Consider $M$, a bounded domain in ${{\mathbb{R}}^{d}}$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigen-functions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of $M$.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, of conductor $N$ and without complex multiplication. For any positive integer $l$, let ${{\phi }_{1}}$ be the Galois representation associated to the $l$-division points of $E$. From a celebrated 1972 result of Serre we know that ${{\phi }_{1}}$ is surjective for any sufficiently large prime $l$. In this paper we find conditional and unconditional upper bounds in terms of $N$ for the primes $l$ for which ${{\phi }_{1}}$ is not surjective.

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of ${{S}^{3}}$ branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of ${{S}^{3}}$ branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the ${{5}_{2}}$ knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Let $G$ be a finite group and $H$ be a subgroup. We say that $H$ is degree homogeneous if, for each $x\,\in \,\text{Irr}\left( G \right)$, all the irreducible constituents of the restriction ${{x}_{H}}$ have the same degree. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question by E.M. Zhmud’, we investigate general properties of such subgroups. It appears unlikely that degree homogeneous subgroups can be characterized entirely by abstract group properties, but we providemixed criteria (involving both group structure and character properties) which are both necessary and sufficient. For example, $H$ is degree homogeneous in $G$ if and only if the derived subgroup ${H}'$ is normal in $G$ and, for every pair $\alpha $, $\beta $ of irreducible $G$-conjugate characters of ${H}'$, all irreducible constituents of ${{\alpha }^{H}}$ and ${{\beta }^{H}}$ have the same degree.

Let $A$ be the inductive limit of a system

$${{A}_{1\,}}\,\xrightarrow{{{\phi }_{1,\,2}}}\,{{A}_{2}}\,\xrightarrow{{{\phi }_{2,\,3}}}\,{{A}_{3}}\,\to \cdots $$

with ${{A}_{n}}\,=\,\oplus _{i=1}^{{{t}_{n}}}\,{{P}_{n,\,i}}{{M}_{\left[ n,\,i \right]}}(C({{X}_{n,\,i}})){{P}_{n,\,i}}$, where ${{X}_{n,\,i}}$ is a finite simplicial complex, and ${{P}_{n,\,i}}$ is a projection in ${{M}_{[n,i]}}\,\left( C\left( {{X}_{n,i}} \right) \right)$. In this paper, we will prove that $A$ can be written as another inductive limit

$${{B}_{1}}\,\xrightarrow{{{\psi }_{1,\,2}}}\,{{B}_{2}}\,\xrightarrow{{{\psi }_{2,\,3}}}\,{{B}_{3}}\,\to \cdots $$

with ${{B}_{n}}\,=\,\oplus _{i=1}^{{{s}_{n}}}\,{{Q}_{n,i}}{{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}})){{Q}_{n,\,i}}$, where ${{Y}_{n,\,i}}$ is a finite simplicial complex, and ${{Q}_{n,\,i}}$ is a projection in ${{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}}))$, with the extra condition that all the maps ${{\psi }_{n,n+1}}$ are injective. (The result is trivial if one allows the spaces ${{Y}_{n,\,i}}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple $\text{AH}$ algebras. The special case that the spaces ${{X}_{n,\,i}}$ are graphs is due to the third author.

We study countable splitting of Markushevich bases in weakly Lindelöf Banach spaces in connection with the geometry of these spaces.

For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

We obtain a complete description of the conjugacy classes ${{C}_{1,\ldots ,}}{{C}_{n}}$ in $SU\left( 2 \right)$ with the property that ${{C}_{1}}\cdots {{C}_{n}}\,=\,SU(2)$. The basic instrument is a characterization of the conjugacy classes ${{C}_{1,\ldots ,}}{{C}_{n+1}}$ in $SU\left( 2 \right)$ with ${{C}_{1}}\cdots {{C}_{n+1}}\,\backepsilon \,I$, which generalizes a result of $\left[ \text{Je-We} \right]$.

We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of ${{\text{J}}^{*}}$-algebras. We prove that if $P$ is a normal bimodule idempotent and $\left\| P \right\|\,<\,2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension $n\,\ge \,3$. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D\,=\,{{2}^{h}}c$ where $c\,>\,1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be ${{2}^{h}}$. At the end of the paper, we also address the case where $D\,=\,c$ is odd and the central norm of $\sqrt{D}$ is equal to 2.

If $f$ is a real-valued function on $\left[ -\pi ,\,\pi \right]$ that is Henstock-Kurzweil integrable, let ${{u}_{r}}(\theta )$ be its Poisson integral. It is shown that ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,o\left( 1/\left( 1-r \right) \right)$ as $r\,\to \,1$ and this estimate is sharp for $1\,\le \,p\,\le \,\infty $. If $\mu $ is a finite Borel measure and ${{u}_{r}}(\theta )$ is its Poisson integral then for each $1\,\le \,p\,\le \,\infty $ the estimate ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,O\left( {{\left( 1-r \right)}^{1/p-1}} \right)$ as $r\,\to \,1$ is sharp. The Alexiewicz norm estimates $\left\| {{u}_{r}} \right\|\,\le \,\left\| f \right\|$$\left( 0\,\le \,r\,<\,1 \right)$ and $\left\| {{u}_{r}}-f \right\|\,\to 0\left( r\to 1 \right)$ hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when $u$ is in the harmonic Hardy space associated with the Alexiewicz norm and when $f$ is of bounded variation.

We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field $\mathbb{I}$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincaré-type equations and the connection between the solutions of these functional equations and the generalized Heine series.