We consider the free linear Schrödinger equation on a torus 𝕋d , perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping:

Here u = u(t, x), x ∈ 𝕋d , 0 < ν ≪ 1, q∗ ∈ ℕ, f is a positive continuous function, ρ is a positive parameter and are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation

where the nonlinearity F(u) is made out of the resonant terms of the monomial |u| 2q∗ u.

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

Using recent results by Măcinic, Papadima and Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^{1}(F(G),\mathbb{C})$, where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement associated to an irreducible complex reflection group $G$.

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math.94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$. When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$. For $r=\dim \overline{X}$, we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$-cycles with modulus.