We show that the manifold resulting from $r$-surgery on the knot ${{5}_{2}}$, which is the two-bridge knot corresponding to the rational number $3/7$, has a left-orderable fundamental group if the slope $r$ satisfies $0\,\le \,r\,\le \,4$.

We give conditions under which a vector valued Ornstein Uhlenbeck process has continuous sample paths in lp for 1 ≦ p < ∞. We also show when the space lp is not entered at all, i.e., when it has zero capacity.

E. Cohen [1], [2] has considered an analogue of the famous Goldbach problem in certain finite rings. In the following another analogue is considered in the ring Jn of residue classes over the integers modulo n.

As 'primes' in Jn we take all residue classes coprime to n. That is, all elements p ϵ Jn which are not factors of zero in Jn. We denote this set by Rn.

The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

L. Moser [3] recently gave a very simple proof that

1.

has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that

2.

has no solutions. It is easy to see on the other hand that

3.

has infinitely many non-trivial solutions.

Let us consider the well-known Hermite-Fejér interpolating process on the interval [—1,1] i.e. let

(1.1)

(sometimes we omit the second indices),

(1.2)

Milnor showed that non-singular bilinear products on Rn exist only if n = 1, 2, 4, 8 using topological methods. In this note, we give a proof of this result by purely analytical methods.

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

In this paper we examine the examples of isospectral but non-isometric Riemannian manifolds given by Milnor, Ikeda, and Vignéras. Of these, only Milnor's example is accounted for by Sunada's method of constructing isospectral manifolds, and even then only as an "unnatural" construction.

The General Selection Principle referred to in the title is really Tychonoff's theorem on products of compact spaces, but in a somewhat disguised form (c.f. Theorem 2.1, below). It is believed that this form is one which lends itself very well to many applications. More specifically, one of the immediate corollaries of the main theorem is a theorem due to Rado (c.f. Cor 2.1.1.), which has been used by Erdos and de Bruijn [2], Luxemburg [7], and the author [9] to give simple proofs of a variety of results.

Recent papers have shown that ${{C}^{1}}$ maps $F:\,{{\mathbb{R}}^{2}}\,\to {{\mathbb{R}}^{2}}$ whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or $F$ is a polynomial. Specifically, $F\,=\,(u,\,v)$ must take the form

$$u\,=\,ax\,+\,by\,+\,\beta \phi (\alpha x\,+\,\beta y)\,+\,e$$

$$v\,=\,cx\,+\,dy\,-\,\alpha \phi \,(\alpha x\,+\,\beta y)\,+\,f$$

for some constants $a,\,b,\,c,\,d,\,e,\,f,\,\alpha ,\,\beta $ and a ${{C}^{1}}$ function $\phi $ in one variable. If, in addition, the function $\phi $ is not affine, then

1

$$\alpha \beta (d\,-\,a)\,+\,b{{\alpha }^{2}}\,-\,c{{\beta }^{2}}\,=\,0.$$

This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.

It is shown that if a function in the Dirichlet space is inveritible then it is cyclic with respect to the operator of multiplication by the identity function.

We prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate $p$-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the $p$-Laplacian.

We show that, for most of the elliptic curves $\text{E}$ over a prime finite field ${{\mathbb{F}}_{p}}$ of $p$ elements, the discriminant $D\left( E \right)$ of the quadratic number field containing the endomorphism ring of $\text{E}$ over ${{\mathbb{F}}_{p}}$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over ${{\mathbb{F}}_{p}}$.

In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner– Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

Beineke and Harary gave an example of a family of tournaments Tn such that every subtournament of Tn is irreducible or transitive. We characterize all tournaments with this property.

We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated with non-ordinary modular forms.

Since the solution of Post' s Problem by Friedberg and Muchnik, a good deal of abstract knowledge about the semilattice of recursively enumerable degrees has been developed, especially in recent papers of G. E. Sacks. In this note, we show that one specific class of simple sets contains only sets of degree 0'; no contribution to the general theory is claimed. One of the sets belonging to the special class considered is the original simpie-but-not-hypersimple S of Post [1]. According to information received from Sacks, J. R. Myhill gave a proof, in 1953, of the completeness of S. However, as far as we know, Myhill' s proof (presented in a seminar) does not exist in published form.

We show that, if X and Y are Banach spaces such that X has the Mazur's intersection property and such that there exists T, an operator from Y into X so that T* and T** are injective, then there exists on Y an equivalent norm which has the Mazur's intersection property.

We deduce from this result and from a result of M. Talagrand that there exists on the long James space J(η) an equivalent norm which has the Mazur's intersection property.

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x)\approx m(x,y,x)\approx m(x,x,y)\approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.