Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial.

We define generalized Mellin transforms of mixed cusp forms, show their convergence, and prove that the function obtained by such a Mellin transform of a mixed cusp form satisfies a certain functional equation. We also prove that a mixed cusp form can be identified with a holomorphic form of the highest degree on an elliptic variety.

We construct two non-isomorphic nuclear, stably finite, real rank zero ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ for which there is an isomorphism of ordered groups $\Theta :{{\oplus }_{n\ge 0}}{{K}_{\bullet }}(E;\mathbb{Z}/n)\to {{\oplus }_{n\ge 0}}{{K}_{\bullet }}({E}';\mathbb{Z}/n)$ which is compatible with all the coefficient transformations. The ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz’s algebra ${{\mathcal{O}}_{\infty }}$ one obtains a pair of non-isomorphic, real rank zero, purely infinite ${{C}^{*}}$-algebras with similar properties.

In this paper we prove: for any positive integers $a$ and $q$ with $\left( a,\,q \right)\,=\,1$, we have uniformly

$$\sum\limits_{\begin{matrix} n\le N \\ (n,q)=1,n\bar{n}\equiv 1(\,\bmod \,q) \\ \end{matrix}}{\mu (n)e(\frac{a\bar{n}}{q})\ll Nd(q)\left\{ \frac{{{\log }^{\frac{5}{2}}}N}{{{q}^{\frac{1}{2}}}}+\frac{{{q}^{\frac{1}{5}}}{{\log }^{\frac{13}{5}}}N}{{{N}^{\frac{1}{5}}}} \right\}.}$$

This improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].

We give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of the Wheeler-Shakmatov theorem, which states that if $X$ is normal and $\dim\left( X \right)\,\le \,1$, then every quasi-measure on $X$ extends to a measure.

We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

We give a “wall-crossing” formula for computing the topology of the moduli space of a closed $n$-gon linkage on ${{\mathbb{S}}^{2}}$. We do this by determining the Morse theory of the function ${{\rho }_{n}}$ on the moduli space of $n$-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first $\left( n\,-\,1 \right)$ side-lengths are fixed. We obtain a Morse function on the $\left( n\,-\,2 \right)$-torus with level sets moduli spaces of $n$-gon linkages. The critical points of ${{\rho }_{n}}$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ${{\rho }_{n}}$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section 2 the equivalence between Shimogaki’s theorem and some martingale inequalities will be established, and in Section 3 the equivalence between Boyd’s theorem and martingale inequalities with change of probability measure will be established.

We derive a necessary and sufficient condition for $\text{HNN}$-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of $\text{HNN}$-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of $\text{HNN}$-extensions of nilpotent groups with cyclic associated subgroups.

We define embedding of an $n$-dimensional normed space into ${{L}_{-p}},\,0\,<\,p\,<\,n$ by extending analytically with respect to $p$ the corresponding property of the classical ${{L}_{p}}$-spaces. The well-known connection between embeddings into ${{L}_{p}}$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in ${{L}_{-p}}$ if and only if $\parallel x{{\parallel }^{-p}}$ is a positive definite distribution. We show that the technique of embedding in ${{L}_{-p}}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $P\in [n-3,n),\mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{X}_{i}} \right|}^{-p}})\ge \mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{Y}_{i}} \right|}^{-p}})$, where ${{X}_{1}},...,{{X}_{n}}\,\text{and}\,{{Y}_{1}},...,{{Y}_{n}}$ are jointly $q$-stable symmetric random variables, $0\,<\,q\,\le \,2$, so that, for some $k\,\in \,\mathbb{N},\,1\,\le \,k\,<\,n$, the vectors $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{k}} \right)$ and $\left( {{X}_{k+1}},\,.\,.\,.\,,{{X}_{n}} \right)$ have the same distributions as $({{Y}_{1}},...,{{Y}_{k}})\,\,\text{and}\,\,({{Y}_{k+1}},...,{{Y}_{n}})$, respectively, but ${{Y}_{i}}\,\text{and}\,{{Y}_{j}}$ are independent for every choice of $1\,\le \,i\,\le \,k,\,k\,+\,1\,\le \,j\,\le \,n$.

A preordering $T$ is constructed in the polynomial ring $A=\mathbb{R}[{{t}_{1}},{{t}_{2}},...]$ (countablymany variables) with the following two properties: (1) For each $f\,\in \,A$ there exists an integer $N$ such that $-\,N\,\le \,f\left( p \right)\,\le \,N$ holds for all $P\in \text{Spe}{{\text{r}}_{T}}(A)$. (2) For all $f\,\in \,A$, if $N+f,N-f\in T$ for some integer $N$, then $f\,\in \,\mathbb{R}$. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering $T$ in a finitely generated $\mathbb{R}$-algebra $A$, if property (1) holds, then for any $f\in A,f>0\,\text{on}\,\text{Spe}{{\text{r}}_{T}}(A)\Rightarrow f\in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\sum{{{C}^{2}}}$ (the set of sums of squares) as the preordering.

In this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and $\left( t,\,m,\,s \right)$-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (i.e., necessary conditions) for existence of these objects.

Let $Q$ be a simple Artinian ring with finite dimension over its center. An order $R$ in $Q$ is said to be Prüfer if any one-sided $R$-ideal is a progenerator. We study prime and primary ideals of a Prüfer order under the condition that the center is Prüfer. Also we characterize branched and unbranched prime ideals of a Prüfer order.

A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

A separator of a connected graph $G$ is a set of vertices whose removal disconnects $G$. In this paper we give various conditions for a separator to contain a minimal one. In particular we prove that every separator of a connected graph that has no thick end, or which is of bounded degree, contains a minimal separator.

Let $H$ be an exceptional, adjoint group of type ${{E}_{6}}$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $P{{D}^{\times }}\,\times \,{{G}_{2}}\left( k \right)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of ${{G}_{2}}\left( k \right)$.

Let $R$ be a non-$\text{GPI}$ prime ring with involution and characteristic $\ne 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta \,=\,\rho \,+\,\varepsilon$ where $\varepsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $\left[ K,\,K \right]$, and $\rho$ can be extended to an ordinary derivation of $\left\langle K \right\rangle$ into $RC$, the central closure.

A theorem of Korányi and Wolf displays any Hermitian symmetric domain as a Siegel domain of the third kind over any of its boundary components. In this paper we give a simple proof that an analogous realization holds for any self-adjoint homogeneous cone.