35,041 research outputs found
Clustering, Hamming Embedding, Generalized LSH and the Max Norm
We study the convex relaxation of clustering and hamming embedding, focusing
on the asymmetric case (co-clustering and asymmetric hamming embedding),
understanding their relationship to LSH as studied by (Charikar 2002) and to
the max-norm ball, and the differences between their symmetric and asymmetric
versions.Comment: 17 page
Sublattices of lattices of order-convex sets, I. The main representation theorem
For a partially ordered set P, we denote by Co(P) the lattice of order-convex
subsets of P. We find three new lattice identities, (S), (U), and (B), such
that the following result holds. Theorem. Let L be a lattice. Then L embeds
into some lattice of the form Co(P) iff L satisfies (S), (U), and (B).
Furthermore, if L has an embedding into some Co(P), then it has such an
embedding that preserves the existing bounds. If L is finite, then one can take
P finite, of cardinality at most , where n is the number of
join-irreducible elements of L. On the other hand, the partially ordered set P
can be chosen in such a way that there are no infinite bounded chains in P and
the undirected graph of the predecessor relation of P is a tree
On scattered convex geometries
A convex geometry is a closure space satisfying the anti-exchange axiom. For
several types of algebraic convex geometries we describe when the collection of
closed sets is order scattered, in terms of obstructions to the semilattice of
compact elements. In particular, a semilattice , that does not
appear among minimal obstructions to order-scattered algebraic modular
lattices, plays a prominent role in convex geometries case. The connection to
topological scatteredness is established in convex geometries of relatively
convex sets.Comment: 25 pages, 1 figure, submitte
Sublattices of lattices of convex subsets of vector spaces
For a left vector space V over a totally ordered division ring F, let Co(V)
denote the lattice of convex subsets of V. We prove that every lattice L can be
embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite
lower bounded, then V can be taken finite-dimensional, and L embeds into a
finite lower bounded lattice of the form ,
for some finite subset of . In particular, we obtain a new universal
class for finite lower bounded lattices
The linear isometry group of the Gurarij space is universal
We give a construction of the Gurarij space, analogous to Katetov's
construction of the Urysohn space. The adaptation of Katetov's technique uses a
generalisation of the Arens-Eells enveloping space to metric space with a
distinguished normed subspace. This allows us to give a positive answer to a
question of Uspenskij, whether the linear isometry group of the Gurarij space
is a universal Polish group
Join-semidistributive lattices of relatively convex sets
We give two sufficient conditions for the lattice Co(R^n,X) of relatively
convex sets of n-dimensional real space R^n to be join-semidistributive, where
X is a finite union of segments. We also prove that every finite lower bounded
lattice can be embedded into Co(R^n,X), for a suitable finite subset X of R^n.Comment: 11 pages, first presented on AAA-65 in Potsdam, March 200
Entire curves avoiding given sets in C^n
Let be a proper closed subset of and
at most countable (). We give conditions
of and , under which there exists a holomorphic immersion (or a proper
holomorphic embedding) with .Comment: 10 page
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