In this paper, we establish the existence of positive solution of a nonlinear subelliptic equation involving the critical Sobolev exponent on the Heisenberg group, which generalizes a result of Brezis and Nirenberg in the Euclidean case.

We show that, if a Banach algebra $\mathfrak{A}$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\mathfrak{A}$ implies the $\left( 2m+1 \right)$-weak amenability of $\mathfrak{A}$ for all $m\,\ge \,1$.

An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\le \,x$.

We examine Hilbert bimodules which possess a (generally unbounded) involution. Topics considered include a linking algebra representation, duality, locality, and the role of these bimodules in noncommutative differential geometry

The singular spectrum of $u$ in a convolution equation $\mu *u\,=\,f$, where $\mu$ and $f$ are tempered ultra distributions of Beurling or Roumieau type is estimated by

$$SSu\,\subset \,\left( {{\mathbf{R}}^{n}}\,\times \,\text{Char}\,\mu \right)\,\cup \,SSf$$

The same is done for $S{{S}_{*}}u$.

Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\ell _{1}^{\left( n \right)}$ in an arbitrary real normed space $X$.

The said criterion involves finding ${{2}^{n}}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy ${{2}^{n-1}}$ metric equalities.

The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of ${{\mathbf{C}}^{m}}$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers.

In this paper we prove that there is no nontrivial ${{L}^{q}}$-integrably $p$-harmonic 1-form on a complete manifold with nonnegatively Ricci curvature $\left( 0\,<\,q\,<\,\infty \right)$.

Simple necessary conditions for weak type (1, 1) of invariant operators on $L\left( {{\mathbb{R}}^{d}} \right)$ and their applications to rational Fourier multiplier are given.

Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of ${{\mathbb{C}}^{N}}\left( N\ge 2 \right)$ is removable for holomorphic functions, if its topological dimension is less than or equal to $N\,-\,2$. Besides, they asked whether closed subsets of ${{\mathbb{C}}^{2}}$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question.