21,161 research outputs found
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
On certain isogenies between K3 surfaces
The aim of this paper is to construct "special" isogenies between K3
surfaces, which are not Galois covers between K3 surfaces, but are obtained by
composing cyclic Galois covers, induced by quotients by symplectic
automorphisms. We determine the families of K3 surfaces for which this
construction is possible. To this purpose we will prove that there are
infinitely many big families of K3 surfaces which both admit a finite
symplectic automorphism and are (desingularizations of) quotients of other K3
surfaces by a symplectic automorphism.
In the case of involutions, for any we determine the
transcendental lattices of the K3 surfaces which are isogenous (by a
non Galois cover) to other K3 surfaces. We also study the Galois closure of the
isogenies and we describe the explicit geometry on an example.Comment: 28 page
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