The main goals of the paper are to establish a priori bounds for the prime power divisors of the discriminants of spinor regular positive definite primitive integral ternary quadratic lattices, and to describe a procedure for determining all such lattices.

One-parameter overpartition-theoretic analogues are given of two classical families of partition identities: Andrews' combinatorial generalization of the Gollnitz–Gordon identities and a theorem of Andrews and Santos on partitions with attached odd parts. Geometric counterparts arising from multiple $q$-series identities are also discussed. These involve representations of overpartitions in terms of generalized Frobenius partitions.

If $K$ is an algebraic function field of one variable over an algebraically closed field $k$ and $F$ is a finite extension of $K$, then any element $a$ of $K$ can be written as a norm of some $b$ in $F$ by Tsen's theorem. All zeros and poles of $a$ lead to zeros and poles of $b$, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of $b$ can be restricted in terms of the genus of $K$, respectively $F$. If $k$ is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemann surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic $p>0$ using standard arguments from valuation theory.

A previous conjecture is verified for any normal surface singularity which admits a good ${\mathbb C}^*$-action. This result connects the Seiberg–Witten invariant of the link (associated with a certain ‘canonical’ spin$^c$ structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere.

As an application, a topological interpretation is found of the generalized Batyrev stringy invariant (in the sense of Veys) associated with such a singularity.

The result is partly based on the computation of the Reidemeister–Turaev sign-refined torsion and the Seiberg–Witten invariant (associated with any spin$^c$ structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero.

Let $U^+$ be the plus part of the quantized enveloping algebra of a simple Lie algebra of type $A_n$ and let $\mathcal{B}^*$ be the dual canonical basis of $U^+$. Let $b$, $b'$ be in $\mathcal{B}^*$, and suppose that one of the two elements is a $q$-commuting product of quantum flag minors. It is shown that $b$ and $b'$ are multiplicative if and only if they $q$-commute.

Let ${\mathcal H}_{q}(d)$ be the Iwahori–Hecke algebra of the symmetric group, where $q$ is a primitive $l$th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for $l \geq 4$ the multiplicities in a Specht or dual Specht module filtration of an ${\mathcal H}_q(d)$-module are well defined. A cohomological criterion is given for when an ${\mathcal H}_{q}(d)$-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting $q = 1$, results are obtained for the symmetric group in characteristic $p \geq 5$. These results are false in general for $p = 2$ or $3$.

Ramanujan claimed in his first letter to Hardy (16 January 1913) that $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!}\bigg(\frac{1}{3}+\frac{4}{135(k+\theta(k))}\bigg) \qquad{(k=1,2,{\ldots})},$$ where $\theta(k)$ lies between $2/21$ and $8/45$. This conjecture was proved in 1995 by Flajolet et al. The paper establishes the following refinement. $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!} \bigg(\frac{1}{3}+\frac{4}{135 k}-\frac{8}{2835(k+\theta^*(k))^2}\bigg) \qquad{(k=1,2,{\ldots})},$$ where $$-\frac{1}{3}<\theta^*(k)\leq -1+\frac{4}{\sqrt{21(368-135e)}}=-0.140\,74{\ldots}\,.$$ Both bounds for $\theta^*(k)$ are sharp.

For the $p$-Laplacian $\Delta_p v \,{=}\, {\rm div}\:(|\nabla v|^{p-2}\nabla v)$, $p\,{>}\,1$, the eigenvalue problem $-\Delta_p v + q(|x|)|v|^{p-2}v \,{=}\, \lambda |v|^{p-2}v$ in $\R^n$ is considered under the assumption of radial symmetry. For a first class of potentials $q(r)\,{\to}\,\infty$ as $r\,{\to}\,\infty$ at a sufficiently fast rate, the existence of a sequence of eigenvalues $\lambda_k\,{\to}\,\infty$ if $k\,{\to}\,\infty$ is shown with eigenfunctions belonging to $L^p(\R^n)$. In the case $p\,{=}\,2$, this corresponds to Weyl's limit point theory. For a second class of power-like potentials $q(r)\,{\to}\,{-}\infty$ as $r\,{\to}\,\infty$ at a sufficiently fast rate, it is shown that, under an additional boundary condition at $r\,{=}\,\infty$, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues $\lambda_k$ with $\lambda_k \,{\to}\,\pm \infty$ if $k\,{\to}\,\pm\infty$. In this case, every solution of the initial value problem belongs to $L^p(\R^n)$. For $p\,{=}\,2$, this situation corresponds to Weyl's limit circle theory.

A class of rational functions and sufficient conditions on it such that the Julia set is a Jordan arc, respectively no Jordan arc, are introduced. Further, it is shown that any rational function which has a Jordan arc as Julia set belongs up to Möbius conjugation to this class.

The theory of higher-dimensional shifts of finite type is still largely an open area of investigation. Recent years have seen much activity, but fundamental questions remain unanswered. In this paper we consider the following basic question. Given a shift of finite type (SFT), under what topological mixing conditions are we guaranteed the existence of Bernoulli (or even $K$, mixing, or weakly mixing) invariant measures?

The space of admissible vector fields, consistent with the structure of a network of coupled dynamical systems, can be specified in terms of the network's symmetry groupoid. The symmetry groupoid also determines the robust patterns of synchrony in the network – those that arise because of the network topology. In particular, synchronous cells can be identified in a canonical manner to yield a quotient network. Admissible vector fields on the original network induce admissible vector fields on the quotient, and any dynamical state of such an induced vector field can be lifted to the original network, yielding an analogous state in which certain sets of cells are synchronized. In the paper, necessary and sufficient conditions are specified for all admissible vector fields on the quotient to lift in this manner. These conditions are combinatorial in nature, and the proof uses invariant theory for the symmetric group. Also the symmetry groupoid of a quotient is related to that of the original network, and it is shown that there is a close analogy with the usual normalizer symmetry that arises in group-equivariant dynamics.

Operator-valued Fourier multiplier theorems are used to establish maximal regularity results for an integro-differential equation with infinite delay in Banach spaces. Results are obtained under general conditions for periodic solutions in the vector-valued Lebesgue and Besov spaces. The latter scale includes in particular the Hölder spaces $C^{\alpha},\,0\,{<}\, \alpha \,{<}\, 1 .$

For any closed oriented manifold $M$, the top degree multi-vector fields transverse to the zero section of $\wedge^{{\rm top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of their zero locus and a finite number of numerical invariants. The group governing the infinitesimal deformations of such multi-vector fields is computed, and an explicit set of generators exhibited. For the sphere $S^n$, a correspondence between certain isotopy classes of multi-vector fields and classes of weighted signed trees is established.

Let $l$ be an oriented link of $d$ components in a homology $3$-sphere. For any nonnegative integer $q$, let $l(q)$ be the link of $d-1$ components obtained from $l$ by performing $1/q$ surgery on its $d$th component $l_d$. The Mahler measure of the multivariable Alexander polynomial $\Delta_{l(q)}$ converges to the Mahler measure of $\Delta_l$ as $q$ goes to infinity, provided that $l_d$ has nonzero linking number with some other component. If $l_d$ has zero linking number with each of the other components, then the Mahler measure of $\Delta_{l(q)}$ has a well defined but different limiting behavior. Examples are given of links $l$ such that the Mahler measure of $\Delta_l$ is small. Possible connections with hyperbolic volume are discussed.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

A necessary and sufficient condition for the almost sure existence of the exact packing measure on the boundary of a Galton–Watson tree is obtained with an explicit exact packing measure. In the case where the exact packing measure does not exist almost surely, a criterion which classifies functions $\ph$ according to whether the $\ph$ packing measure is zero or infinity is given.