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 results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial
let
Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
is considered subject to the boundary conditions
We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.
Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and
Suppose that
If 0$"> for all , then is unbounded.
Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.
This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of cannot tend to zero uniformly on an arc of the unit circle.
Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let be so large that
and |
has at least
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.