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$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and
Suppose that
If 0$"> 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.
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 bilinear Hilbert transform is given by
It satisfies estimates of the type
In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.
In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations
where the 1-periodic function is a smooth function and satisfies sublinearity:
It is also shown, via a connection between the operator and Laguerre functions, that
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.
in a separable, real Hilbert space. We prove that if generates a -semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise. 相似文献
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
Closely related and equally interesting notion is that of , which is the collection of numbers which can be represented as a sum of different elements of :
The goal of this paper is to investigate the structure of and , where is a subset of . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.
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.
In this paper we give exact formulae for for various values of . We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.
Theorem 0.1. For , we have
and