Let $\varphi$ denote the Euler function. In this paper, we show that
for all large $x$ there are more than $x^{0.33}$ Carmichael numbers
$n\le x$ with the property that $\varphi(n)$ is a perfect square. We
also obtain similar results for higher powers.

Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{-1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.

In this paper we study holomorphic maps between almost Hermitian
manifolds. We obtain a new criterion for the harmonicity of such
holomorphic maps, and we deduce some applications to horizontally
conformal holomorphic submersions.

New necessary and sufficient conditions are established for Banach
spaces to have the approximation property; these conditions are
easier to check than the known ones. A shorter proof of a result
of Grothendieck is presented, and some properties of a weak
version of the approximation property are addressed.

We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.

Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the
normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of
the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.

We prove a new upper bound for the smallest zero $\mathbf{x}$
of a quadratic form over a number field with the additional
restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed
hyperplanes. Our bound is polynomial in the height of the quadratic
form, with an exponent depending only on the number of variables but
not on $m$.

We show that for compact orientable hyperbolic orbisurfaces, the
Laplace spectrum determines the length spectrum as well as the
number of singular points of a given order. The converse also holds, giving
a full generalization of Huber's theorem to the setting of
compact orientable hyperbolic orbisurfaces.

Let $C_p$ denote the cyclic group of order $p$, where $p \geq 3$ is
prime. We denote by $V_n$ the indecomposable $n$ dimensional
representation of $C_p$ over a field $\mathbb F$ of characteristic
$p$. We compute a set of generators, in fact a SAGBI basis, for
the ring of invariants $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$.

On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form
$\a$, there is an almost complex structure associated with $\a$. We
show how this implies vanishing theorems for the Gromov--Witten
invariants of $X$. This extends the approach used by Parker and
the author for K\"{a}hler surfaces to higher dimensions.

In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
Christoffel--Darboux formulas are presented for the first time.

We prove there exist exponentially decaying generalized eigenfunctions
on a blow-up of the Sierpinski gasket with boundary. These are used
to show a Borel-type theorem, specifically that for a prescribed jet
at the boundary point there is a smooth function having that jet.

Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with
integer coefficients. In this paper we deal with the curve
$f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable
conditions it is possible to determine all of its rational points.

The solutions to Hilbert's Fourth Problem in the regular case
are projectively flat Finsler metrics. In this paper,
we consider the so-called $(\alpha,\beta)$-metrics defined by a
Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a
necessary and sufficient condition for such metrics to be projectively
flat in dimension $n \geq 3$.

A ring $R$ is said to be $n$-clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$-good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$
of all $\omega\times \omega$ row and column-finite matrices over
any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.

We prove a big Picard type extension theorem for holomorphic maps
$f\from X-A \rightarrow M$, where $X$ is a complex manifold,
$A$ is an analytic subvariety of $X$, and $M$ is the complement of the
union of a set of hyperplanes in ${\Bbb P}^n$ but is not
necessarily hyperbolically imbedded in ${\Bbb P}^n$.