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.

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is finitely generated for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ is finitely generated. In this paper it is shown that if $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is Artinian for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$ is Artinian.

In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

Using ideas of S. Wassermann on non-exact ${{C}^{*}}$-algebras and property $\text{T}$ groups, we show that one of his examples of non-invertible ${{C}^{*}}$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a ${{C}^{*}}$-extension which is not even invertible up to homotopy.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $\mathfrak{k}\,\subset \mathfrak{g}$ be any algebraic subalgebra reductive in $\mathfrak{g}$. For any simple finite dimensional $\mathfrak{k}$-module $V$, we construct simple $\left( \mathfrak{g},\mathfrak{k} \right)$-modules $M$ with finite dimensional $\mathfrak{k}$-isotypic components such that $V$ is a $\mathfrak{k}$-submodule of $M$ and the Vogan norm of any simple $\mathfrak{k}$-submodule $V\prime \subset M,V\prime \ne \,V$, is greater than the Vogan norm of $V$. The $\left( \mathfrak{g},\mathfrak{k} \right)$-modules $M$ are subquotients of the fundamental series of $\left( \mathfrak{g},\mathfrak{k} \right)$-modules.

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.

We extend results of P. M. Gauthier, W. Hengartner and A. A. Nersesyan on simultaneous approximation and interpolation on Arakelian sets.

In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right)$, where $\left| q \right|<1$. He called them mock theta functions, because as $q$ radially approaches any point ${{e}^{2\pi ir}}\left( r\,\text{rational} \right)$, there is a theta function ${{F}_{r}}\left( q \right)$ with $F\left( q \right)-{{F}_{r}}\left( q \right)=O\left( 1 \right)$. In this paper we establish the relationship between two families of mock theta functions.

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 $\text{G}{{\text{L}}_{2n}}$.

We prove Beurling's theorem for rank 1 Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak* Kadec–Klee norm is a three space property.

Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of “coquecigrues,” in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction.

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.

It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all “special” $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-${{l}_{p}}$ subspaces for some $1\,\le \,p\,\le \,\infty $. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy–Liouville–Mirimanoff polynomials to show that the intersection of the Fermat curve of degree $p$ with the line $X+Y=Z$ in the projective plane contains no algebraic points of degree $d$ with $3\le d\le 11$. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of Pólya and Szegö. These conditions are conjecturally also necessary for irreducibility.

We study the structure of the sets of symmetric sequences and spreading models of an Orlicz sequence space equipped with partial order with respect to domination of bases. In the cases that these sets are “small”, some descriptions of the structure of these posets are obtained.

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.

We show that there exists a non-archimedean Fréchet-Montel space $W$ with a basis and with a continuous norm such that any non-archimedean Fréchet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fréchet space is isomorphic to a quotient of some non-archimedean nuclear Fréchet space with a basis and with a continuous norm.

Let $G$ be a compact topological group and let $f:\,G\,\to \,G$ be a continuous transformation of $G$. Define ${{f}^{*}}:G\to G$ by ${{f}^{*}}\left( x \right)\,=\,f\left( {{x}^{-1}} \right)x$ and let $\mu \,=\,{{\mu }_{G}}$ be Haar measure on $G$. Assume that $H\,=\,\text{IM}\,{{f}^{*}}$ is a subgroup of $G$ and for every measurable $C\,\subseteq \,H,\,{{\mu }_{G}}{{\left( \left( {{f}^{*}} \right) \right)}^{-1}}\left( \left( C \right) \right)\,=\,\mu H\left( C \right)$. Then for every measurable $C\,\subseteq \,G$, there exist $S\subseteq C$ and $g\,\in \,G$ such that $f\left( S{{g}^{-1}} \right)\,\subseteq \,C{{g}^{-1}}$ and $\mu \left( S \right)\,\ge \,{{\left( \mu \left( C \right) \right)}^{2}}$.