We show that every separable nuclear residually finite dimensional -algebras satisfying the Universal Coefficient Theorem can be embedded into a unital separable simple AF-algebra.
Assume the dimension of is greater than one. About 20 years ago the author asked the following questions:
Can this aposyndetic decomposition raise dimension? Can it lower dimension? We answer these questions by proving the following theorem.
Theorem. The dimension of the quotient space is one.
The dyadic Carleson embedding operator acting on -valued functions has norm at least . Thus the Carleson Embedding Theorem fails for Hilbert space valued functions.
Theorem 1. The following are equiconsistent:
(i) a Jónsson cardinal;
(ii) a sufficiently elementary submodel of the universe of sets with not homeomorphic to
The reverse direction is a corollary to:
Theorem 2. is Jónsson hereditarily separable, hereditarily Lindelöf, with .
We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.
The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:
1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;
2) those divisors in the Gauss system parametrized by the canonical curve are reducible.
In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:
Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.
Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.
Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.
We will say that a -ideal of subsets of satisfies the condition iff for every , if
then .
Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .
If is a surjective local homomorphism with kernel , such that and the conormal module has a free summand of rank , then the degree central subspace of the homotopy Lie algebra of has dimension greater than or equal to . This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.
where is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Liénard equations with asymmetric nonlinearities.
and its discrete counterpart , , and show that travelling wave solutions of these equations that are bounded between and are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara's Theorem (which is a Tauberian theorem for Laplace transforms).
This paper designs a set of graph operations and proves that starting from , by repeatedly applying these operations, one can construct all graphs with (for ). This can be viewed as an analogue of Hajós' Theorem for the circular chromatic number.