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1.
It is well known that the Green function of the standard discrete Laplacian on ,

exhibits a pathological behavior in dimension . In particular, the estimate

fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian,

and conjectured that the estimate

holds for all . In this paper we prove this conjecture.

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2.
We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength:

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.

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3.

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.

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4.
The Sturm-Liouville equation


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.

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5.
For let be the continued fraction expansion of . Write


We construct some numbers 's with


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6.
In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator

where and are positive constants satisfying the nonresonant condition

and is periodic in the first variable and bounded.

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7.
There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

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.

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8.
Markov's inequality is

for all polynomials . We prove a precise version of this inequality with an arbitrary continuum in the complex plane instead of the interval .

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9.
In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem.

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

satisfies . We show that any polynomial in

and     

has at least

zeros in any disk with center on the unit circle and radius .

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10.
Let be a homogeneous, decomposable continuum that is not aposyndetic. The Aposyndetic Decomposition Theorem yields a cell-like decomposition of into homogeneous continua with quotient space being an aposyndetic, homogeneous continuum.

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.

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