It is shown that given any sequence of automorphisms ${{\left( {{\phi }_{k}} \right)}_{k}}$ of the unit ball ${{\mathbb{B}}_{N}}$ of ${{\mathbb{C}}^{N}}$ such that $\left\| {{\phi }_{k}}\left( 0 \right) \right\|$ tends to 1, there exists an inner function $I$ such that the family of “non-Euclidean translates” ${{\left( I\,\text{o}\,{{\phi }_{k}} \right)}_{k}}$ is locally uniformly dense in the unit ball of ${{H}^{\infty }}\left( {{\mathbb{B}}_{N}} \right)$ .

In this paper we establish that Hankel multipliers of Laplace transform type are bounded from ${{L}^{p}}\left( w \right)$ into itself when $1\,<\,p\,<\infty$ , and from ${{L}^{1}}\left( w \right)$ into ${{L}^{1,\infty }}\left( w \right)$ , provided that $w$ is in the Muckenhoupt class ${{A}^{p}}$ on $\left( \left( 0,\,\infty\right),\,dx \right)$ .

Let ${{A}_{n}}\,=\,\{{{a}_{0}}+{{a}_{1}}z+\,.\,.\,.\,+\,{{a}_{n-1}}{{z}^{n-1}}\,:\,{{a}_{j}}\,\in \,\{0,1\}\}$ , whose elements are called zero-one polynomials and correspond naturally to the ${{2}^{n}}$ subsets of $\left[ n \right]\,:=\,\{0,\,1,\,.\,.\,.\,,\,n-1\}$ . We also let ${{A}_{n,m\,}}\,=\,\{\alpha \left( z \right)\,\in \,{{A}_{n}}\,:\,\alpha \left( 1 \right)\,=\,m\}$ , whose elements correspond to the $\left( _{m}^{n} \right)$ subsets of $\left[ n \right]$ of size $m$ , and let ${{B}_{n}}\,=\,{{A}_{n+1}}\,\backslash \,{{A}_{n}}$ , whose elements are the zero-one polynomials of degree exactly $n$ .

Many researchers have studied norms of polynomials with restricted coefficients. Using $\|\alpha {{\|}_{p}}$ to denote the usual ${{L}_{p}}$ norm of $\alpha$ on the unit circle, one easily sees that $\alpha \left( z \right)\,=\,{{a}_{0}}+{{a}_{1}}z+.\,.\,.+{{a}_{N}}{{z}^{N}}\,\in \,\mathbb{R}\left[ z \right]$ satisfies $\|\alpha \|_{2}^{2}\,=\,{{c}_{0}}$ and $\|\alpha \|_{4}^{4}\,=\,c_{0}^{2}\,+\,2\left( c_{1}^{2}\,+\,.\,.\,.\,+\,c_{N}^{2} \right)$ , where ${{c}_{k}}\,:=\,\sum _{j=0}^{N-k}\,{{a}_{j}}{{a}_{j+k}}\,\text{for}\,\text{0}\le \,\text{k}\,\le N$ .

If $\alpha \left( z \right)\,\in \,{{A}_{n,m}}$ , say $\alpha \left( z \right)\,=\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{1}}}}\,+\,.\,.\,.\,+\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{m}}}}$ where ${{\beta }_{1}}\,<\,.\,.\,.\,<\,{{\beta }_{m}}$ , then ${{c}_{k}}$ is the number of times $k$ appears as a difference ${{\beta }_{i}}\,-\,{{\beta }_{j}}$ . The condition that $\alpha \,\in \,{{A}_{n,m}}$ satisfies ${{c}_{k}}\,\in \,\{0,1\}$ for $1\,\le \,k\,\le \,n\,-\,1$ is thus equivalent to the condition that $\{{{\beta }_{1}},\,.\,.\,.\,,\,{{\beta }_{m}}\}$ is a Sidon set (meaning all differences of pairs of elements are distinct).

In this paper, we find the average of $\left\| \alpha\right\|_{4}^{4}$ over $\alpha \,\in \,{{A}_{n}}$ , $\alpha \,\in \,{{B}_{n}}$ and $\alpha \,\in \,{{A}_{n,m}}$ . We further show that our expression for the average of $\left\| \alpha\right\|_{4}^{4}$ over ${{A}_{n,m}}$ yields a new proof of the known result: if $m\,=\,o\left( {{n}^{1/4}} \right)$ and $B\left( n,\,m \right)$ denotes the number of Sidon sets of size $m$ in $\left[ n \right]$ , then almost all subsets of $\left[ n \right]$ of size $m$ are Sidon, in the sense that ${{\lim }_{n\to \infty }}\,B\left( n,\,m \right)/\left( _{m}^{n} \right)\,=\,1$ .

We study the topological 4-dimensional surgery problem for a closed connected orientable topological 4-manifold $X$ with vanishing second homotopy and ${{\pi }_{1}}\left( X \right)\,\cong \,A\,*\,F\left( r \right)$ , where $A$ has one end and $F\left( r \right)$ is the free group of rank $r\,\ge \,1$ . Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{r}},\,d \right)$ , stabilize when $r\,\ge \,d$ . We give a complete characterization of the effective divisors on ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{d}},\,d \right)$ . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

We prove that the neighborhood complex $\text{N}\left( G \right)$ , the box complex $\text{B}\left( G \right)$ , the homomorphism complex $\text{Hom}\left( {{K}_{2}},\,G \right)$ and the Lovász complex $\text{L}\left( G \right)$ have the same simple ${{\mathbb{Z}}_{2}}$ -homotopy type in the sense of Whitehead. We show that these graph complexes are simple ${{\mathbb{Z}}_{2}}$ -universal.

A Mauldin–Williams graph $M$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a ${{C}^{*}}$ -algebra ${{O}_{M}}\left( K \right)$ with a Mauldin–Williams graph $M$ and the invariant set $K$ , laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $M$ satisfies the open set condition in $K$ , and $G$ is irreducible and is not a cyclic permutation, then the associated ${{C}^{*}}$ -algebra ${{O}_{M}}\left( K \right)$ is simple and purely infinite. We calculate the $K$ -groups for some examples including the inflation rule of the Penrose tilings.

In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner– Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

We prove, among other things, that if ${{C}_{p}}\left( X \right)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

We use the mutually stationary sets of Foreman and Magidor as a tool to establish the validity of the two-cardinal version of the diamond principle in some special cases.

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\text{GL}\left( n,\,\mathbb{C} \right)$ is isomorphic to another. As a consequence we discover families of Littlewood–Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

We show that limits of CS iterations of the $n$ -Silver forcing notion have the $n$ -localization property.

It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

Using a canonical linear embedding of the algebra ${{G}^{\infty }}\left( \Omega\right)$ of Colombeau generalized functions in the space of $\overline{\mathbb{C}}$ -valued $\mathbb{C}$ -linear maps on the space $D\left( \Omega\right)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${{G}^{\infty }}$ . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

In this paper we derive formulas for summation of series involving J. Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H. M. Srivastava. These series are expressed in terms of the Riemann $\zeta$ function and Dirichlet functions $\eta$ , $\lambda$ , $\beta$ , and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.