We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.

Let $A$ be any one of the three elliptic curves over $\mathbb{Q}$ with conductor 11. We show that $A$ has Mordell–Weil rank zero over its field of 5-division points. In each case we also compute the 5-primary part of the Tate–Shafarevich group. Our calculations make use of the Galois equivariance of the Cassels–Tate pairing.

Given an algorithm for resolution of singularities that satisfies certain conditions (‘a good algorithm’), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_j$, such that the induced family on each part $T_j$ is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families.

Let $R$ be a ring and $M_R$ be an $R$-module. We characterize the existence of ${\rm Add} M$-covers and ${\rm Add} M$-envelopes in terms of finiteness conditions of $M$ over its endomorphism ring. We then present some applications related to the existence of well-behaved direct sum decompositions for direct products of copies of $M$. Our results can be viewed as natural extensions of classical theorems of Bass and Chase on coherent and perfect rings.

We prove the following: if a group $\Gamma$ is torsion-free, and relatively hyperbolic (with the Bounded Coset Penetration property), relative to a subgroup admitting a finite classifying space, then $\Gamma$ admits a finite classifying space. In this case, if the subgroup admits a boundary in the sense of $\mathcal{Z}$-structures, we prove that $\Gamma$ admits a boundary. This extends classical results of Rips, and of Bestvina and Mess to the relative case.

We study Rouquier blocks of symmetric groups and Schur algebras in detail, and obtain explicit descriptions for the radical layers of the principal indecomposable, Weyl, Young and Specht modules of these blocks. At the same time, the Jantzen filtrations of the Weyl modules are shown to coincide with their radical filtrations. We also address the conjectures of Martin, Lascoux–Leclerc–Thibon–Rouquier and James for these blocks.

The definition of Thom polynomials for Lagrange, Legendre and critical point function singularities is given. The approach is based on the notion of classifying space of singularities. This approach provides a universal method for computing Thom polynomials. Characteristic classes of complex Lagrange and Legendre singularities of small codimension are computed. Two independent methods for these computations are presented. The first is the direct one based on the resolution of singularities, the Gysin homomorphism, and the notion of adjacency exponents. The second uses the existence theorem for Thom polynomials and is based on Rimányi's idea of using symmetries. The expressions for the resulting polynomials reduced modulo 2 agree with those obtained by Vassiliev for the real case.

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well-known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest.

For any non-empty subset $\mathcal I$ of the natural numbers, let $\Lambda_{\mathcal I}$ denote those numbers in the unit interval whose continued fraction digits all lie in $\mathcal I$. Define the corresponding transfer operator

$$ \mathcal L_{\mathcal I, \beta} f(z) = \sum_{n \in \mathcal I} \left( \frac{1}{n + z} \right)^{2 \beta} f\left( \frac{1}{n + z} \right) $$

for $\text{Re} (\beta) > \max (0, \theta_{\mathcal I})$, where $\text{Re} (\beta) = \theta_{\mathcal I}$ is the abscissa of convergence of the series $\sum_{n \in \mathcal I}n^{-2 \beta}$.

When acting on a certain Hilbert space $\mathcal H_{\mathcal I, \beta}$, we show that the operator $\mathcal L_{\mathcal I, \beta}$ is conjugate to an integral operator $\mathcal K_{\mathcal I, \beta}$. If furthermore $\beta$ is real, then $\mathcal K_{\mathcal I, \beta}$ is selfadjoint, so that $\mathcal L_{\mathcal I, \beta} : \mathcal H_{\mathcal I, \beta} \to \mathcal H_{\mathcal I, \beta}$ has purely real spectrum. It is proved that $\mathcal L_{\mathcal I, \beta}$ also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space $C^\omega [0, 1]$, and on the Fréchet space $C^\infty [0, 1]$.

The analytic properties of the map $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ are investigated. For certain alphabets $\mathcal I$ of an arithmetic nature (for example, $\mathcal I = \{\text{primes}\}$, $\mathcal I = \{\text{squares}\}$, $\mathcal I$ an arithmetic progression, $\mathcal I$ the set of sums of two squares) it is shown that $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ admits an analytic continuation beyond the half-plane $\text{Re} (\beta) > \theta_{\mathcal I}$.

We derive absolute stability results of Popov-type for infinite-dimensional systems in an input-output setting. Our results apply to feedback systems where the linear part is the series interconnection of an $L^2$-stable linear system and an integrator, and the non-linearity satisfies a sector condition which allows for non-linearities with lower gain equal to zero (such as saturation, or more generally, bounded non-linearities). These results are used to prove convergence and stability properties of low-gain integral feedback control applied to $L^2$-stable linear systems subject to actuator and sensor non-linearities. The class of actuator/sensor non-linearities under consideration contains standard non-linearities which are important in control engineering such as saturation and deadzone. Moreover, we use the input-output theory developed to derive state-space results on absolute stability and low-gain integral control for strongly stable well-posed infinite-dimensional linear systems.