Let $\mathcal{A}$ denote the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$ and satisfy the normalization conditions $f(0) = 0$ and $f'(0) = 1$. This paper investigates the inverse logarithmic coefficients $Γ_n$, which are defined by the expansion $\log(f^{-1}(w)/w) = 2\sum_{n=1}^{\infty} Γ_n w^n$. We establish sharp upper and lower bounds for the difference of the moduli of the first two inverse logarithmic coefficients, $|Γ_2| - |Γ_1|$, for several significant subclasses of univalent functions. Specifically, we derive sharp estimates for functions belonging to the class of starlike functions with respect to symmetric points ($\mathcal{S}_S^*$), convex functions with respect to symmetric points ($\mathcal{K}_S$), and functions associated with the lune domain ($\mathcal{S}_{\leftmoon}^*$ and $\mathcal{C}_{\leftmoon}$). The results are obtained by employing subordination techniques and utilizing sharp estimates for the coefficients of Schwarz functions. In each case, the extremal functions that attain these bounds are explicitly identified. Our findings provide further insights into the geometric properties of inverse mappings and extend recent research on coefficient functionals in geometric function theory.

Conditions ensuring optimal parameter estimation in the presence of missing data are well established in inference, typically relying on the Missing-at-Random (MAR) assumption. In prediction, similar principles are often assumed to apply. However, methods considered biased in inference, such as pattern sub-modelling or unconditional imputation, have been shown to achieve optimal predictive performance under any missingness mechanism, including non-MAR (MNAR). To explain this apparent contradiction, we introduce a new formal framework for describing missingness in prediction. Central to this framework is a distinction between two prediction targets, defined according to whether or not the indicator of observation of the predictors is exploited to predict the outcome. This distinction leads to a classification of the missingness mechanisms describing the conditions under which these targets are equal, and when consistent prediction of each is achievable. A key result is that both targets may be consistently predicted under conditions weaker than MAR. We discuss the implications of this paradigm for handling missing data in prediction, distinguishing between missingness at development, validation and deployment of a forecaster. The findings are illustrated using simulated data and a real-world application with the prediction of significant injury after trauma upon arrival at the emergency department.

In this paper we give a fully combinatorial description of the zero entropy periodic patterns on trees. Unlike previously known characterizations of such patterns, our criterion is independent of any particular topological realization of the pattern and provides, thus, a practical and fast algorithm to test zero entropy. As an application, consider a $k$-star $T$ (a tree with $k$ edges attached at a unique branching point of valence $k$) and the set $\mathcal{F}_{n,k}$ of all continuous maps $\map{f}{T}$ having a periodic orbit of period $n$ properly contained in $T$ (each edge of $T$ contains at least one point of the orbit). We find all pairs $(n,k)$ such that $\mathcal{F}_{n,k}$ contains maps of entropy zero, and we describe the patterns of such zero-entropy orbits.

Along with the increased availability and capabilities of quantum computers comes the core question: how can access to quantum computing be allocated in a responsible way? This report introduces a general legal-ethical framework that providers of access to quantum computing can apply to develop robust access policies tailored to their specific context. We demonstrate the applicability of this general legal-ethical framework in the specific context of a small, 16-qubit quantum computer that will be hosted by SURF (the Dutch IT cooperative for research and education), integrated with Snellius (the Dutch national supercomputer), and operated jointly as part of the EuroSSQ-HPC consortium, procured in partnership with the EU-wide EuroHPC Joint Undertaking.

We investigate fractional revival in graphs with respect to the adjacency, Laplacian, and signless Laplacian matrices. We observe that, under certain conditions, fractional revival is preserved under graph complementation. Then we establish a connection between fractional revival in a graph and in its double cover, and obtain a complete characterization of pretty good plus state transfer in cycles and their complements. This leads to characterizations of pretty good vertex state transfer in weighted paths with potential.

Defects, fluctuations, degenerate states and correlated interactions facilitate the emergence of exotic properties in condensed matter systems while also inducing atomic-scale local correlated structures that deviate from the average long-range order. Establishing the structure-property relationship from the perspective of these atomic-scale local correlated structures remains ambiguous and controversial due to the lack of direct methods for identifying such local correlated structures. In this work, based on the photoexcited ultrafast structural response, we propose a Bragg scattering phase breaking regime to identify the sub-angstrom local correlated structures in quantum materials. With this regime, we unambiguously identify the many-body-interaction driven local correlated structures with static off-center Ag displacements of 0 to 0.5 angstrom in the low temperature ground state of AgCrSe2. As temperature rising, these static local correlated structures transform to a dynamic state where the thermal fluctuations overwhelm the multiple localized quantum states, signifying the strong anharmonicity of the local structures. The state-of-the-art density functional theory simulation well reproduces the intrinsic many-body-interaction driven local correlated structures. These unique local correlated structures evidence the first many-body localization with topological order characteristic in real material systems and provide a unified scenario for the versatile quantum properties in single crystalline AgCrSe2. Our work not only offers a universal approach to characterize sub-angstrom local correlated structures across a wide range of quantum materials but also deepens our understanding of the fundamental mechanism behind exotic properties from the perspective of atomic-scale local correlated structures.

Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally (across all scales and locations) well-approximated by harmonic functions. A similar characterisation is obtained which further takes into account the local oscillations of the approximating harmonic functions. The first characterisation is new even when $X = \mathbb{R}^n$; the second characterisation is a version of Dorronsoro's Theorem in RCD spaces and gives a new proof of (a special case) of this theorem in Euclidean space.

We propose our framework as an effective field theory valid below the cut-off scale $Λ$, in which we explain the tiny scale of neutrino masses by integrating a non-invertible symmetry with the dynamical determination of the cut-off scale. In our model, we introduce three families of SU(2)$_L$ quintet fermions ($Σ_R$) and a quartet scalar ($ϕ_4$), both of which are charged under the Fibonacci fusion rule (FFR). A central feature of this construction is that the vacuum expectation value (VEV) $v_4$ of $ϕ_4$ is induced at the one-loop level via dynamical symmetry breaking. To resolve the inherent arbitrariness of the cut-off scale $Λ$ in loop-induced VEV models, we identify $Λ$ with the scale at which the SU(2)$_L$ gauge coupling $g_2$ encounters the renormalization group evolution (RGE) of $g_2$, naturally fixing the physical cut-off within the range of approximately $10^5$ to $10^7$ GeV. Quantitatively, this framework yields $v_4\sim 0.07-0.1$ GeV, which in turn leads to neutrino Yukawa couplings on the order of $10^{-3}$. This result provides a significantly more natural explanation for the neutrino mass hierarchy compared to standard seesaw models, which typically require couplings as small as $10^{-6}$ or less. Notably, our approach maintains a relatively simple particle content and does not necessitate additional (gauge) bosons for symmetry breaking.

High-pressure experiments using diamond anvils have revealed novel properties and phase behavior of H2O under extreme conditions. When contained in diamond-anvil cells, the H2O samples are usually in direct contact with the diamond anvil. However, the extent to which this interface affects measured pressure-induced properties and behavior, including coexistence lines of ice phases, remains unknown. Combining artificial neural network methods and active learning schemes with large-scale molecular dynamics simulations, we elucidate the interfacial effects on various properties of high-pressure ice phases, including superionic states, solid-solid phase transitions, and melting. The results reveal that the presence of this interface can significantly lower the hydrogen superionic transition temperature. Remarkably, the interface can also induce a spontaneous transition from bcc- to fcc-based ice following the inverse Bain mechanism. Further, we redefined a stability field of bcc and fcc ice below the melting line and predicted the existence of fcc ice at much lower pressures than previously thought. More broadly, the results emphasize the importance of interface effects in understanding a wide range of phenomena reported in experimental studies of ice under pressure, including inconsistencies between theoretical and experimental results of this fundamental system.

This paper investigates the asymptotic behavior of a hyperbolic relaxation system designed for homogeneous two-phase flows in the limit of vanishing relaxation time. The governing equations comprise conservation laws for mixture mass and momentum, supplemented by a transport equation for the gas phase mass that includes a stiff relaxation source term. This source term drives the system toward local thermodynamic equilibrium. Under the assumptions of constant liquid density and an ideal isothermal gas phase, we demonstrate that, as the relaxation parameter \(ε\rightarrow 0\), a subsequence of solutions \((p^ε,u^ε)\) converges strongly in \(L_{\mathrm{loc}}^{1}\) to an entropy solution of the equilibrium Euler system. The proof integrates several analytical techniques: the construction of a suitable entropy pair and associated energy estimates, a transport equation approach for representing the error, commutator estimates, and the theory of compensated compactness. This work provides a rigorous justification of the relaxation limit for the homogeneous two-phase flows model.

Triangles are abundant in real-world networks but rare in standard null models for sparse graphs. Existing explanations typically rely on explicit triadic closure mechanisms or geometry-based connection rules. We propose an alternative hypothesis: the frequent appearance of triangles may arise naturally from the requirement of dynamic stability that maintains coexistence of species in Lotka-Volterra systems with competitive interactions. To evaluate this idea, we show that, across all possible interaction graphs, coexistence is guaranteed whenever the coupling strength is below the reciprocal of the graph's maximum degree. We also show that coexistence can persist up to a critical coupling strength of 1, which leaves a large gap that is unexplained by the graph degrees alone. These lower and upper bounds are achieved for star and complete graphs respectively. To investigate what structural properties of the interaction graph control the critical coupling within the gap, we optimise networks algorithmically while keeping the degree sequence fixed. We find that networks supporting stronger interaction strengths consistently exhibit higher clustering coefficients in several network models. Moreover, in real-world grassland plant networks, we observe higher clustering and stronger stability than those expected from a configuration model with the same degree sequence. Our result suggests that triangles, and clustering in general, may emerge as a structural signature of stabilising competition.

Quantum-matter theory (QMT), based on the Schrödinger or Dirac equations, is firmly established for both intra- and intermolecular interactions. However, there are two key issues with QMT. First, its applicability to large molecular complexes is hindered by the relatively high computational cost of the calculations required to achieve high accuracy. Second, fields are also quantum objects that produce many intriguing effects beyond standard QMT approaches to molecular systems. This review focuses on recent developments in quantum-field theory (QFT) approaches to both covalent and non-covalent interactions for molecules in vacuum and subject to environments such as cavities and solvents. QFT provides a rich playground for novel chemical theories and insights. For example, chemical reactions and van der Waals interactions can be manipulated by cavities, boundaries, and optical excitations; novel interactions emerge when molecules interact with quantized fields; systems with millions of atoms could soon be treated with coarse-grained QFT formalisms; and unexpected scaling laws for atomic and molecular properties can emerge when QFT is applied to sets of chemical systems. This review sets the stage for an exciting QFT-driven path for further development of chemical theory.

We take quantum theory and replace $\mathbb{C}$ by $\mathbb{C}[\varepsilon]$ where $\varepsilon^2=0$, i.e. we extend quantum theory to the ring of dual complex numbers. The aim is to develop a common language in which to treat continuous quantum physics and discrete quantum models in a unified manner, including their symmetries. Since quantum theory is linear, introducing $\varepsilon$ is enough to model infinitesimals. A first objection to this programme is that $\mathbb{C}[\varepsilon]$ is not a field, since division by $\varepsilon$ is undefined, while quantum mechanics typically relies on division. A second objection concerns whether unitarity still makes sense given $\varepsilon^2 = 0$. Hence, the core of this work is dedicated to proving that \dual quantum theory remains fully consistent. In particular, norm is preserved at all times, and renormalization never requires dividing by an infinitesimal. An equivalence with conventional quantum theory is demonstrated: the \dual extension of a parametrized quantum operation automatically provides a linear treatment of its first-order variations. As a first example application, we provide a unified description of both the Dirac equation in the continuum and the Dirac Quantum Walk in the discrete. We establish the discrete Lorentz covariance of the latter.

Large Language Models (LLMs) have demonstrated strong capabilities in biomedical question answering, yet their tendency to generate plausible but unverified claims poses serious risks in clinical settings. To mitigate these risks, the TREC 2025 BioGen track mandates grounded answers that explicitly surface contradictory evidence (Task A) and the generation of narrative driven, fully attributed responses (Task B). Addressing the absence of target ground truth, we present a proxy-based development framework using the SciFact dataset to systematically optimize retrieval architectures. Our iterative evaluation revealed a "Simplicity Paradox": complex adversarial dense retrieval strategies failed catastrophically at contradiction detection (MRR 0.023) due to Semantic Collapse, where negation signals become indistinguishable in vector space. We further identify a Retrieval Asymmetry: filtering dense embeddings improves contradiction detection but degrades support recall, compromising reliability. We resolve this via a Decoupled Lexical Architecture built on a unified BM25 backbone, balancing semantic support recall (0.810) with precise contradiction surfacing (0.750). This approach achieves the highest Weighted MRR (0.790) on the proxy benchmark while remaining the only viable strategy for scaling to the 30 million document PubMed corpus. For answer generation, we introduce Narrative Aware Reranking and One-Shot In-Context Learning, improving citation coverage from 50% (zero-shot) to 100%. Official TREC results confirm our findings: our system ranks 2nd on Task A contradiction F1 and 3rd out of 50 runs on Task B citation coverage (98.77%), achieving zero citation contradict rate. Our work transforms LLMs from stochastic generators into honest evidence synthesizers, showing that epistemic integrity in biomedical AI requires precision and architectural scalability isolated metric optimization.

Ranking individuals based on their performance in different coalitions is a problem emerging in various domains (teams sports, scientific evaluation, argumentation, etc.). Often, for practical reasons, the number of comparable coalitions is limited. Therefore, the foundational principles of ranking solutions must support realistic interpretations in contexts where only certain coalitions can be compared. To address this issue, in this paper we present an axiomatic analysis of solutions for the social ranking problem centered on the notion of consistency. More precisely, we show that an appropriate notion of consistency, which specifies how to combine rankings on individuals across different rankings on coalitions, plays a key role in any axiomatic characterization, representing the true distinguishing feature of each solution. This role is further highlighted by the taxonomy of the complementary axioms used in our characterizations, which boil down to well-studied properties of invariance with respect to the label of players or coalitions, and also with respect to minor changes in a coalitional ranking. By showing the logical independence of the axioms used in each characterization, as well as a rigorous analysis of alternative notions of consistency with respect to the majority of solutions from the literature, this work attempts to provide a first systematic study of the social ranking problem over a variable domain of coalitions.

High-resolution spectroscopy provides unique constraints on exoplanet atmospheric composition and dynamics. The past decade of ground-based campaigns has accumulated extensive public archives, yet many observations remain unanalysed. We present a homogeneous blind-search analysis of 50 SPIRou transits spanning 14 exoplanets, ranging from super-Earths to ultra-hot Jupiters, combining data from large program and public archived observations. Using automated data reduction and atmospheric retrieval via Nested Sampling validated by Cross Correlation Function analysis, we confirm previous H2O and CO detections in HD 189733 b, WASP-76 b, and WASP-127 b, report tentative H2S detections in HD 189733 b and TOI-1807 b, tentative detection of GJ 3470 b's atmosphere, and provide upper limits for non-detections. This work demonstrates a scalable method for systematic archive analysis, providing a first step toward ground-based support of large space-based atmospheric characterization programs and the study of atmospheric diversity across exoplanet populations from a statistical perspective.

We present high-speed photometry of the eclipsing cataclysmic variable CSS131106 J052412+004148. We determine the system parameters by modelling the eclipse lightcurve using the photometric eclipse method, in which the mass ratio is determined from the relative timings of the white dwarf and bright spot eclipses. Despite the blended white dwarf and bright spot ingress, typical of longer period cataclysmic variables, we perform simulations that show we are able to reliably constrain the component masses. We find a mass ratio of $q = 0.81 \pm 0.06$ and inclination $i = 78.5 \pm 0.7$ degrees. The white dwarf and donor masses were found to be $M_{w} = 0.72 \pm 0.04 \, M_{\odot}$ and $M_{d} = 0.58 \pm 0.06 \, M_{\odot}$ respectively. The white dwarf temperature was estimated to be $T_{\rm eff} = 18~500 \pm 2~000$ K, implying a moderate accretion rate of $\dot{M} = 3 \pm 1 \times 10^{-10} M_{\odot}$ yr$^{-1}$. The donor in CSS131106 J052412+004148 joins two other long-period cataclysmic variables (IP Peg and HS 0220+0031) in being unusually small for its mass, even when compared to detached M-dwarfs. The donors in all three systems are also unusually cool for their mass. We discuss possible explanations for the small radii and cool temperatures of the donors in these systems, but find no viable explanation for their properties.

Understanding relaxation dynamics in open quantum systems is a central problem in nonequilibrium quantum physics. Here we investigate the quantum Mpemba effect in the spin-boson model. In the weak-coupling Markovian regime we show that the occurrence of the effect strongly depends on the choice of distance measure at low temperature: while it appears in the trace distance, it can disappear in the quantum relative entropy. Going beyond the weak-coupling approximation, numerically exact simulations of the full system-bath dynamics reveal that increasing coupling enhances the effect in the trace distance and restores it in the quantum relative entropy. For all spin-bath couplings prior to delocalized-localized quantum phase transition, we uncover a simple geometric structure of the effect on the Bloch sphere: within the excited-state hemisphere, pairs of states related by rotations generically exhibit relaxation-order inversion. These results highlight the role of geometry and system-environment correlations in anomalous quantum relaxation.

The pulsar timing array systems have reported a detection of a nanohertz-band stochastic gravitational wave background in our galaxy. It is of interest to use this observation to probe modified gravity and to forecast the sensitivity with which certain deviations can be tested in the coming years. In this paper, we focus on the modified dispersion relation of the tensor modes and its effect on the overlap reduction function of the timing residual cross-correlations. We perform a comprehensive forecast of the phase velocity uncertainty, $σ_v$, using a Fisher analysis validated by a mock-data study to account for potential non-Gaussian behavior. We also take into account the sample variance effect and provide an observational timeline for future PTA sensitivity: detecting a $10\%$ or $-1\%$ deviation from the speed of light at the $3σ$ level requires $\mathcal{O}(30)$ years of observations.

The article concerns damped driven Jaynes-Cummings equation which describes quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators, and their structures correspond to the theory of completely positive and trace preserving generators (CPTP) of Lindblad and Kossakowski & al. Our main result is the construction of global generalised solutions with values in the Hilbert space of nonnegative Hermitian Hilbert-Schmidt operators in the case of time-dependent pumping. The proofs rely on finite-dimensional approximations of the annihilation and creation operators.

Pulsar wind nebulae (PWNe), formed when the wind originating from a rapidly rotating neutron star flows out into its surroundings, have now been observed across the electromagnetic spectrum from the radio to the PeV gamma-ray regime. For most of these sources, leptonic processes, where electrons interacting with background photon fields produce high-energy photons through inverse Compton scattering, are believed to be the origin of associated very-high-energy gamma-ray emission. As such, these objects cannot contribute significantly to the galactic hadronic cosmic ray flux at ~TeV-PeV energies. However, in a handful of cases, the possibility for an energetically sub-dominant hadron population being accelerated and producing very to ultra-high energy gamma-rays through pion decay has not yet been comprehensively excluded. Such scenarios have received renewed attention in the light of recent results from the Large High Altitude Air Shower Observatory (LHAASO). In this review we explore the theoretical background positing hadronic acceleration in galactic PWNe, considering cases where the hadrons escape from the pulsar surface and/or are accelerated in the wind, as well as potential 'shock mixing' scenarios. We also explore current and future possible constraints on a hadronic component to PWNe from observations.

We show that, for any given $k \ge 2$, every sufficiently large number appears as the Steiner--Wiener $k$ index of a graph.

Non-Hermitian physics traditionally relies on active gain--loss modulation or non-reciprocal couplings, which often introduce significant complexity, compromise stability, and offer very limited scalability in conservative systems. Here we propose an attenuation-gauge duality paradigm in which non-Hermitian topology emerges within fully passive, conservative systems through coupling to a structured reservoir. We derive that a spatially varying reservoir can establish an attenuation-gauge duality, where the spatial variation manifests as an emergent imaginary gauge field in the effective dynamics. It drives the boundary accumulation of skin modes while preserving energy conservation, analogous to Feshbach projection in quantum open systems. We validate this universal wave paradigm via macroscopic mechanical metamaterials, demonstrating that the direction of the skin effect can be reversed by tuning a single passive coupling parameter$t_\perp$, driven by a topological phase transition characterized by the spectral winding number. This framework also allows for a nonlinear extension, where amplitude-dependent coupling can induce intrinsic topological transitions.

The article addresses the damped driven Jaynes-Cummings for quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators. Our main result is the construction of a contraction dynamical semigroup in the Hilbert space of Hermitian Hilbert-Schmidt operators in the case of a nonpositive dissipation operator and time-independent pumping. All trajectories of the semigroup are generalised solutions to the Jaynes-Cummings equations. As a key example, we prove nonpositivity for the basic dissipation operator of Quantum Optics.

Integer weighing matrices (IW-matrices for short) are integer valued orthogonal square matrices. One usecase of these is to create classical weighing matrices with various block structures. In this paper we study and classify the space $IW(n,k)$ of the integer weighing matrices of small size $n\times n$ and weight $k$. Our classification includes a full list of all inequivalent matrices up to Hadamard equivalence and automorphism groups. We then continue to a secondary classification of the symmetric and antisymmetric IW up to symmetric Hadamard equivalence. We apply this to the case of projective space weighing matrices. Next we use the classification to count the cardinality of the spaces of all $IW(n,k)$ as well as the symmetric and anti-symmetric subspace. We supply practical algorithms and implement them in \texttt{Sagemath}. Finding an (anti-)symmetric IW matrix in a given Hadamard class can be done for significantly higher orders. In particular we solve some open cases: Symmetric $W(23,16)$, $W(28,25)$ and $W(30,17)$, and an anti-symmetric $W(28,25)$. We conclude by showing a detailed classification of $IW(7,25)$. We have also improved the \texttt{NSOKS} algorithm to find all possible representations of an integer $k$ as a sum of $n$ integer squares.

To understand the complicated formation processes of disk galaxies, we carry out a comparative study for NUV-r blue and red spiral galaxies drawn from a parent sample of u-r red spirals with $M_{*} > 10^{10.5} M_{\odot}$ at 0.02 < z < 0.07, based on the optical data from the Sloan Digital Sky Survey (SDSS) and the ultraviolet (UV) data from the Galaxy Evolution Explorer (GALEX). The analyses of the images and surface brightness profiles in the NUV and optical bands show that the differences between NUV-r blue and red spirals mainly occur in the outer disks (1-3 $R_{\rm e}$), and the contrast in NUV band is much larger than that in the optical bands. Both the positions on the star formation main sequence diagram and the NUV-r color profiles suggest that NUV-r red spirals have been fully quenched, whereas NUV-r blue spirals host quenched bulges and inner disks, as well as star-forming outer disks. Particularly, the disk mass-size relations indicate that, at a given disk mass, NUV-r blue spirals possess larger optical disks than NUV-r red spirals, by a factor of $\sim 1.20$. The environments and optical morphologies are consistent with the scenario that NUV-r blue spirals obtained fresh fuel for star formation either by interacting or merging with gas-rich galaxies or through accreting surrounding HI gas.

Quantifying model mismatch in a control-relevant manner is fundamental in robust control. A well-known metric for this purpose is the $ν$-gap, or Vinnicombe metric, which measures the discrepancy between a nominal model and the real system from a closed-loop viewpoint. However, its computation typically requires explicit knowledge of the true system. In this letter, we propose an identification-free, data-driven method to estimate the $ν$-gap between discrete-time SISO systems directly from input-output experiments. The method is complemented by a data-driven winding-number test, based on Welch-type averaging, to verify a required topological condition for the computation of the metric. Numerical simulations on heavy-duty gas-turbine models and a textbook example show that the proposed estimate closely matches MATLAB$^©$ \texttt{gapmetric}, while correctly detecting cases in which the admissibility conditions fail.

We simplify the proof of the optimality of the Shortest Lower-Bound First (SLF) algorithm, introduced by Gupta, Kaplan, Lindermayr, Schlöter, and Yingchareonthawornchai [FOCS'25], for minimizing the total flow time in the $\varepsilon$-clairvoyant setting.

This work builds on the theoretical frameworks presented in "Liquidity pools as mean field games: A new framework" and "Liquidity pools as mean field games with transaction costs" by the same author, where the strategic interactions among traders in a constant-product market-making protocol were modelled using mean field games (MFG), first without transaction costs and then incorporating them. Here we present the formulation of a more complete model that integrates three types of agents: traders, liquidity providers (LPs), and arbitrageurs. While we do not establish existence results for this general model, the formulation identifies the main technical difficulties and lays the groundwork for future work. The LP acts as a dominating player in the sense of 'Mean field games with a dominating player' by Bensoussan, Chau, and Yam: its strategy influences the mean field distribution of the traders, and the equilibrium is sought as a solution to the coupled system of three problems that constitute a Major-Minor game. The arbitrageurs operate by solving the optimization problem presented in "An analysis of uniswap markets" by Angeris et al., and their impact on the LP is captured through the loss-versus-rebalancing of "Automated market making and loss-versus-rebalancing" by Milionis et al. The material in this article should be read as an open research proposal rather than a collection of closed results.

The Lyndon array stores, at each position of a word, the length of the longest maximal Lyndon subword starting at that position, and plays an important role in combinatorics on words, for example in the construction of fundamental data structures such as the suffix array. In this paper, we introduce the Inverse Lyndon Array, the analogous structure for inverse Lyndon words, namely words that are lexicographically greater than all their proper suffixes. Unlike standard Lyndon words, inverse Lyndon words may have non-trivial borders, which introduces a genuine theoretical difficulty. We show that the inverse Lyndon array can be characterized in terms of the next greater suffix array together with a border-correction term, and prove that this correction coincides with a longest common extension (LCE) value. Building on this characterization, we adapt the nearest-suffix framework underlying Ellert's linear-time construction of the Lyndon array to the inverse setting, obtaining an O(n)-time algorithm for general ordered alphabets. Finally, we discuss implications for suffix comparison and report experiments on random, structured, and real datasets showing that the inverse construction exhibits the same practical linear-time behavior as the standard one.