We say a measure is @math @math -rectifiable if there is a countable union of @math @math -surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in @math to be @math @math -rectifiable, with @math . We assume a priori the measure to have positive and finite upper density. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the @math constants depend on such a function. Along the way we also give sufficient conditions for @math parametrizations for one-sided Reifenberg flat sets in terms of the same square function. Key tools for the proof come from Guy David and Tatiana Toro's Reifenberg parametrizations of sets with holes in the Holder and Lipschitz categories.
In the last few years, C. Fefferman, A. Israel, and G.K. Luli @cite_52 have been investigating Whitney type extension problems for @math maps, finding conditions to fit smooth functions to data.
@cite_52
2038964811 2147668104 2023763411 2106856555
In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center. Abstract We consider boundary value problems for semilinear hyperbolic systems of the type ∂ t u j + a j ( x , λ ) ∂ x u j + b j ( x , λ , u ) = 0 , x ∈ ( 0 , 1 ) , j = 1 , … , n , with smooth coefficient functions a j and b j such that b j ( x , λ , 0 ) = 0 for all x ∈ [ 0 , 1 ] , λ ∈ R , and j = 1 , … , n . We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on λ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov–Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used. We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 .