomials

We study the computational complexity of a criterion and an algorithm for diagonalization of multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. They are based on Harrison's center theory and only require solving linear and quadratic systems of equations. Detailed descriptions and computational complexity of each step of the algorithm are provided. The complexity analysis focuses on the impacts of problem sizes, including the number of variables and the degree of given polynomials. We show that this algorithm runs in polynomial time and validate it through numerical experiments. Other diagonalization algorithms are reviewed and compared in terms of complexity

2-Amino-4,6-dimethyl­pyrimidinium chloro­acetate

There are two cations and two anions in the asymmetric unit of the title compound, C6H10N3 +·C2H2ClO2 −. In the crystal, the components are linked by inter­molecular N—H⋯O and N—H⋯N hydrogen bonds to form a two-dimensional network. Additional stabilization is provided by weak inter­molecular C—H⋯O inter­actions