Research Council of Norway

We prove that for any two lattices L, M C_ Rd of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set E C_ Rd such that E tiles Rd when translated by L or by M. A consequence of this is that the indicator function of E forms a Weyl-Heisenb erg (Gabor) orthogonal basis of L2(Rd) when translated by L and modulated by M & lowast;, the dual lattice of M. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

A framework for risk-averse optimization problems is introduced that is resilient to ambiguities in the true form of the underlying probability distribution. The focus is on problems with partial differential equations (PDEs) as constraints, although the formulation is more broadly applicable. The framework is based on combining risk measures with problem relaxation techniques, and it builds off of previous advances for risk-neutral problems. This work advances the existing theory with strengthened Γ-convergence results, novel existence results and first-order optimality criteria. In particular, the theoretical approach naturally accommodates infinite-dimensional probability spaces; no finite-dimensional noise assumption is needed. The framework blends aspects of both distributionally robust optimization (DRO) and distributionally optimistic optimization (DOO) approaches. The DRO aspect facilitates strong out-of-sample performance, while the DOO aspect takes care of adversarial and outlier data, as illustrated with numerical examples.

We prove a Littlewood-Paley formula for the Hardy space of Dirichlet series ℋ^p with 1≤ p<∞ in terms of almost every vertical limit function. This significantly strengthens previous results, which hold either only as an average over the vertical limit functions or under additional assumptions of uniform convergence. As part of our approach, we obtain a Hardy-Stein identity for the derivative of the p-mean of almost every vertical limit. We further show that the mean counting function exists for any f in ℋ^p in terms of almost all of its vertical limit functions. This is done by establishing a version of Jensen's formula in this setting. In the process, we also deduce ergodic versions of Fatou's lemma and the monotone and dominated convergence theorems for the Kronecker flow.

We revisit the problem of rigorously and deterministically finding elements of large order in the multiplicative group of integers modulo a natural number $N$. Solving this problem is an essential step in several recent deterministic algorithms for factoring $N$, including the currently fastest ones. In 2018, the second author gave an algorithm that for a given target order $D \geq N^{2/5}$, finds either an element of order exceeding $D$, or a nontrivial divisor of $N$, or proves that $N$ is prime. The running time was \[ O\left(\frac{D^{1/2}}{(\log \log D)^{1/2}} \log^2 N \right) \] bit operations, asymptotically the same as the cost of computing the order of a single element using Sutherland's optimisation of the classical babystep-giantstep method. Subsequent work by several authors weakened the hypothesis $D \geq N^{2/5}$ to $D \geq N^{1/6}$. In this paper, we show that the hypothesis may be dropped altogether. Moreover, if $N$ is prime, we can guarantee returning an element of order exceeding $D$, rather than a proof that $N$ is prime.

Renewable energy power plants are increasingly expected to provide ancillary services to the grid. Yet, the variability of sources such as solar photovoltaics requires additional operational flexibility to deliver these services. In this context, hybrid energy storage systems (HESSs) offer an appropriate solution, because different technologies with complementary characteristics can be applied and leveraged to share the power demand and improve the overall performance. Yet, a proper power sharing requires a detailed model of the storage elements. This aspect has barely been studied in the literature, where most models consider constant efficiencies. Moreover, most formulations are based on optimisation problems that are computationally too demanding to be solved in real time. In this paper, a controller is proposed to minimise the losses of a HESS considering detailed power-dependent efficiency curves of each storage technology. The power sharing method is based on the analytical verification of the Karush-Kuhn-Tucker (KKT) conditions, which makes it computationally efficient and suitable for real-world deployment. The proposed controller is applied to a test case consisting of a 10 MW PV power plant with a HESS based on a lithium-ion battery and a redox-flow battery, each with 2.5 MW power and 5 MWh capacity. The main contributions are verified via numerical simulations performed in MATLAB/Simulink, whereas a real-time implementation deployed in OPAL-RT demonstrates the viability of the algorithm for real-time applications.