The Cuntz-Krieger algebra ${{\mathcal{O}}_{B}}$ is defined for an arbitrary, possibly infinite and infinite valued, matrix $B$. A graph ${{C}^{*}}$-algebra ${{G}^{*}}\left( E \right)$ is introduced for an arbitrary directed graph $E$, and is shown to coincide with a previously defined graph algebra ${{C}^{*}}\left( E \right)$ if each source of $E$ emits only finitely many edges. Each graph algebra ${{G}^{*}}\left( E \right)$ is isomorphic to the Cuntz-Krieger algebra ${{\mathcal{O}}_{B}}$ where $B$ is the vertex matrix of $E$.

For a modular elliptic curve $E/\mathbb{Q}$, we show a number of links between the primes $\ell $ for which the mod $\ell $ representation of $E/\mathbb{Q}$ has projective dihedral image and congruence primes for the newform associated to $E/\mathbb{Q}$.

Let ${{\mathbf{P}}^{n}}$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic 0. For an integer $t\,\ge \,3$ consider the invertible sheaf $O\left( t \right)$ on ${{\mathbf{P}}^{n}}$ (Serre twist of the structure sheaf). Let $N\,=\,\left( \underset{n}{\mathop{t+n}}\, \right)$, the dimension of the space of global sections of $O\left( t \right)$, and let $k$ be an integer satisfying $0\,\le \,k\,\le \,N\,-\,\left( 2n\,+\,2 \right)$. Let ${{P}_{1}},\ldots ,{{P}_{k}}$ be general points on ${{\mathbf{P}}^{n}}$ and let $\pi :\,X\,\to \,{{\mathbf{P}}^{n}}$ be the blowing-up of ${{\mathbf{P}}^{n}}$ at those points. Let ${{E}_{i}}\,=\,{{\pi }^{-1}}\left( {{P}_{i}} \right)$ with $1\,\le \,i\,\le \,k$ be the exceptional divisor. Then $M={{\pi }^{*}}\left( O(t) \right)\,\otimes \,{{O}_{X}}\left( -{{E}_{1}}-\cdots -{{E}_{k}} \right)$ is a very ample invertible sheaf on $X$.

On explicite une classe de champ de vecteurs polynomiaux non analytiquement linéarisables à l'aide de la correction introduite par Écalle-Vallet. Notamment, on étend des résultats de Schuman sur la trivialité des hamiltoniens homogàenes isochrones.

In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

A partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree 5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type ${{E}_{8}}$.

Exact analytical expressions for the inverse Laplace transforms of the functions $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order $n$. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

For an integer $n\,\ge \,3$, let ${{M}_{n}}$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then ${{M}_{n}}$ is a Fano manifold of complex dimension $n\,-\,3$. Let ${{\Theta }_{{{M}_{n}}}}$ be the sheaf of germs of holomorphic sections of the tangent bundle $T{{M}_{n}}$. In this paper, we prove ${{H}^{q}}\left( {{M}_{n}},\,{{\Theta }_{{{M}_{n}}}} \right)\,=\,0$ for all $q\,\ge \,0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on ${{M}_{n}}$ consists of a point. Thus the complex structure on ${{M}_{n}}$ is locally rigid.

We give an upper bound on the essential dimension of the group $\mathbb{Z}/q\,\rtimes \,{{\left( \mathbb{Z}/q \right)}^{*}}$ over the rational numbers, when $q$ is a prime power.

The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations ${{X}^{2}}-D{{Y}^{2}}\,=\,c$ and ${{x}^{2}}-D{{y}^{2}}=-c$ when $c\,\in \,\mathbb{Z}$, and $D\,\in \,\mathbb{N}$ is not a perfect square. This continues work in [6]–[8].

In this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system ${{B}_{2}}$.