We propose a manifestly duality-invariant, Lorentz-invariant, and local action to describe quantum electrodynamics theory in the presence of magnetic monopoles that derives from Sen's formalism. By employing field strengths as the dynamical variables, rather than potentials, this formalism resolves longstanding ambiguities in prior frameworks. Our analysis finds consistent outcomes at both tree and loop levels using the established principles of quantum field theory, obviating the need for external assumptions or amendments. We clarify the mechanisms of charge renormalisation and demonstrate the renormalisation group invariance of the charge quantisation condition. Our approach can be useful for phenomenological studies and in quantum field theories with strong-weak dualities.

This paper introduces PRIMETIME, a synthetic generator that supports both benchmarking and fine-tuning of two primitive operations underlying temporal reasoning in Large Language Models (LLMs): parsing and arithmetic on datetimes. Existing temporal benchmarks assume simplified canonical datetime forms, conflate arithmetic, composition, and world knowledge into a single aggregate score, and offer no direct path to remediation. The first contribution is methodological: the PRIMETIME synthetic generator delivers non-conflated, uncontaminated, and unlimited datetime exemplars that enable a decompositional evaluation strategy for each primitive in isolation. The generator is extensible to support complex datetime tasks and is publicly released, alongside generated benchmarks. The second contribution is diagnostic: under this evaluation strategy, the primitives themselves prove individually unreliable, with per-primitive accuracy ranging from near-zero to perfect across models and prompting conditions. The third contribution is constructive: the same generator used for diagnosis also produces new training exemplars for fine-tuning, and the resulting models show that the primitives are fully learnable and the composed Event Planning task reaches frontier-level accuracy using small quantized LoRA transformers. The broader takeaway is that a single synthetic generator can serve both diagnosis and production-ready deployment. This methodological pattern may apply beyond temporal reasoning.

We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian, \begin{equation}\tag{$\mathcal{P}_{λ,\varepsilon}$} -Δ_p u+\varepsilon(-Δ_p)^s u=λ|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }Ω,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus Ω, \end{equation} where $Ω\subset\mathbb{R}^N$ is a bounded open set, $ε\in(0,1]$, $0<s<1<q<p<N$, and $p^*=\frac{Np}{N-p}$, and $λ\in \mathbb{R}$ is a parameter. For $λ\leq 0$, we show that ($\mathcal{P}_{λ,\varepsilon}$) has no nontrivial solution. For $λ>0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $Λ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has a positive minimal solution for $0<λ<Λ_\varepsilon$, a positive solution for $λ=Λ_\varepsilon$ and no positive solution for $λ>Λ_\varepsilon$. We also prove the existence of $0<λ^\#\leqΛ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has at least two positive solutions for $λ\in(0,λ^\#)$ provided $\varepsilon$ small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from $p=2$ to the general $1<p<N$. Additionally, it extends the classical result of Azorero and Peral (Indiana Univ. Math. J., 43(3):947-957, 1994) to the mixed local-nonlocal quasilinear problems. Moreover, our results complements the multiplicity results for nonnegative solutions in da Silva, et al. (J. Differential Equations, 408:494-536, 2024).

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.

We study the enumeration of graph orientations under local degree constraints. Given a finite graph $G = (V, E)$ and a family of admissible sets $\{\mathsf P_v \subseteq \mathbb{Z} : v \in V\}$, let $\mathcal N (G; \prod_{v \in V} \mathsf P_v)$ denote the number of orientations in which the out-degree of each vertex $v$ lies in $P_v$. We prove a general duality formula expressing $\mathcal N(G; \prod_{v \in V} \mathsf P_v)$ as a signed sum over edge subsets, involving products of coefficient sums associated with $\{\mathsf P_v\}_{v \in V}$, from a family of polynomials. Our approach employs gauge transformations, a technique rooted in statistical physics and holographic algorithms. We also present a probabilistic derivation of the same identity, interpreting the orientation-generating polynomial as the expectation of a random polynomial product. As applications, we obtain explicit formulas for the number of even orientations and for mixed Eulerian-even orientations on general graphs. Our formula generalizes a result of Borbényi and Csikvári on Eulerian orientations of graphs.

Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These important properties, however, are not enjoyed by general probability distributions. This paper surveys what is known about them, providing systematic constructions of examples and counterexamples as well as necessary and sufficient conditions. Novel sufficient conditions for both properties are derived in the context of discrete random variables via information-theoretic tools.

The training of large language models (LLMs) requires substantial computational resources, complex software stacks, and carefully designed workflows to achieve scalability and efficiency. This report presents best practices and insights gained from the OpenGPT-X project, a German initiative focused on developing open, multilingual LLMs optimized for European languages. We detail the use of high-performance computing (HPC) systems, primarily JUWELS Booster at JSC, for training Teuken-7B, a 7-billion-parameter transformer model. The report covers system architecture, training infrastructure, software choices, profiling and benchmarking tools, as well as engineering and operational challenges. It includes measured throughput data of various configurations of 3D parallelism during training and the impact of features such as flash attention.

Being one of the first exoplanets observed by the James Webb Space Telescope, WASP-39 b has become an iconic target and many transit spectra recorded with different instruments (NIRISS, NIRCAM, NIRSpec G395H, NIRSpec PRISM and MIRI) are currently available, allowing in-depth studies of its atmosphere. We present here a novel approach to interpret WASP-39 b's transit spectroscopic data, consisting of a multi-step process where ab initio equilibrium chemistry models and blind retrievals are used iteratively to find physically robust, optimal solutions. Following this approach, we have identified a new scenario to explain WASP-39 b's atmospheric composition, in which silicon-based chemistry plays a major role. In this scenario, SiO may explain the spectral absorption at 4.1 $μ$m, currently interpreted as being due to SO$_2$. SiO and the other gas species identified by the retrieval models, i.e. H$_2$O, CO$_2$, Na and K, are consistent with an atmosphere in chemical equilibrium with a temperature-pressure profile constrained by H$_2$O and CO$_2$ absorption bands. In addition, silicate clouds and hazes can produce the spectral features observed by MIRI in the spectral window 5-12 $μ$m. While we advocate the need for more data, possibly at higher spectral resolution, to confirm our results for WASP-39 b's atmospheric composition, we highlight a refined atmospheric retrieval strategy with pre-selection and post-reconstruction to guide the next generation of transit spectroscopy.

Let $G(x):=\{1/x\}$ be the Gauss map. By $g_n(x)=\frac{1}{x+n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_n$ by limiting the collection of functions $g_k$, $k\in\mathbb N$, to the first $n$ elements, meaning that $G_n = \{g_k \}_{k=1}^n$. We are interested in the asymptotics of the Hausdorff measure of the limit set $J_n$ i. e. set consisting of irrational elements of $[0,1]$ having continued fraction expansion with entries at most $n$. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_n$ is the limit set of the piecewise-linear analogue of $G_n$, $h_n$ is its Hausdorff dimension and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{π^2}, $$ where $J_n$ is the limit set of the system $G_n$, i.e. the set consisting of irrational numbers in $[0,1]$ that continued fraction expansion with entries not exceeding $n$. Similarly as for the piecewise linear map, $h_n$ is the Hausdorff dimension of $J_n$ and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$.

A central puzzle in strongly correlated electronic phases is strange metallic transport, marked by $T$-linear resistivity and $B$-linear magnetoresistance, in sharp contrast with quadratic scalings observed in conventional metals. Here, we demonstrate that proximity to quantum critical points, a recurring motif in the phase diagrams of strange metal candidates, can explain both transport anomalies. We construct and solve a minimal microscopic model by coupling electronic excitations at the Fermi surface to quantum critical bosons via a spatially disordered Yukawa interaction, as well as static pinned domains of density wave order. The resultant transport relaxation rate scales as $k_B T/\hbar$ at low magnetic fields, and as an effective Bohr magneton $\tildeμ_B B/\hbar$ at low temperatures. Further, the magnetoresistance in our model shows a scaling collapse upon rescaling the magnetic field and the resistance by temperature, in agreement with experimental observations.

Digital quantum matter -- realized when discrete quantum gates approximate continuous time evolution -- is susceptible to heating into chaotic, structureless states. If digitization errors are adequately suppressed, a long-lived transient regime of approximately energy-conserving dynamics can be observed on gate-based quantum computers. Conservation of energy, in turn, enables the exploration of a wide variety of complex behaviors observed in equilibrium systems, ranging from the nontrivial microscopic origins of thermalization itself to the stabilization of effective models hosting exotic emergent properties. Here, we use Quantinuum's system model H2 quantum computer to simulate digitized dynamics of the quantum Ising model, suppressing digitization errors well enough to observe thermalization on timescales that severely challenge classical simulation methods. Relaxation of an inhomogeneous state reveals an emergent hydrodynamics due to approximate energy conservation, and we compute the associated diffusion constant. By reprogramming our simulations to take place on a triangular lattice with periodic boundary conditions, we observe thermalization consistent with emergent gauge and topological constraints resulting from lattice frustration. Our results were enabled by continued advances in two-qubit gate quality (native partial entangler fidelities of $99.94(1)\%$), and establish digital quantum computers as powerful tools for studying (effectively) continuous-time dynamics.

A semi-Lagrangian discontinuous finite element scheme based on the characteristic Galerkin method (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of the transport equation and its adjoint equation. First, the existence and stability of CSLDG are proven, along with the uniqueness of the numerical solution. Subsequently, in contrast to the commonly used interpolation-based dimensional splitting schemes (IBS) within the CSLDG framework, a separated-variable dimensional splitting approach based on the tensor product (SVS) is proposed and applied to the two-dimensional case. Numerical experiments show comparable accuracy between methods, but SVS demonstrates superior computational efficiency to IBS, especially on large-scale meshes.

We apply Epstein's construction of hypersurfaces in the hyperbolic disk $\mathbb D$ to prove identities between the Schwarzian action on $\operatorname{PSL}_2(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, the length of the corresponding Epstein curve in $\mathbb D$, and the area enclosed by the Epstein curve. These results are inspired by the holographic duality between Jackiw--Teitelboim gravity and Schwarzian field theory. We also show that the horocycle truncation used in the construction of the Epstein curve defines a renormalized length of hyperbolic geodesics in $\mathbb D$, which coincides with the logarithm of the bi-local observable of Schwarzian field theory. The construction of the Epstein curve also extends to the coadjoint orbits $\operatorname{PSL}_2^{(n)}(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, and we obtain the same identities for the analog of the Schwarzian action on these coadjoint orbits. Furthermore, we show that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve. This energy is the action functional of Schramm--Loewner evolutions and holographically expressed as a renormalized volume in hyperbolic $3$-space. As a by-product of these relations, we obtain two immediate proofs of the non-negativity of the Schwarzian action using the isoperimetric inequality and the monotonicity of the Loewner energy.

Human-AI collaboration is often proposed to improve high-stakes decision-making, yet the influence of increased stakes and imperfect AI on decision-making strategies is not fully understood. Studying such behavior in realistic settings is challenging, as application-grounded evaluations are costly, rely on experts, or lack meaningful consequences for decision errors. To address this, we introduce Blockies, a parametric dataset generator for visual diagnostic tasks, and conduct an empirical study examining how perceived stakes influence reliance calibration and behavior. Results show that raised stakes lead to longer deliberation, but less calibrated reliance, with participants increasingly deferring to incorrect AI advice as decision time increased. These findings highlight that increased effort under higher stakes does not necessarily improve reliance calibration and show the importance of accounting for stakes when evaluating human-AI decision-making.

This paper proposes a novel method to generate bid bounds that can serve as offer caps for energy storage in electricity markets to help reduce system costs and regulate potential market power exercises. We derive the bid bounds based on a tractable multi-period economic dispatch chance-constrained formulation that systematically incorporates the uncertainty and risk preference of the system operator. The key analytical results verify that the bounds effectively cap storage bids across all uncertainty scenarios with a guaranteed confidence level. We show that bid bounds decrease as the state of charge increases but rise with greater netload uncertainty and risk preference. We test the effectiveness of the proposed pricing mechanism based on the 8-bus ISO-NE test system, including agent-based storage bidding models. Simulation results demonstrate that the proposed bid bounds effectively align storage bids with the social welfare objective and outperform existing deterministic bid bounds. Under 30% renewable capacity and 20% storage capacity, the bid bounds contribute to an average reduction of 0.17% in system cost, while increasing storage profit by an average of 10.16% across various system uncertainty scenarios and bidding strategies. These benefits scale up with increased storage economic withholding and storage capacity.

We present an inclusive search for anomalous production of single-photon events from neutrino interactions in the MicroBooNE experiment. The search and its signal definition are motivated by the previous observation of a low-energy excess of electromagnetic shower events from the MiniBooNE experiment. We use the Wire-Cell reconstruction framework to select a sample of inclusive single-photon final-state interactions with a final efficiency and purity of 7.0% and 40.2%, respectively. We leverage simultaneous measurements of sidebands of charged current $ν_μ$ interactions and neutral current interactions producing $π^{0}$ mesons to constrain signal and background predictions and reduce uncertainties. We perform a blind analysis using a dataset collected from February 2016 to July 2018, corresponding to an exposure of $6.34\times10^{20}$ protons on target from the Booster Neutrino Beam (BNB) at Fermilab. In the full signal region, we observe agreement between the data and the prediction, with a goodness-of-fit $p$-value of 0.11. We then isolate a sub-sample of these events containing no visible protons, and observe $93\pm22\text{(stat.)}\pm35\text{(syst.)}$ data events above prediction, corresponding to just above $2σ$ local significance, concentrated at shower energies below 600 MeV.

We prove Gamma conjecture I for all flag varieties by following a strategy proposed by Galkin and Iritani. The main new ingredient is showing that the totally positive part of the Rietsch mirror is mirror to the $\widehatΓ$-class and contains the critical point of the superpotential that corresponds to the Perron-Frobenius eigenvalue on the A-side.

IQPopt is a software package designed to optimize large-scale instantaneous quantum polynomial circuits on classical hardware. By exploiting an efficient classical simulation algorithm for expectation value estimation, circuits with thousands of qubits and millions of gates can be optimized, provided the relevant objective function has an efficient description in terms of Pauli-Z type observables. Since sampling from instantaneous quantum polynomial circuits is widely believed to be hard for classical computers, this provides a method to identify powerful circuit instances before deployment and sampling on quantum hardware, where computational advantages may exist. The package leverages automatic differentiation in JAX, can be accelerated with access to hardware accelerators such as graphics processing units, and contains a dedicated module that can be used to train and evaluate quantum generative models via the maximum mean discrepancy.

A novel machine-learning-based flavor-tagging algorithm combining same-side and opposite-side tagging is used to obtain the equivalent of 27$\,$000 tagged B$^0_\mathrm{s}$ $\to$ J/$ψ\, ϕ$(1020) decays from pp collisions at $\sqrt{s}$ $=$ 13 TeV, collected by the CMS experiment and corresponding to an integrated luminosity of 96.5 fb$^{-1}$. A time- and flavor-dependent angular analysis of the $μ^+μ^-$K$^+$K$^-$ final state, consistent with a $ϕ$(1020) $\to$ K$^+$K$^-$ decay, is used to measure parameters of the $\mathrm{B}^0_\mathrm{s}-\overline{\mathrm{B}}^0_\mathrm{s}$ system. The weak phase is measured to be $ϕ_\mathrm{s}$ = $-$73 $\pm$ 22 (stat) $\pm$ 10 (syst) mrad, which, combined with the $\sqrt{s}$ $=$ 8 TeV CMS result, gives $ϕ_\mathrm{s}$ = $-$75 $\pm$ 23 mrad. This value differs from zero by 3.2 standard deviations, providing the first evidence for mixing-induced $CP$ violation in B$^0_\mathrm{s}$ $\to$ J/$ψ\,ϕ$(1020) decays. All measured physics parameters are found to agree with standard model predictions where available.

We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direction. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.

This book offers a comprehensive introduction to spectral networks from a unified viewpoint that bridges geometry with the physics of supersymmetric gauge theories. It provides the foundational background needed to approach the frontiers of this rapidly evolving field, treating geometric and physical aspects in parallel. After surveying fundamental topics in algebra and geometry, a detailed introduction to higher-rank Teichmüller theory is developed, including Fock-Goncharov theory for Hitchin representations, maximal representations and the more recent notion of $Θ$-positivity. Spectral networks are subsequently introduced, emphasizing their utility in the study of character varieties via the abelianization and non-abelianization maps they define. In parallel, key aspects of four-dimensional gauge dynamics with eight supercharges are explored, including electric-magnetic duality, Seiberg-Witten theory, and class $\mathcal S$ theories. The role of spectral networks as a framework for determining and analyzing BPS spectra in class $\mathcal S$ theories is then examined. The final chapter outlines recent applications of spectral networks across a range of contemporary research areas. This volume is intended for researchers and advanced students in either mathematics or physics who wish to enter the field.

We consider the problem of testing the mean of a bounded real random variable. We introduce a notion of optimal classes for e-variables and e-processes, and establish the optimality of the coin-betting formulation among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean. As a consequence, we provide a direct and explicit characterisation of all valid e-variables and e-processes for this testing problem. In the language of classical statistical decision theory, we fully describe the set of all admissible e-variables and e-processes, and identify the corresponding minimal complete class.

Inexact proximal augmented Lagrangian methods (ipALMs) have been widely used for solving linearly constrained convex optimization problems, owing to their strong theoretical guarantees and excellent numerical performance. In practice, however, existing ipALMs typically employ Rockafellar-type absolute error criteria for solving the subproblems, which require delicate problem-dependent tuning of error-tolerance sequences. In this paper, we propose ripALM, a relative-type ipALM whose subproblem error criterion has only a \textit{single} tolerance parameter in $[0,1)$. This makes the method simpler to implement and less sensitive to parameter tuning in practice. On the other hand, the use of such a relative-type error criterion renders the convergence of our ripALM beyond the scope of the convergence theory of existing ipALMs. To address this gap, we develop a new analysis framework under which ripALM is shown to admit desirable global convergence properties and it achieves an asymptotic (super)linear convergence rate under a standard error bound condition. While there exist other relative-type inexact pALMs, to ensure convergence, they require additional correct steps that generally impede the convergence speed. To the best of our knowledge, ripALM is the first relative-type inexact version of the vanilla pALM that avoids both summable tolerance parameter sequences and correction steps, while retaining rigorous convergence guarantees. Numerical experiments on quadratically regularized optimal transport and basis pursuit denoising problems demonstrate the effectiveness and robustness of our proposed method.

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation snarks, i.e. cycle permutation graphs that do not admit a $3$-edge-colouring. This allows us to generate all cycle permutation graphs up to order $34$ and all permutation snarks up to order $46$, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order $6 \bmod 8$ or a smallest permutation snark of girth at least $6$ and give more evidence in support of a conjecture of Goddyn. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to conjectures by Jackson and Zhang.

The $Λ$ Cold Dark Matter ($Λ$CDM) cosmological model has been highly successful in predicting cosmic structure and evolution, yet recent precision measurements have highlighted discrepancies, especially in the Hubble constant inferred from local and early-Universe data. Gamma-ray bursts (GRBs) present a promising alternative for cosmological measurements, capable of reaching higher redshifts than traditional distance indicators. This work leverages GRBs to refine cosmological parameters independently of the $Λ$CDM framework. Using the Platinum compilation of long GRBs, we calibrate the Dainotti relations-empirical correlations among GRB luminosity properties-as standard candles through artificial neural networks (ANNs). We analyze both the 2D and 3D Dainotti calibration relations, leveraging an ANN-driven Markov Chain Monte Carlo approach to minimize scatter in the calibration parameters, thereby achieving a stable Hubble diagram. This ANN-based calibration approach offers advantages over Gaussian processes, avoiding issues such as kernel function dependence and overfitting. Our results emphasize the need for model-independent calibration approaches to address systematic challenges in GRB luminosity variability, ultimately extending the cosmic distance ladder in a robust way. By addressing redshift evolution and reducing systematic uncertainties, GRBs can serve as reliable high-redshift distance indicators, offering critical insights into current cosmological tensions.

We characterise the Schatten class $S^p$ properties of commutators $[b,T]$ of singular integrals and pointwise multipliers in a general framework of (quasi-)metric measure spaces. This covers, unifies, and extends a range of previous results in different special cases. As in the classical results on $\mathbb R^d$, the characterisation has three parts: (1) For $p>d$, we have $[b,T]\in S^p$ if and only if $b$ is in a suitable Besov (or fractional Sobolev) space. (2) For $p\leq d$, we have $[b,T]\in S^p$ if and only if $b$ is constant. (3) For $p=d$, we have $[b,T]\in S^{d,\infty}$ (a weak-type Schatten class) if and only if $b$ is in a first-order Sobolev space. Result (1) extends to all spaces of homogeneous type as long as there are appropriate singular integrals, but for the more delicate properties (2) and (3), we assume a complete doubling metric space supporting a suitable Poincaré inequality, which is still very general. These latter results depend on new characterisations of constant functions and Sobolev spaces over such spaces obtained in a companion paper of the author with R. Korte. Even when specialised to various concrete domains considered earlier, the present results extend ones available in the literature by covering a larger class of operators with minimal kernel assumptions, removing a-priori assumptions on the pointwise multiplier $b$, and allowing Schatten classes on the weighted spaces $L^2(w)$ with an arbitrary Muckenhoupt weight $w\in A_2$. Even on $\mathbb R^d$, such weighted results were previously known for a few special operators $T$ only, and on all other domains, they are completely new.

We consider the canonical Wiener-Hopf factorisation of $2 \times 2$ symmetric matrices $\mathcal M$ with respect to a contour $Γ$. For the case that the quotient $q$ of the two diagonal elements of $\mathcal M$ is a rational function, we show that due to the symmetric nature of the matrix $\mathcal M$, the second column in each of the two matrix factors that arise in the factorisation is determined in terms of the first column in each of these matrix factors, by multiplication by a rational matrix, and we give a method for determining the second columns of these factors. We illustrate our method with two examples in the context of a Riemann-Hilbert approach to obtaining solutions to the Einstein field equations.

This paper proposes a novel distributed approach for solving a cooperative Constrained Multi-agent Reinforcement Learning (CMARL) problem, where agents seek to minimize a global objective function subject to shared constraints. Unlike existing methods that rely on centralized training or coordination, our approach enables fully decentralized online learning, with each agent maintaining local estimates of both primal and dual variables. Specifically, we develop a distributed primal-dual algorithm based on actor-critic methods, leveraging local information to estimate Lagrangian multipliers. We establish consensus among the Lagrangian multipliers across agents and prove the convergence of our algorithm to an equilibrium point, analyzing the sub-optimality of this equilibrium compared to the exact solution of the unparameterized problem. Furthermore, we introduce a constrained cooperative Cournot game with stochastic dynamics as a test environment to evaluate the algorithm's performance in complex, real-world scenarios.

Accurate prediction of remaining useful life under creep conditions is essential for the structural reliability of high-temperature components in critical engineering systems. Traditional approaches based on deterministic parametric models often overlook the substantial variability inherent in experimental data, compromising the accuracy and robustness of long-term predictions. This study introduces a probabilistic framework to quantify uncertainties in creep rupture time prediction. Robust regression techniques are first applied to mitigate the influence of outliers and enhance the stability of model estimates. Global sensitivity analysis using Sobol indices is then employed to identify the dominant contributors to model uncertainty, followed by Monte Carlo simulations to propagate these uncertainties and estimate the distribution of the remaining useful life. Finally, model selection is guided by statistical criteria, including the Akaike and Bayesian information criteria, to identify the most reliable predictive model. The proposed framework not only enables the definition of safe operational limits with quantifiable confidence levels but is also general and extensible to other time-dependent degradation phenomena, such as fatigue and creep-fatigue interaction.

A phylogenetic tree is an edge-weighted binary tree, with leaves labelled by a collection of species, that represents the evolutionary relationships between those species. For such a tree, a phylogenetic diversity index is a function that apportions the biodiversity of the collection across its constituent species. The diversity index polytope is the convex hull of the images of phylogenetic diversity indices. We study the combinatorics of phylogenetic diversity indices to provide a minimal compact description of the diversity index polytope. Furthermore, we discuss extensions of the polytope to expand the study of biodiversity measurement.