For holomorphic functions $f$ in the unit disk $ \mathbb{D} $ with $f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity ${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function ${\mathrm{d} }_{\mathrm{e} } f(r \mathbb{D} )/ r$ is increasing for $0\lt r\lt 1$.

Let $G$ denote a finite group and $\mathrm{cd} (G)$ the set of irreducible character degrees of $G$. Huppert conjectured that if $H$ is a finite nonabelian simple group such that $\mathrm{cd} (G)= \mathrm{cd} (H)$, then $G\cong H\times A$, where $A$ is an abelian group. He verified the conjecture for many of the sporadic simple groups and we complete its verification for the remainder.

We study natural families of $\bar {\partial } $-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

In this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

In this paper we obtain some sharp ${L}^{p} - {L}^{q} $ estimates and the restricted weak-type endpoint estimates for the multiplier operator of negative order associated with conic surfaces in ${ \mathbb{R} }^{3} $ which have finite type degeneracy.

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation

$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$

$x, y, z$

$f(n)\gt 0$

$n\geq 2$

$f(n)$

$f(p)$

$n$

$p$

$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$

Let $K$ be a commutative ring with unity, $R$ an associative $K$-algebra of characteristic different from $2$ with unity element and no nonzero nil right ideal, and $f({x}_{1} , \ldots , {x}_{n} )$ a multilinear polynomial over $K$. Assume that, for all $x\in R$ and for all ${r}_{1} , \ldots , {r}_{n} \in R$ there exist integers $m= m(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ and $k= k(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ such that $\mathop{[{x}^{m} , f({r}_{1} , \ldots , {r}_{n} )] }\nolimits_{k} = 0$. We prove that: (1) if $\text{char} (R)= 0$ then $f({x}_{1} , \ldots , {x}_{n} )$ is central-valued on $R$; and (2) if $\text{char} (R)= p\gt 2$ and $f({x}_{1} , \ldots , {x}_{n} )$ is not a polynomial identity in $p\times p$ matrices of characteristic $p$, then $R$ satisfies ${s}_{n+ 2} ({x}_{1} , \ldots , {x}_{n+ 2} )$ and for any ${r}_{1} , \ldots , {r}_{n} \in R$ there exists $t= t({r}_{1} , \ldots , {r}_{n} )\geq 1$ such that ${f}^{{p}^{t} } ({r}_{1} , \ldots , {r}_{n} )\in Z(R)$, the center of $R$.

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis on factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${B}_{n} / n$ when $n$ is divisible by $p- 1$. Using these, we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${B}_{n} / n$ when $n$ is divisible by $p- 1$.

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore, we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite (a key step in the proof is showing that every duality of a thick finite projective plane admits an absolute point). Our results also hold for all finite irreducible spherical buildings of rank at least 3, and imply that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.

We study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

Let $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

Extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a two-parameter collection of kernel functions integrated against measures on the torus. For specific choices of the parameters many families from classical geometric function theory are included. These families include the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others. The main result introduces a surprising new class of extreme points.

Let $R$ be a commutative Noetherian ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$ be an ideal of $R$ such that $\mathfrak{a}M\not = M$. We show among the other things that, if $c$ is a nonnegative integer such that ${ H}_{\mathfrak{a}}^{i} (M)= 0$ for all $i\lt c$, then there is an isomorphism $\mathrm{End} ({ H}_{\mathfrak{a}}^{c} (M))\cong { \mathrm{Ext} }_{R}^{c} ({ H}_{\mathfrak{a}}^{c} (M), M)$; and if $c$ is a nonnegative integer such that ${ H}_{\mathfrak{a}}^{i} (M)= 0$ for all $i\not = c$, there are the following isomorphisms:

(i) $~\quad{ H}_{\mathfrak{b}}^{i} ({ H}_{\mathfrak{a}}^{c} (M))\cong { H}_{\mathfrak{b}}^{i+ c} (M)$ and

(ii) $\quad{ \mathrm{Ext} }_{R}^{i} (R/ \mathfrak{b}, { H}_{\mathfrak{a}}^{c} (M))\cong { \mathrm{Ext} }_{R}^{i+ c} (R/ \mathfrak{b}, M)$

for all $i\in { \mathbb{N} }_{0} $ and all ideals $\mathfrak{b}$ of $R$ with $\mathfrak{b}\supseteq \mathfrak{a}$. We also prove that if $\mathfrak{a}$ and $\mathfrak{b}$ are ideals of $R$ with $\mathfrak{b}\supseteq \mathfrak{a}$ and $c: = \mathrm{grade} (\mathfrak{a}, M)$, then there exists a natural homomorphism from $\mathrm{End} ({ H}_{\mathfrak{a}}^{c} (M))$ to $\mathrm{End} ({ H}_{\mathfrak{b}}^{c} (M))$, where $\mathrm{grade} (\mathfrak{a}, M)$ is the maximum length of $M$-sequences in $\mathfrak{a}$.

We show that complete uniform visibility manifolds of finite volume with sectional curvature $- 1\leq K\leq 0$ have positive simplicial volume. This implies that their minimal volume is nonzero.

Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

We investigate island systems with continuous height functions and strongly laminar systems which are laminar systems containing sets with disjoint boundaries. In the discrete case, we show that for a maximal rectangular system of islands $ \mathcal{H} $ on an $m$ by $n$ rectangular grid we have $\lceil \min (m, n)/ 4\rceil \leq \vert \mathcal{H} \vert \leq \lceil m/ 2\rceil \lceil n/ 2\rceil $. In the continuous case we show that under some conditions maximal strongly laminar systems $ \mathcal{H} $ have cardinality ${\aleph }_{0} $ or ${2}^{{\aleph }_{0} } $ and present examples with $\vert \mathcal{H} \vert = {\aleph }_{0} $.