Recently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb{R}\times {{S}^{n-1}}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${{\mathbb{C}}^{n}}$ is foliated by round $\left( n-1 \right)$ -spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${{\mathbb{C}}^{n}}$ . The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma _{0}^{+}\left( p \right)\,\,:={{\Gamma }_{0}}\left( p \right)+{{w}_{p}}$ , where $p$ is a prime and ${{w}_{p}}$ is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on ${{\Gamma }_{1}}\left( p \right)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$ -Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Let $M$ be a symplectic 4-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

We study two sufficient conditions that imply global injectivity for a ${{C}^{1}}$ map $X:{{\mathbb{R}}^{2}}\to {{\mathbb{R}}^{2}}$ such that its Jacobian at any point of ${{\mathbb{R}}^{2}}$ is not zero. One is based on the notion of half-Reeb component and the other on the Palais–Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais–Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais–Smale condition it is not a necessary condition for the global injectivity of the map $X$ .

We extend the notion of linking number of an ordinary link of two components to that of a singular link with transverse intersections, in which case the linking number is a half-integer. We then apply it to simplify the construction of the Seifert matrix, and therefore the Alexander polynomial, in a natural way.

Beginning with a seminal paper of Rényi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daróczy and Kátai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

We show that, for most of the elliptic curves $\text{E}$ over a prime finite field ${{\mathbb{F}}_{p}}$ of $p$ elements, the discriminant $D\left( E \right)$ of the quadratic number field containing the endomorphism ring of $\text{E}$ over ${{\mathbb{F}}_{p}}$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over ${{\mathbb{F}}_{p}}$ .

We will show that any extension of a product of two Cantor minimal $\mathbb{Z}$ -systems is affable in the sense of Giordano, Putnam and Skau.

Let $X$ be a smooth complex projective curve of genus $g\ge 1$ . Let $\xi \in {{J}^{1}}\left( X \right)$ be a line bundle on $X$ of degree 1. Let $W=\text{Ex}{{\text{t}}^{1}}\left( {{\xi }^{n}},{{\xi }^{-1}} \right)$ be the space of extensions of ${{\xi }^{n}}$ by ${{\xi }^{-1}}$ . There is a rational map ${{D}_{\xi }}:G\left( n,W \right)\to S{{U}_{X}}\left( n+1 \right)$ , where $G\left( n,W \right)$ is the Grassmannian variety of $n$ -linear subspaces of $W$ and $S{{U}_{X}}\left( n+1 \right)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$ , then ${{D}_{\xi }}$ is everywhere defined and is injective.

In the present paper, we introduce a modification of the Meyer-König and Zeller $\left( \text{MKZ} \right)$ operators which preserve the test functions ${{f}_{0}}\left( x \right)=1$ and ${{f}_{2}}\left( x \right)={{x}^{2}}$ , and we show that this modification provides a better estimation than the classical $\left( \text{MKZ} \right)$ operators on the interval $\left[ \frac{1}{2},1 \right)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r$ -th order generalization of our operators and study their approximation properties.

Let ${{G}_{1}}$ and ${{G}_{2}}$ be $p$ -adic groups. We describe a decomposition of Ext-groups in the category of smooth representations of ${{G}_{1}}\times {{G}_{2}}$ in terms of Ext-groups for ${{G}_{1}}$ and ${{G}_{2}}$ . We comment on $\text{Ext}_{G}^{1}\left( \pi ,\pi\right)$ for a supercuspidal representation $\pi$ of a $p$ -adic group $G$ . We also consider an example of identifying the class, in a suitable $E\text{x}{{\text{t}}^{1}}$ , of a Jacquet module of certain representations of $p$ -adic $\text{G}{{\text{L}}_{2n}}$ .

Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D.$ We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$ . This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

We prove that a separable, nuclear, purely infinite, simple ${{C}^{*}}$ -algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$ -groups are direct sums of cyclic groups.

This paper presents some results on the simple exceptional Jordan algebra over an algebraically closed field $\Phi$ of characteristic not 2. Namely an example of simple decomposition of $H\left( {{O}_{3}} \right)$ into the sumof two subalgebras of the type $H\left( {{Q}_{3}} \right)$ is produced, and it is shown that this decomposition is the only one possible in terms of simple subalgebras.

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${{\text{R}}^{n}}$ and $H$ be the mean curvature of $M$ . We obtain some low geometric estimates of the total squaremean curvature $\int\limits_{M}{{{H}^{2}}d\sigma }$ . The low bounds are geometric invariants involving the volume of $M$ , the total scalar curvature of $M$ , the Euler characteristic and the circumscribed ball of $M$ .