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 $\varphi \left( n \right)$ is a perfect square. We also obtain similar results for higher powers.

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

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 ${{C}^{*}}$-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 ${{C}^{*}}$-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 ${{\Gamma }_{0}}{{\left( f \right)}^{+}}$ the normalizer of ${{\Gamma }_{0}}\left( f \right)$ in $\text{SL}\left( 2,\,\mathbb{R} \right)$. We find the analogues of the cusp form $\Delta$ for the groups ${{\Gamma }_{0}}{{\left( f \right)}^{+}}$.

We prove a new upper bound for the smallest zero $x$ of a quadratic form over a number field with the additional restriction that $x$ 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 $p\,\ge \,3$ 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 $\mathbb{F}{{\left[ {{V}_{2}}\,\oplus \,{{V}_{2}}\,\oplus \,{{V}_{3}} \right]}^{{{C}_{p}}}}$.

If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

On a compact Kähler manifold $X$ with a holomorphic 2-form $\alpha$, there is an almost complex structure associated with α. 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 Kähler 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 $\text{q}\,\text{=}\,\text{2,}\,\text{3}$ and $f\left( X,\,Y \right)$, $g\left( X,\,Y \right)$, $h\left( X \right)$ be polynomials with integer coefficients. In this paper we deal with the curve $f{{\left( X,\,Y \right)}^{\text{q}}}\,=\,h\left( X \right)g\left( X,\,Y \right)$ , and we show that under some favourable conditions it is possible to determine all of its rational points.

We prove in $\text{ZFC}$ that if $\text{cf}\left( \text{ }\lambda \text{ } \right)>{{\aleph }_{0}}$ and ${{2}^{\text{cf}\left( \text{ }\!\!\lambda\!\!\text{ } \right)}}\,<\,\text{ }\!\!\lambda\!\!\text{ }$, then $\text{ }\!\!\lambda\!\!\text{ }\,\to \,{{\left( \text{ }\!\!\lambda\!\!\text{ ,}\,\omega \,\text{+}\,\text{1} \right)}^{2}}$.

The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $\left( \alpha ,\,\beta \right)$-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 $n\,\ge \,3$.

$\text{A}$ 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 $B\left( R \right)$ of all $\omega \,\times \,\omega$ row and column-finite matrices over any ring $R$ is 2-clean. Finally, the group ring $R{{C}_{n}}$ is considered where $R$ is a local ring.

We prove a big Picard type extension theoremfor holomorphic maps $f\,:\,X\,-\,A\,\to \,M$, 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 ${{\mathbb{P}}^{n}}$ but is not necessarily hyperbolically imbedded in ${{\mathbb{P}}^{n}}$.