1,541,260 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
Vita Laudanda: Essays in Memory of Ulrich S. Leopold
Erich R. W. Schultz, Editor. Waterloo: Wilfrid Laurier University Press, 1976
Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range
We investigate how the phase diagram of a repulsive soft-core attractive potential, with a liquid-liquid phase transition in addition to the standard gas-liquid phase transition, changes by varying the parameters of the potential. We extend our previous work on short soft-core ranges to the case of large soft-core ranges, by using an integral equation approach in the hypernetted-chain approximation. We show, using a modified van der Waals equation we recently introduced, that if there is a balance between the attractive and repulsive part of the potential this potential has two fluid-fluid critical points well separated in temperature and in density. This implies that for the repulsive (attractive) energy
U
R
(
U
A
)
and the repulsive (attractive) range
w
R
(
w
A
)
the relation
U
R
∕
U
A
∝
w
R
∕
w
A
holds for short soft-core ranges, while
U
R
∕
U
A
∝
3
w
R
∕
w
A
holds for large soft-core ranges
Angle-deformations in Coxeter groups
The isomorphism problem for Coxeter groups has been reduced to its
'reflection preserving version' by B. Howlett and the second author. Thus, in
order to solve it, it suffices to determine for a given Coxeter system (W,R)
all Coxeter generating sets S of W which are contained in R^W, the set of
reflections of (W,R). In this paper, we provide a further reduction: it
suffices to determine all Coxeter generating sets S in R^W which are
sharp-angled with respect to R.Comment: 23 pages, 6 figures, submitted to AG
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