Let $X,\,Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\,\times \,Y\,\cong \,X\,\times \,W$ . Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\,\times \,{{X}^{\sigma }}\,\cong \,W\,\times \,{{W}^{\sigma }}$ (where ${{A}^{\sigma }}$ denotes the complex conjugate of any variety $A$ ) is at most countable.

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.

We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$ . We also introduce a natural partial order on $\mathfrak{g}:\,x\,\le y$ if $f(K\,\cdot \,x)\,\subseteq \,f(K\,\cdot \,y)$ for all $f\,\in \,{{\mathfrak{g}}^{*}}$ , the complex dual of $\mathfrak{g}$ . This partial order is $K$ -invariant and induces a partial order on the orbit space $\mathfrak{g}/K$ . We prove that, under some restrictions on $\mathfrak{g}$ , the set $f(K\,\cdot \,x)$ is star-shaped with respect to the origin.

Let $q\,=\,{{p}^{r}}$ with $p$ an odd prime, and ${{\mathbf{F}}_{q}}$ denote the finite field of $q$ elements. Let $\text{Tr}\,:\,{{\mathbf{F}}_{q}}\,\to \,{{\mathbf{F}}_{p}}$ be the usual trace map and set ${{\zeta }_{p}}\,=\,\exp (2\pi i/p)$ . For any positive integer $e$ , define the (modified) Gauss sum ${{g}_{r}}(e)$ by

$${{g}_{r}}(e)=\underset{x\in {{\mathbf{F}}_{q}}}{\mathop{\sum }}\,\zeta _{p}^{\text{Tr}{{x}^{e}}}$$

Recently, Evans gave an elegant determination of ${{g}_{1}}(12)$ in terms of ${{g}_{1}}(3),\,{{g}_{1}}(4)$ and ${{g}_{1}}(6)$ which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum ${{g}_{r}}(12)$ .

We give an algorithm for a surgery description of a $p$ -fold cyclic branched cover of ${{B}^{3}}$ branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple ${{C}^{*}}$ -algebras. The first result of this kind was shown by Powers for the $\text{UHF}$ algbras some 30 years ago.

We study the farthest-point distance function, which measures the distance from $z\,\in \,\mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane.

The logarithm of this distance is subharmonic as a function of $z$ , and equals the logarithmic potential of a unique probability measure with unbounded support. This measure ${{\sigma }_{E}}$ has many interesting properties that reflect the topology and geometry of the compact set $E$ . We prove ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$ -gon for some odd $n$ . Also we show ${{\sigma }_{E}}(E)\,=\,\frac{1}{2}$ for smooth convex sets of constant width. We conjecture ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for all $E$ .

It is known that a unital simple ${{C}^{*}}$ -algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital ${{C}^{*}}$ -algebras with tracial topological rank zero that have real rank other than zero.

Let $0\,\to \,J\,\to \,E\,\to A\,\to \,0$ be a short exact sequence of ${{C}^{*}}$ -algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a ${{C}^{*}}$ -algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$ -space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

We determine when an automorphism of a subfactor of type $\text{II}{{\text{I}}_{0}}$ with finite index is nonstrongly free in the sense of $\text{C}$ . Winsløw in terms of the modular endomorphisms introduced by M. Izumi.

On a smooth $n$ -dimensional complete variety $X$ over $\mathbb{C}$ we show that the trace map ${{\bar{\theta }}_{X}}\,:\,{{H}^{n}}(X,\,\Omega _{X}^{n})\,\to \,\mathbb{C}$ arising from Lipman's version of Grothendieck duality in $\left[ \text{L} \right]$ agrees with

$${{(2\pi i)}^{-n}}{{(-1)}^{n(n-1)/2}}\,\int_{X}{:\,H_{DR}^{2n}\,(X,\,\mathbb{C})\,\to \,\mathbb{C}}$$

under the Dolbeault isomorphism.

Let ${{A}_{\theta }}$ be the universal ${{C}^{*}}$ -algebra generated by two unitaries $U,\,V$ satisfying $VU\,=\,{{e}^{2\pi i\theta }}UV$ and let $\Phi $ be the antiautomorphism of ${{A}_{\theta }}$ interchanging $U$ and $V$ . The $K$ -theory of ${{R}_{\theta }}\,=\,\{a\,\in \,{{A}_{\theta }}\,:\,\Phi (a)\,=\,{{a}^{*}}\}$ is computed. When $\theta $ is irrational, an inductive limit of algebras of the form ${{M}_{q}}(C(\mathbb{T}))\,\oplus \,{{M}_{{{q}'}}}(\mathbb{R})\,\oplus \,{{M}_{q}}(\mathbb{R})$ is constructed which has complexification ${{A}_{\theta }}$ and the same $K$ -theory as ${{R}_{\theta }}$ .

In the sequel we construct simple, unital, separable, stable, amenable ${{C}^{*}}$ -algebras for which the ordered ${{K}_{0}}$ -group is strongly perforated and group isomorphic to $Z$ . The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised.

A theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.