538,887 research outputs found
More Set-theory around the weak Freese-Nation property
In this paper, we introduce a very weak square principle which is even weaker
than the similar principle introduced by Foreman and Magidor. A
characterization of this principle is given in term of sequences of elementary
submodels of H(\chi). This is used in turn to prove a characterization of
kappa-Freese-Nation property under the very weak square principle and a weak
variant of the Singular Cardinals Hypothesis.
A typical application of this characterization shows that under
2^{\aleph_0}<\aleph_\omega and our very weak square for \aleph_\omega, the
partial ordering [omega_\omega]^{<\omega} (ordered by inclusion) has the
aleph_1-Freese-Nation property.
On the other hand we show that, under Chang's Conjecture for \aleph_\omega
the partial ordering above does not have the aleph_1-Freese-Nation property.
Hence we obtain the independence of our characterization of the
kappa-Freese-Nation property and also of the very weak square principle from
ZFC
On squares, outside guessing of clubs and I_{<f}[lambda]
Suppose that lambda = mu^+. We consider two aspects of the square property on
subsets of lambda. First, we have results which show e.g. that for aleph_0 <=
kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a
sufficient condition for the set of elements of lambda whose cofinality is
bounded by kappa, to be split into the union of mu sets with squares. Secondly,
we introduce a certain weak version of the square property and prove that if mu
is a strong limit, then this weak square property holds on lambda without any
additional assumption
Weak Hopf Algebras I: Integral Theory and C^*-structure
We give an introduction to the theory of weak Hopf algebras proposed recently
as a coassociative alternative of weak quasi-Hopf algebras. We follow an
axiomatic approach keeping as close as possible to the "classical" theory of
Hopf algebras. The emphasis is put on the new structure related to the presence
of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the
role of non-commutative numbers in many respects. A theory of integrals is
developed in which we show how the algebraic properties of A, such as the
Frobenius property, or semisimplicity, or innerness of the square of the
antipode, are related to the existence of non-degenerate, normalized, or Haar
integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique
Haar measure h in A and of a canonical grouplike element g in A implementing
the square of the antipode and factorizing into left and right algebra
elements. Further discussion of the C^*-case will be presented in Part II.Comment: 40 pages, LaTeX, to appear in J. Algebr
Mass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the
cost function is given by the square of the sub-Riemannian distance. Under
appropriate assumptions, we generalize Brenier-McCann's Theorem proving
existence and uniqueness of the optimal transport map. We show the absolute
continuity property of Wassertein geodesics, and we address the regularity
issue of the optimal map. In particular, we are able to show its approximate
differentiability a.e. in the Heisenberg group (and under some weak assumptions
on the measures the differentiability a.e.), which allows to write a weak form
of the Monge-Amp\`ere equation
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