We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$ , then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

The aim of this note is to provide a conceptually simple demonstration of the fact that repetitive model sets are characterized as the repetitive Meyer sets with an almost automorphic associated dynamical system.

The Jacobi coefficients $c_{j}^{\ell }\left( \alpha ,\,\beta\right)\,\left( 1\,\le \,j\,\le \,\ell ,\,\alpha ,\,\beta \,>\,-1 \right)$ are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomials $P_{k}^{\left( \alpha ,\,\beta\right)}\,\left( k\,\ge \,0,\,\alpha ,\,\beta \,>\,-1 \right)$ into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.

Associated with any closed quantum subgroup $G\,\subset \,U_{N}^{+}$ and any index set $I\,\subset \,\{1,\,.\,.\,.\,,\,N\}$ is a certain homogeneous space ${{X}_{G,I}}\subset S_{\mathbb{C},+}^{N-1},$ called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds $X\subset S_{\mathbb{C},+}^{N-1}$ that can appear in this way.

For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$ . In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.

For an analytic function $f$ on the unit disk $\mathbb{D}$ , we show that the ${{L}^{2}}$ integral mean of $f$ on $\text{c}\,\text{}\,\text{ }\!\!|\!\!\text{ z }\!\!|\!\!\text{ }\,\text{}\,\text{r}$ with respect to the weighted area measure ${{\left( 1\,-\,|z{{|}^{2}} \right)}^{\alpha }}dA\left( z \right)$ is a logarithmically convex function of $r$ on $\left( c,\,1 \right)$ , where $-3\,\le \,\alpha \,\le \,0\,\text{and}\,\text{c}\,\in \,[\,0,\,1)$ . Moreover, the range $[-3,\,0]$ for $\alpha $ is best possible. When $c\,=\,0$ , our arguments here also simplify the proof for several results we obtained in earlier papers.

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number ${{p}_{m\times n}}$ of distinct multiple self-avoiding polygons in the $m\,\times \,n$ rectangular grid on the square lattice. For $m\,=\,2,\,{{p}_{2\times n}}\,=\,{{2}^{n-1}}\,-1$ . And for integers $m,\,n\,\ge \,3$ ,

$${{2}^{m+n-3}}\left( \frac{17}{10} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}\,\le \,{{p}_{m\times n}}\,\le \,{{2}^{m+n-3}}\left( \frac{31}{16} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}.$$

Given two monic polynomials $f$ and $g$ with coefficients in a number field $K$ , and some $\alpha \,\in \,K$ , we examine the action of the absolute Galois group $Gal\left( \bar{K}/K \right)$ on the directed graph of iterated preimages of $\alpha $ under the correspondence $g\left( y \right)\,=\,f\left( x \right)$ , assuming that $\deg \left( f \right)\,>\,\deg \left( g \right)$ and that $\gcd \left( \deg \left( f \right),\deg \left( g \right) \right)\,=1$ . If a prime of $K$ exists at which $f$ and $g$ have integral coefficients and at which $\alpha $ is not integral, we show that this directed graph of preimages consists of finitely many $Gal\left( \bar{K}/K \right)$ -orbits. We obtain this result by establishing a $p$ -adic uniformization of such correspondences, tenuously related to Böttcher’s uniformization of polynomial dynamical systems over $\mathbb{C}$ , although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

On a real hypersurface $M$ in a complex two-plane Grassmannian ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ we have the Lie derivation $\mathcal{L}$ and a differential operator of order one associated with the generalized Tanaka–Webster connection ${{\widehat{\mathcal{L}}}^{\left( k \right)}}$ . We give a classification of real hypersurfaces $M$ on ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ satisfying $\widehat{\mathcal{L}}_{\xi }^{\left( k \right)}\,S\,=\,{{\mathcal{L}}_{\xi }}S$ , where $\xi$ is the Reeb vector field on $M$ and $s$ the Ricci tensor of $M$ .

Let $G$ be the $n$ -fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $G$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

In this paper, we first discuss the relation between $\text{VB}$ -Courant algebroids and $\text{E}$ -Courant algebroids, and we construct some examples of $\text{E}$ -Courant algebroids. Then we introduce the notion of a generalized complex structure on an $\text{E}$ -Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\text{gl}\left( V \right)\oplus V$ correspond to complex Lie algebra structures on $V$ .

We prove an interpolation formula for the values of certain $p$ -adic Rankin-Selberg $L$ -functions associated with non-ordinary modular forms.

A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance

$$d_{N}^{2}:=\underset{{{A}_{N}}}{\mathop{\inf }}\,\frac{1}{2\pi }\int _{-\infty }^{\infty }{{\left| 1-\zeta {{A}_{N}}\left( \frac{1}{2}+it \right) \right|}^{2}}\frac{dt}{\frac{1}{4}+{{t}^{2}}},$$

where the infimum is over all Dirichlet polynomials

$${{A}_{N}}\left( s \right)\,=\,\sum\limits_{n=1}^{N}{\frac{{{a}_{n}}}{{{n}^{s}}}}$$

of length $N$ . In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

In this paper we give some generalizations and improvements of the Pavlović result on the Holland–Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in ${{\text{R}}^{m}}$ .

It is shown that the unit ball in ${{\mathbb{C}}^{n}}$ is the only complexmanifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

If $B$ is the Blachke product with zeros $\{{{z}_{n}}\},\,\text{then}\,\left| {B}'\left( z \right) \right|\,\le \,{{\Psi }_{B}}\left( z \right)$ , where

$${{\Psi }_{B}}\,=\,\sum\limits_{n}{\frac{1-{{\left| {{z}_{n}} \right|}^{2}}}{{{\left| 1-{{\overline{z}}_{n}}z \right|}^{2}}}.}$$

Moreover, it is a well-known fact that, for $0\,<\,p\,<\,\infty $ ,

$${{M}_{p\left( r,{B}' \right)}}\,=\,{{\left( \frac{1}{2\pi }\,\int_{0}^{2\pi }{{{\left| {B}'\left( \text{r}{{\text{e}}^{i\theta }} \right) \right|}^{p}}d\theta } \right)}^{{1}/{p}\;}},\,0\,\le \,r\,<\,1,$$

is bounded if and only if ${{M}_{p}}\left( r,\,{{\Psi }_{B}} \right)$ is bounded. We find a Blaschke product ${{B}_{0}}$ such that ${{M}_{p}}\left( r,\,{{{{B}'}}_{0}} \right)$ and ${{M}_{p}}\left( r,{{\Psi }_{{{B}_{0}}}} \right)$ are not comparable for any $\frac{1}{2}\,<\,p\,<\,\infty $ . In addition, it is shown that, if $0\,<\,p\,<\,\infty$ , $B$ is a Carleson-Newman Blaschke product and a weight $\omega $ satisfies a certain regularity condition, then

$${{\int }_{\mathbb{D}}}{{\left| {B}'\left( z \right) \right|}^{p}}\omega \left( z \right)dA\left( z \right)~\asymp {{\int }_{\mathbb{D}}}{{\Psi }_{B}}{{\left( z \right)}^{p}}\omega \left( z \right)dA\left( z \right),$$

where $d\,A\left( z \right)$ is the Lebesgue area measure on the unit disc.

The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree $b\,\ge \,4$ , where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of degree $b$ and three of its total inflectional tangents.

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.