74,565 research outputs found
An extremal problem on crossing vectors
For positive integers and , two vectors and from
are called -crossing if there are two coordinates and
such that and . What is the maximum size of
a family of pairwise -crossing and pairwise non--crossing vectors in
? We state a conjecture that the answer is . We prove
the conjecture for and provide weaker upper bounds for .
Also, for all and , we construct several quite different examples of
families of desired size . This research is motivated by a natural
question concerning the width of the lattice of maximum antichains of a
partially ordered set.Comment: Corrections and improvement
On an extremal problem connected with simplices
In this note we investigate the behavior of the volume that the convex hull
of two congruent and intersecting simplices in Euclidean -space can have. We
prove some useful equalities and inequalities on this volume. For the regular
simplex we determine the maximal possible volume for the case when the two
simplices are related to each other via reflection at a hyperplane intersecting
them.Comment: 11 page
On an extremal problem for poset dimension
Let be the largest integer such that every poset on elements has a
-dimensional subposet on elements. What is the asymptotics of ?
It is easy to see that . We improve the best known upper
bound and show . For higher dimensions, we show
, where is the largest
integer such that every poset on elements has a -dimensional subposet on
elements.Comment: removed proof of Theorem 3 duplicating previous work; fixed typos and
reference
An extremal problem for integer sparse recovery
Motivated by the problem of integer sparse recovery we study the following
question. Let be an integer matrix whose entries are in
absolute value at most . How large can be if all
submatrices of are non-degenerate? We obtain new upper and lower bounds on
and answer a special case of the problem by Brass, Moser and Pach on
covering -dimensional grid by linear subspaces
On an extremal problem for nonoverlapping domains *
The paper considers the problem of finding the range of functional I = J f (z
0), f (z 0), F ( 0), F ( 0) , defined on the class M of pairs
functions (f (z), F ()) that are univalent in the system of the disk and
the interior of the disk, using the method of internal variations. We establish
that the range of this functional is bounded by the curve whose equation is
written in terms of elliptic integrals, depending on the parameters of the
functional I
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