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$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem
admits at least strong solutions in .
In this paper we consider the following property:
For every function there are functions
(for ) such that
We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .
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 .
In particular, we deal with the Dirichlet boundary condition
where , 2$">, or with the following normal derivative boundary conditions:
where , 2$">, 0$"> and is the unit outward normal to the boundary .
To such a matrix and unit complex number there corresponds a signature,
Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.
If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.
To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.
where of class satisfies the natural growth
for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).
The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .
The optimal in view of this asymptotic relation lower estimate for the sums has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of that depends on the ratio of the volume to the moment of inertia of .
under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).
for and is less than one whenever either (i) or (ii) and certain assumptions on the mutual disposition of the sets and are satisfied.
for all . Here is defined by
It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.
satisfy a Strong Maximum Principle on any open connected .