We construct meromorphic functions with asymptotic power series expansion in ${{z}^{-1}}$ at $\infty$ on an Arakelyan set $A$ having prescribed zeros and poles outside $A$. We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

A Banach space $X$ is said to be quotient hereditarily indecomposable if no infinite dimensional quotient of a subspace of $X$ is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space ${{X}_{GM}}$ constructed by W. T. Gowers and B. Maurey in $[\text{GM}]$. Then we provide an example of a reflexive hereditarily indecomposable space $\hat{X}$ whose dual is not hereditarily indecomposable; so $\hat{X}$ is not quotient hereditarily indecomposable. We also show that every operator on ${{\hat{X}}^{*}}$ is a strictly singular perturbation of an homothetic map.

We show that a martingale problem associated with a competing species model has a unique solution. The proof of uniqueness of the solution for the martingale problem is based on duality technique. It requires the construction of dual probability measures.

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types {3, 6} and {6, 3}.

When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).

We study the correspondence of representations arising by restricting the minimal representation of the linear group of type ${{E}_{7}}$ and relative rank 4. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of ${{G}_{2}}$ and $\text{P}{{\text{U}}_{3}}\left( D \right)$.

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a ${{C}^{1}}$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$-dimensional points is contained in an $n$-dimensional ${{C}^{1}}$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of $\text{G}$. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

Let $\tilde{M}$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \mathbb{Z}_{2}^{d}$ and branch set a trivalent graph $\Gamma $; such a cover is determined by a coloring of the edges of $\Gamma $ with elements of $G$. For each index-2 subgroup $H$ of $G,\,{{M}_{H}}=\tilde{M}/H$ is a double branched cover of $M$. Sakuma has proved that ${{H}_{1}}\left( {\tilde{M}} \right)$ is isomorphic, modulo 2-torsion, to ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$, and has shown that ${{H}_{1}}\left( {\tilde{M}} \right)$ is determined up to isomorphism by ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover ${{M}_{H}}\to M$ is connected, and when $d=3$ and $\Gamma $ is the complete graph ${{K}_{4}}$. We prove this for a larger class of coverings: when $d=2$, for any coloring of a connected graph; when $d=3\,\text{or}\,\text{4}$, for an infinite class of colored graphs; and when $d=5$, for a single coloring of the Petersen graph.

Lattices and $\mathbb{Z}$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain $4D$ examples that are related to $4D$ root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

Let $L$ be a simple algebraic group and $P$ a parabolic subgroup with Abelian unipotent radical ${{P}^{u}}$ . Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of $P$-orbits in ${{P}^{u}}$ . We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra $k[{{P}^{u}}]$ is a free module over the algebra of covariants.

Let $m$ be a point of the maximal ideal space of ${{H}^{\infty }}$ with nontrivial Gleason part $P\left( m \right)$. If ${{L}_{m}}\,:\,\text{D}\,\to \,\text{P(m)}$ is the Hoffman map, we show that ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ is a closed subalgebra of ${{H}^{\infty }}$. We characterize the points $m$ for which ${{L}_{m}}$ is a homeomorphism in terms of interpolating sequences, and we show that in this case ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ coincides with ${{H}^{\infty }}$. Also, if ${{I}_{m}}$ is the ideal of functions in ${{H}^{\infty }}$ that identically vanish on $P\left( m \right)$, we estimate the distance of any $f\,\in \,{{H}^{\infty }}\,\text{to}\,{{I}_{m}}$.

The Sierpiński carpets first considered by C.McMullen and later studied by Y. Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i) calculates the Hausdorff, box (or Minkowski), and packing dimensions of the modified Sierpiński carpets and (ii) shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.

We prove a conjecture of Kudla and Rallis about the first occurrence in the theta correspondence, for dual pairs of the form $\left( U\left( p,q \right),\,U\left( r,s \right) \right)$ and most representations.

The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

Let $U\left( n,\,n \right)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U\left( n,\,n \right)$ has at most simple poles at the integers or half integers in certain strip of the complex plane.

An isomorphism invariant is an expression, defined for a configuration in the projective plane, which takes the same value for all isomorphic configurations. Examples are given as well as a general method (Nehring sequences) for constructing such invariants.

Let $L={{L}_{0}}+{{L}_{1}}$ be a ${{\mathbb{Z}}_{2}}$ -graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that ${{L}_{0}}$ is abelian and $L$ is generated by finitely many homogeneous elements ${{a}_{1}},.\,.\,.,{{a}_{k}}$ such that every commutator in ${{a}_{1}},.\,.\,.,{{a}_{k}}$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite ${2}'$ -group $G$ admitting an involutory automorphism $\phi $ with ${{C}_{G}}\left( \phi \right)$ abelian is locally finite.

Partial answers are given to two questions. When does a lattice $\Lambda $ contain a sublattice ${\Lambda }'$ of index $N$ that is geometrically similar to $\Lambda $? When is the sublattice “clean”, in the sense that the boundaries of the Voronoi cells for ${\Lambda }'$ do not intersect $\Lambda $?

Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that

$${{b}^{2}}-4ac=d,a>0,\gcd (a,b,c)=1.$$

The value of $\left| \eta ((b+\sqrt{d})/\left. 2a) \right| \right.$ is determined explicitly, where $\eta \left( z \right)$ is Dedekind’s eta function

$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$, the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$, and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega \right])$.