A in the unit @math -cube is a set @math such that for every @math and @math in @math we either have @math for all @math , or @math for all @math . We consider subsets, @math , of the unit @math -cube @math that satisfy [ card (A C) k, , for all chains , C [0,1]^n , , ] where @math is a fixed positive integer. We refer to such a set @math as a @math -antichain. We show that the @math -dimensional Hausdorff measure of a @math -antichain in @math is at most @math and that the bound is asymptotically sharp. Moreover, we conjecture that there exist @math -antichains in @math whose @math -dimensional Hausdorff measure equals @math and we verify the validity of this conjecture when @math .
When @math this conjecture is clearly true, and when @math it is observed in @cite_0 that the validity of Conjecture is an immediate consequence of the following, well-known, result. Recall that a singular function @math is a strictly decreasing function whose derivative equals zero almost everywhere.
@cite_0
2110785107 2067941796 2119973367 2953316345
We consider the functions @math defined as the @math th partial derivative of Lebesgue's singular function @math with respect to @math at @math . This sequence includes a multiple of the Takagi function as the case @math . We show that @math is continuous but nowhere differentiable for each @math , and determine the Holder order of @math . From this, we derive that the Hausdorff dimension of the graph of @math is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for @math using the binary expansion of @math , and use this to find the sets of points where @math and @math take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of @math and @math . We consider the behavior of the GMRES method for solving a linear system @math when @math is singular or nearly so, i.e., ill conditioned. The (near) singularity of @math may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular @math , we give conditions under which the GMRES iterates converge safely to a least-squares solution or to the pseudoinverse solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to GMRES. A practical procedure is outlined for efficiently and reliably detecting singularity or ill conditioning when it becomes a threat to the performance of GMRES. We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota. In this paper, we deduce the vanishing of Selmer groups for the Rankin-Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated @math -value, thus establishing the rank 0 case of the Bloch-Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and @math -adic @math -functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin's method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur's conjecture (as proven in weight 2 by Cornut-Vatsal), and as a consequence of our results, we deduce from Nekovar's work a proof of the parity conjecture in this setting.