39 research outputs found
Optimal bounds for a colorful Tverberg--Vrecica type problem
We prove the following optimal colorful Tverberg-Vrecica type transversal
theorem: For prime r and for any k+1 colored collections of points C^l of size
|C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color
classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the
collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane
that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k)
is even or k=0.
Along the proof we obtain three results of independent interest: We present
two alternative proofs for the special case k=0 (our optimal colored Tverberg
theorem (2009)), calculate the cohomological index for joins of chessboard
complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant
bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).Comment: Substantially revised version: new notation, improved results,
additional references; 12 pages, 2 figure
Joint distribution for the Selmer ranks of the congruent number curves
summary:We determine the distribution over square-free integers of the pair , where is a curve in the congruent number curve family, is the image of isogeny , , and is the isogeny dual to
A result on the size of iterated sumsets in
In this paper we give a different approach to determining the cardinality of
-fold sumsets when has elements. This
enables us to provide more general result with a shorter and simpler proof. We
also obtain an upper bound for the value of when
is a set of elements with simplicial hull.Comment: It was noticed that the proof of the main Theorem could be improved
to give a more general result. Also some motivating examples are provide
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans