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.

Fundamental solutions to second-order parabolic and elliptic equations with lower-order terms are studied. New sufficient conditions on the coefficients of the lower-order terms are given that guarantee the validity of two-sided Gaussian bounds on the fundamental solution to the parabolic equation, locally and globally in time. As an application, the problem of existence and non-existence of positive solutions to a class of semi-linear equations is studied.

A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group $G$ and a finitely generated subgroup $H$ of $G$ through a monoid extension that is almost an HNN extension. The FHT property of the extension is closely related to the ${\rm FP}_2$ property for the subgroup $H$, while the FDT property of the extension is related to the finite presentability of $H$. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group.

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.

On the symbolic space endowed with a metric given by a Gibbs measure, it is shown that, for any invariant probability measure $\mu$ other than the given Gibbs measure, the set of $\mu$-generical points satisfies a ‘zero-infinity law’ (in particular, its Hausdorff and packing measure are infinite). This extends a result of R. Kaufman on Besicovitch–Eggleston sets, and applies to level sets of Birkhoff averages and certain subsets of self-similar sets.

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.

A $5$-local characterization of the Lyons sporadic simple group is presented.

A computer construction of the Monster is used to prove that PSL(2, 59) is a subgroup of the Monster.

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 integral closure and the Buchsbaum-Rim multiplicity are computed of a wide class of submodules of $\mathcal O_n^p$ through suitable Newton polyhedra. The result thus obtained is an extension to submodules of $\O^p$ of the works of Yoshinaga and Saia on the characterization of Newton non-degenerate functions and ideals, respectively.

The paper deals with the well posedness of a class of ordinary differential equations. The vector field depends on the solution to a scalar conservation law, whose flux function is assumed to have a single inflection point (from whence ‘nonconvex’ is derived). Filippov solutions to the ordinary differential equations are considered, and Hölder continuous dependence on the initial data is proved. The motivation for the problem is a model of traffic flow.

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.

The classification theorem of compact surfaces states that every topological orientable compact surface is homeomorphic to a sphere or to a ‘torus’ of genus $g$, with $g\,=\,1,2,\dots$. It is proved in the paper that every hyperbolic Riemann surface except for $\bold D\setminus\{0\}$ can be decomposed into basic pieces of only a few different types: Y-pieces, funnels and half-disks. As a corollary of this result, the generalization of the classification theorem to non-compact surfaces is obtained.

Topological properties of dynamical systems admitting a local product structure are studied. The results obtained are applied to hyperbolic systems on covers. The main theorem characterizes transitivity on solvable covers in terms of Frobenius elements of periodic orbits.

A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum energy configuration is studied. As an application, the wandering vectors of the shift operator in the Dirichlet spaces associated with finitely atomic measures are also studied. In particular, the zero locus of a wandering vector is discussed.

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 $C^{\alpha},\,0\,{<}\, \alpha \,{<}\, 1 .$

The $(C,F)$-construction from a previous paper of the first author is applied to produce a number of funny rank one infinite measure preserving actions of discrete countable Abelian groups $G$ with ‘unusual’ multiple recurrence properties. In particular, the following are constructed for each $p\in\Bbb N\cup\{\infty\}$:

1. (i) a $p$-recurrent action $T=(T_g)_{g\in G}$ such that (if $p\ne\infty$) no one transformation $T_g$ is $(p+1)$-recurrent for every element $g$ of infinite order;

2. (ii) an action $T=(T_g)_{g\in G}$ such that for every finite sequence $g_1,\dots,g_r\in G$ without torsion the transformation $T_{g_1}\times\cdots\times T_{g_r}$ is ergodic, $p$-recurrent but (if $p\ne\infty$) not $(p+1)$-recurrent;

3. (iii) a $p$-polynomially recurrent $(C,F)$-transformation which (if $p\ne\infty$) is not $(p+1)$-recurrent.

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.

It is proved that a Banach space $X$ is a subspace of a weakly compactly generated Banach space if and only if, for every $\varepsilon\,{>}\,0$, $X$ can be covered by a countable collection of bounded closed convex symmetric sets where the weak$^*$ closure in $X^{**}$ of each of them lies within the distance $\varepsilon$ from $X$. A new short functional-analytic proof of the known result that a continuous image of an Eberlein compact is Eberlein is given as a corollary.

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.