If $n$ and $m$ are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring $R$ with exactly $n$ elements and exactly $m$ prime ideals. Next, assuming the Axiom of Choice, it is proved that if $R$ is a commutative ring and $T$ is a commutative $R$-algebra which is generated by a set $I$, then each chain of prime ideals of $T$ lying over the same prime ideal of $R$ has at most ${{2}^{\left| I \right|}}$ elements. A polynomial ring example shows that the preceding result is best-possible.

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ${{\mathbb{R}}^{3}}$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in ${{H}^{3}}$ or ${{S}^{3}}$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action.

We consider ${{C}^{*}}$ -algebras which are inductive limits of finite direct sums of copies of $C([0,\,1])\otimes {{\mathcal{O}}_{2}}$ . For such algebras, the lattice of closed two-sided ideals is proved to be a complete invariant.

The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains that pinch at an exponential rate, weaker, ${{L}^{2}}$ bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity.

We present a new, simple proof of the classical Abel-Jacobi theorem using some elementary gauge theoretic arguments.

A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

Suppose $m$ and $n$ are integers such that $1\,\le \,m\,\le \,n$. For a subgroup $H$ of the symmetric group ${{S}_{m}}$ of degree $m$, consider the generalized matrix function on $m\,\times \,m$ matrices $B\,=\,\left( {{b}_{ij}} \right)$ defined by ${{d}^{H}}\left( B \right)\,=\,\sum{_{\sigma \in H}\,\prod{_{j=1}^{m}}\,{{b}_{j\sigma \left( j \right)}}}$ and the generalized numerical range of an $n\,\times \,n$ complex matrix $A$ associated with ${{d}^{H}}$ defined by

$${{W}^{H}}\left( A \right)\,=\,\left\{ {{d}^{H}}\left( {{X}^{*}}\,AX \right)\,:\,X\,is\,n\,\times \,m\,\text{such}\,\text{that}\,{{X}^{*}}X\,\text{=}\,{{\text{I}}_{m}} \right\}$$

It is known that ${{W}^{H}}\left( A \right)$ is convex if $m\,=\,1$ or if $m\,=\,n\,=\,2$. We show that there exist normal matrices $A$ for which ${{W}^{H}}\left( A \right)$ is not convex if $3\,\le \,m\,\le \,n$. Moreover, for $m\,=\,2\,<\,n$, we prove that a normal matrix $A$ with eigenvalues lying on a straight line has convex ${{W}^{H}}\left( A \right)$ if and only if $vA$ is Hermitian for some nonzero $v\,\in \,\mathbb{C}$. These results extend those of Hu, Hurley and Tam, who studied the special case when $2\,\le \,m\,\le \,3\,\le \,n$ and $H\,=\,{{S}_{m}}$.

If $A$ is a prime ${{\text{C}}^{*}}$-algebra, $a\,\in \,A$ and $\lambda $ is in the numerical range $W\left( a \right)$ of $a$, then for each $\varepsilon \,>\,0$ there exists an element $h\,\in \,A$ such that $\left\| h \right\|\,=\,1$ and $\left\| {{h}^{*}}(a-\lambda )h \right\|\,<\,\varepsilon $. If $\lambda $ is an extreme point of $W\left( a \right)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general ${{\text{C}}^{*}}$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the ${{\text{C}}^{*}}$-convex hull of $a$ are characterized.

We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on ${{L}^{2}}$ if the Cantor set has positive Hausdorff dimension.

Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U\,=\,G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

We prove that the ${{L}^{2}}$ restriction theorem, and ${{L}^{p}}\,\to \,{{L}^{{{p}'}}}\,,\,\frac{1}{p}\,+\,\frac{1}{{{p}'}}\,=\,1$, boundedness of the surface averages imply certain geometric restrictions on the underlying hypersurface. We deduce that these bounds imply that a certain number of principal curvatures do not vanish.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\mathbb{C}$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

The results herein continue observations on norm form equations and continued fractions begun and continued in the works [1]−[3], and [5]−[6].

This paper contains a geometric proof of an estimate for a restricted x-ray transform. The result complements one of A. Greenleaf and A. Seeger.

The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type I and Type II spectral flow.

Let $\mathbb{T}$ denote the unit circle in the complex plane, and let $X$ be a Banach space that satisfies Burkholder’s $\text{UMD}$ condition. Fix a natural number, $N\,\in \,\mathbb{N}$ . Let $\mathcal{P}$ denote the reverse lexicographical order on ${{\mathbb{Z}}^{N}}$ . For each $f\,\in \,{{L}^{1}}({{\mathbb{T}}^{N}},X)$ , there exists a strongly measurable function $\tilde{f}$ such that formally, for all $\mathbf{n}\,\in \,{{\mathbb{Z}}^{N}},\,\hat{\tilde{f}}\,\left( \mathbf{n} \right)\,=\,-i\,{{sgn }_{\mathcal{P}}}\left( \mathbf{n} \right)\hat{f}\left( \mathbf{n} \right)$ . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of $\text{UMD}$ spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension $N$.