In this paper, we investigate the Schur positivity of modified Hall--Littlewood polynomials indexed by two-column partitions under the action of the $\nabla$ operator. Specifically, we resolve two conjectures posed by Bergeron, Garsia, Haiman, and Tesler in the two-column case. Furthermore, our approach demonstrates that these results can be extended to arbitrary powers $\nabla^k$ for all integers $k\geq 1$.

In this paper, we study a particular conservative standard map in complex dimension 2. In this example, Siegel disks can be visualized and analyzed numerically as to the smoothness of their boundaries. We formulate and numerically support some conjectures.

Bitcoin is the cryptocurrency with the largest market capitalisation, but its widespread adoption is fundamentally limited by the scalability constraints of its consensus algorithm, which requires every transaction to be confirmed onchain. To address this, several Layer-2 scalability solutions have been proposed to move payments offchain -- most notably, the Lightning Network. However, their deployment remains hindered by cumbersome setup requirements: users must lock funds onchain to participate and engage in complex auxiliary protocols (e.g., for channel rebalancing, top-ups, and routing). Other solutions, like payment pools, sidechains and rollups, cannot be implemented in a non-custodial way on Bitcoin due to its limited scripting capabilities, or require all protocol participants to update the offchain state. In this work, we present Ark, the first Bitcoin-compatible commit-chain. Ark enables offchain transactions of virtual UTXOs (VTXOs), through an untrusted operator who aggregates them into succinct onchain commitments. A distinctive feature of Ark is its ease of deployment: users can receive offchain payments without locking any funds beforehand and Ark state updates can be performed only requiring the users involved in that update. We formally define the Ark protocol and prove its security. During this process, we identified two attacks affecting the testnet implementation, which we responsibly disclosed and proposed fixes for, which have been now integrated into the mainnet implementation. Our experimental evaluation demonstrates that Ark can commit onchain to batches of arbitrarily many VTXOs with a constant-sized footprint of approximately 200 vB. Cooperative exits add one output per user, while unilateral exits require $\mathcal{O}(\log n)$ transactions of roughly 150 vB per VTXO for a batch of $n$ VTXOs.

In Section 6 of the paper ``Fra\"ıssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic permutation has finite big Ramsey degrees. Unfortunately, this reduct was incorrectly identified as the generic partial order. We are grateful to Jan Hubička for bringing this error to our attention. In this note we correct the statements that rely on this misidentification and demonstrate that the reduct in question is in fact the generic 2-dimensional partial order. We emphasize that the arguments presented in Section 6 remain valid, with the sole exception of the Claim in the proof of Theorem 6.4, whose role was to (incorrectly) identify the reduct of the generic permutation as the generic partial order. This correction has an unexpected positive consequence. Rather than reproving a well-known result whose existing proof is already notably elegant, this note demonstrates that our general framework can be used to establish that a previously unexplored class of generic relational structures has finite big Ramsey degrees. This observation opens a potentially new direction for further research in the thriving area of big Ramsey combinatorics. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.

Let $r,\ell\geq2$ be integers. Given $r$-graphs $G$ and $F_1,\dots,F_\ell$, we write $G\to(F_1,\dots,F_\ell)$ if every $\ell$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\leq i\leq\ell$, otherwise we write $G\not\to(F_1,\dots,F_\ell)$. The Ramsey number $R(F_1,\dots,F_\ell)$ is the minimum number of vertices in an $r$-graph $G$ satisfying $G\to(F_1,\dots,F_\ell)$. In this note we prove that for any integers $t_1\geq\dots\geq t_\ell>r$, there exists an $r$-graph $G$ such that $G\not\to(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})$ but $G\to(K^{(r)}_s,K^{(r)}_{t_\ell-1})$, where $s=R(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})-1$. This extends recent work by Mendonça, Miralaei, and Mota, who established the statement for $r=2$.

In distributed optimization, multiple parties collaborate to find an optimal solution to a problem. Privacy-preserving distributed optimization uses techniques, such as secure multi-party computation (MPC), to protect the private inputs of each party. In time-critical settings, the runtime overhead introduced by privacy-preserving computations may prevent the optimization from finishing within the deadline. This paper presents an approach for privacy-preserving distributed optimization in time-critical settings that combines evolutionary algorithms for solution search and MPC for the evaluation of solutions. The approach reduces the impact of privacy-preserving computations on runtime and allows to return solution within the deadline. Obfuscation of evaluation results provides additional protection for private inputs from an honest-but-curious platform provider, but introduces a potential trade-off between protection and solution quality. This trade-off is investigated in experiments using a genetic algorithm for both the single-objective assignment problem and the traveling salesperson problem, as well as NSGA-II for the multi-objective assignment problem.

Missing data is pervasive in many scientific domains such as public health, environmental science, and the social sciences. Recoverability from missing data is typically studied using fully specified variable-level missingness models despite that, in many applications, only coarse structural information is available, for instance when variables are grouped into clusters due to limited knowledge or interpretability reasons. In this paper, we investigate recoverability from such abstract representations. We introduce two classes of cluster-based missingness graphs: the m-C-DMG, which retains variable-specific missingness indicators, and the cm-C-DMG, which aggregates missingness mechanisms at the cluster level. We formalize the notion of compatibility between these abstract graphs and underlying variable-level missingness models, and study how this abstraction affects the recoverability of probabilistic and causal queries. In particular, we give graphical conditions of recovering the joint distribution as well as graphical conditions of recovering a macro causal effect. Overall, our results clarify when cluster-level missingness information is sufficient for valid inference, and when finer-grained modeling is necessary.

Brain computer interface (BCI) enables the brain to directly control an external device by converting neural signals into actionable outputs. However, effective real-time translation of brain activity strongly depends on the quality of neural communication between the brain and the external device. 6G is the next generation of wireless communication, expected to provide unprecedented levels of data rates, data security, and automation capabilities. In this context, integrating 6G into BCI systems would not only enhance the performance of brain-device communication, but would also create new opportunities for innovative applications. This work provides a comprehensive study on how BCI technology can be built effectively on top of 6G wireless networks by introducing several technical aspects and use cases. We first provide an overview of BCI and 6G, following their progression from early development to convergence through cognitive communication and advanced neural interfaces. We then highlight the need for the upcoming 6G systems toward BCI technology in every aspect, including 6G technologies such as intelligent edge and zero-touch networks, and 6G use cases such as digital twin, immersive communication, and internet of minds. Furthermore, we identify key technical challenges, open issues, and future research directions related to the 6G-enabled BCI paradigm.

Active matter systems under confinement display persistent surface motion and a strong boundary affinity. However, despite extensive studies of their positional dynamics, much less attention has been given to the corresponding orientational behavior. Here, using molecular simulations of an active Brownian particle confined within a hard circular boundary and the Fokker-Planck equation, we show that the positional distribution of the particle is directly coupled to orientational fluctuations, as characterized by the conditional orientational distribution. Confinement generates two preferred tangential orientational states connected by stochastic flipping pathways: rapid boundary-localized switching and slower bulk-mediated excursions. Further, the positional distribution exhibits a nontrivial power-law decay with distance from the boundary that is closely linked to curvature-induced bistable orientational states and the variance of the associated conditional distribution. The mean waiting time between flips exhibits power-law dependence on the confinement strength. Our results establish that the interplay between orientational fluctuations, bistability, positional accumulation, and stochastic switching governs the observed dynamics of active particles under confinement, providing a framework for understanding transport, exploration, and escape processes in confined active systems.

Spike-type III burst pairs represent a distinct class of solar radio emissions in which clusters of spike-like bursts appear atop the highfrequency onset of type III bursts. Using high time-frequency resolution data from the Chashan Broadband Solar radio spectrometer at meter wavelengths (CBSm), we present the largest statistical study to date of such events, comprising 502 spike-type III pairs from 35 events recorded between November 2023 and October 2025. We find that spike-like clusters systematically precede their associated type III bursts by 0.5-3 s in time (~87% of pairs) and by 3-30 MHz in frequency (~80%), a temporal and spectral offset that differs from earlier reports. The spike-like clusters exhibit diverse morphologies, including point-like, blob-like, drifting, and diffuse structures, with durations of ~0.5-5 s and bandwidths of 15-150 MHz. Bi-directional drifting structures with rates of ~20-100 MHz s-1 are observed, consistent with source motion both toward and away from the Sun. Furthermore, spike emission is predominantly strongly circularly polarized, with more than 64% of clusters showing maximum polarization exceeding 0.6, in stark contrast to the generally weak polarization of type III bursts. These findings point to an origin of the spike radiation in a multiscale, inhomogeneous, and highly dynamic electron-acceleration region, providing novel observational constraints on the mechanisms underlying coherent solar radio bursts.

Let $f$ and $g$ be two Hénon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.

The suppression of the neutron superfluid fraction in the inner crust of a cold neutron star is mitigated by the quantum zero-point motion of ions about their equilibrium position. In turn, the crustal dynamics is altered by the presence of the neutron superfluid. These effects are studied self-consistently to assess the validity of the usual assumption of a perfect rigid lattice. To this end, fully three-dimensional band-structure calculations of the superfluid fraction are carried out in the weak-coupling approximation, considering body- and face-centered cubic lattices. In both cases, the superfluid fraction is still found to be strongly suppressed in the intermediate region of the inner crust. In turn, the effective mass of the ions is dramatically increased, thus further damping the ion fluctuations. These results are of relevance for the rotational and thermal evolutions of neutron stars.

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

Spectra from the cosmic dawn obtained with JWST/NIRSpec (James Webb Space Telescope near-infrared spectroscopy) in the SMACS0723 Early Release Observations are now available. Analyses carried out by different teams indicate poor to extremely low oxygen metallicities (log(O/H)+12<8.0), a characteristic feature of pristine galaxies. In this work, we present new modelling of spectra emitted by objects in the redshift range 2.16<=z<= 8.68, including their recently corrected spectra in the z = 2-9 range. The models account for both photoionisation and shock processes. Our aim is to identify similarities and differences with respect to local galaxies by searching for possible remnants of pristine galaxies among low-z objects. We analyse selected emission-line ratios in relation to elemental abundances and physical parameters. We find that the gaseous clouds within galaxies at cosmic dawn have preshock densities at least a factor of 100 higher than those in local galaxies, but comparable to those calculated for local metal-poor galaxies at 0.005<z<0.05. The metallicities log(O/H)+12 and log(Ne/H)+12 range between 7.9 and 8.55, and between 7.0 and 7.48, respectively. Uncorrected observed Hg/Hb line ratios are mostly >0.5, indicating high temperatures (>10^5K) in the emitting gas. Clear affinities are evident between the high-z galaxy spectra and those of local metal-poor galaxies at 0.005<z<0.05. However, in order to reproduce all the observed line ratios for each spectrum - including Hg/Hb values as high as ~0.8 - emission from cloud fragments was added to that from the main clouds in the pluri-cloud models. We suggest that fragments close to pristine galaxies were destroyed by events that occurred between z>6 and z<0.05, whereas some cloud remnants of pristine galaxies survived and are now found embedded, for example, at 0.005<z<0.05.

We characterize spin-polarized lasing in a two-dimensional photonic lattice fabricated from a GaAs/InGaAs semiconductor microcavity sample. The lattice is defined by a staggered arrangement of rounded rectangular micrometric mesas that laterally confine and couple the optical modes. Polarization-, angle-, and energy-resolved micro-photoluminescence measurements reveal the transition from the strong-coupling regime to photon lasing, accompanied by extended spatial coherence across several lattice unit cells. Under circularly polarized nonresonant excitation, the emitted light acquires a controllable circular polarization whose handedness follows that of the pump. These results establish photonic-lattice VCSELs as a platform for spin-controlled coherent emission in extended optical systems.

Long-term memory systems enable conversational agents based on large language models (LLMs) to retain, retrieve, and apply user-specific information across multi-session interactions. However, existing evaluations mainly assess outcome-level performance or temporal updating, providing limited insight into how systems retrieve and rank temporally valid, factually correct, and contextually applicable memory evidence under conflicting alternatives. To address this gap, we propose MemConflict, a diagnostic framework that treats memory validity as a query-conditioned fitness-for-use problem. MemConflict formalizes dynamic, static, and conditional conflicts over temporal validity, factual correctness, and contextual applicability. It simulates controlled long-horizon histories from structured user profiles, introduces cross-session conflicts, and injects semantically similar distractors to create competition among memory candidates. The resulting multi-session dialogue benchmark supports black-box evaluation of final answers and white-box analysis of supporting-memory retrieval and ranking. Experiments on six representative long-term memory systems show uneven strengths across conflict types, with answer correctness often diverging from memory retrieval and ranking. Sensitivity analyses reveal that longer histories, distractors, implicit queries, and larger conflict distances degrade performance. Diagnostics show failures from missing supporting memories and ineffective use of retrieved memories. Collectively, MemConflict advances principled long-term memory governance through retrieval-aware, conflict-aware reliability assessment.

The integration of machine learning (ML) with traditional physics-based models is reshaping the landscape of weather and climate prediction. On their own, ML-based and physics-based approaches each have significant benefits - but also challenges. Deploying both these approaches side by side has the potential to accelerate the pull through of emerging science in a trusted and practical way. But there are many choices that can be made to how we "blend" ML and established physics-based modelling systems to get the optimal benefits. This paper aims to provide a typology of blended modelling approaches and discusses some of the strategic benefits that come with them. It can be used not just to classify modelling systems, but also identify routes to gradual, incremental or wholesale development and implementation of new and emerging capabilities. These approaches provide a practical path to innovation by combining the speed and adaptability of machine learning with the robustness, trust, and interpretability of physics-based systems. By adopting a structured vocabulary and outlining the benefits and limitations of each approach, this framework supports informed decision-making and strategic planning, and can be used by the wider community to navigate the transition to next-generation prediction systems.

In transportation network analysis, various types of road network data can be used even when focusing on the same region. Since different road network datasets can make different performance in analyses, it is necessary to compare them and make appropriate selections in a qualitative manner. However, many of the existing methods for comparing road network datasets are limited to specific topological evaluations and do not consider transportation. This study proposes a method for quantitative comparison of different road network datasets with explicit consideration for traffic flows on them. The method first conducts a static traffic assignment with hypothetical demand for each dataset, and then compare the results using Wasserstein distance on two dimensional plane. Case study on different sources of road network datasets and their simplifications suggests the potential use of the proposed method in evaluating and selecting road network datasets.

We introduce a high-performance linear-scaling electronic structure method that employs chromatic superposition states (CSS) as a low-dimensional, high-fidelity representation, which can be orders of magnitude smaller than the full Hilbert space. Grounded in the system's finite correlation length, the CSS representation aggregates the uncorrelated orbitals into a single basis via a graph-coloring scheme, and is independent of the system size yet accurately preserves all sparse operators in solving the Kohn-Sham equations. The projection onto CSSs is efficiently computed by employing the block-Lanczos Krylov method which features high hardware efficiency and linear-scaling cost, enabling fast calculation of large-scale Kohn-Sham density matrix. We show that this method already outperforms previous linear-scaling density matrix purification method by more than one order of magnitude in computational speed at even small scale, while preserving high accuracy. The practical utility of the CSS method is demonstrated through molecular dynamics simulation of a 10000 $H_2O$, and self-consistent calculation of a 1-million $H_2O$ with modest resources.

We study chaos-integrability transition purely within a BPS subspace of a specific supersymmetric model that interpolates between the chaotic $\mathcal{N}=2$ SYK model and an integrable $\mathcal{N}=2$ "commuting" SYK model. Using the framework of BPS chaos, we analyze the spectrum of an operator projected onto the BPS subspace. We numerically find that its spectral statistics exhibit random-matrix behavior near the SYK limit and smoothly transitions to Poisson statistics near the integrable limit. Our results provide a direct example of a chaos-integrability crossover diagnosed solely from BPS states.

An important step in the proof of the Herman invariant tori conjecture was the introduction of a normal form with poles along the resonance loci, replacing the Birkhoff normal form, which we call the Hamiltonian normal form. This paper is extracted from previous versions (arXiv:1206.1245, arXiv:1909.06053) and aims to present this Hamiltonian normal form in its simplest form. It is expected that, not only theoretically but also in numerical computations, it will provide better approximations than the standard Birkhoff normal form.

We explore the symmetry properties of Stinespring dilations of single-qubit Pauli channels, addressing both the generic case and the specific examples of phase damping and depolarizing channels. For each scenario, we derive the representation of the Pauli group acting on the Hilbert space of the environment. We then focus on dilations that are continuous in time and driven by a time-independent Hamiltonian, and on collision models that generate a Pauli dynamical semigroup in the limit of fast collisions. First, we complement some recent general results on these types of dilations (M. Cattaneo, Phys. Rev. A 111, 022209 (2025)) with some additions and clarifications, including the case of covariant channels with strongly conserved quantities. Next, we show that the covariance property of Pauli channels impose strong constraints on both the dilation Hamiltonian and the initial state of the environment, and demonstrate how these constraints can be exploited to explicitly construct the time-dependent dilations in all considered cases. Our results are relevant for the quantum simulation of Pauli channels via unitary dilations and of Pauli semigroups via collision models, both in the laboratory and on quantum computers.

Secure containers isolate each container with its own kernel, mitigating shared-kernel attacks prevalent in traditional container systems. However, existing designs still face a fundamental isolation--performance trade-off. Nested-cloud deployments amplify the cost of VM exits and page-table management, while emerging agentic workloads expose bursty memory demand that requires fine-grained elasticity. We attribute this trade-off to two root causes. First, existing designs lack lightweight intra-container isolation primitives for frequent container user--kernel transitions. Second, the host treats container memory management as opaque, forcing reactive secondary faults and coarse-grained huge page mappings to amortize their cost. This paper presents ParaCell, a paravirtualized secure container runtime built on two insights. First, intra-address-space hardware protection primitives can provide lightweight intra-container isolation. ParaCell uses MPK-based XGates to isolate the container user and container kernel within a single address space, turning frequent user--kernel transitions into direct domain switches. Second, container kernel allocators already encode memory-management intent. ParaCell introduces Pager to interpose on allocation and free events, batch proactive GPA to HPA bindings and unbindings, and avoid reactive shadow page-table faults while preserving fine-grained memory elasticity. ParaCell is implemented as a drop-in replacement for RunV. Our experiments demonstrate that, across traditional cloud and emerging agent applications, ParaCell reduces latency by up to 57% and 79% over PVM, and by up to 33% and 88% over RunV, in bare-metal and nested setups, respectively. On agent workloads, ParaCell saves up to 35.6% memory compared with the state-of-the-art VM memory reclamation technique, HyperAlloc.

We introduce the extra slow Tamari lattices, a new family of lattices defined on faithfully balanced tableaux. These tableaux arise naturally from the representation theory of type \( A \) quivers, and our construction extends the classical Tamari lattice and the slow Tamari lattice. We explicitly describe meets and joins in the extra slow Tamari lattices, and then prove that they are lattices. We then show that they are semidistributive, trim, polygonal, and congruence uniform. Their join-irreducible elements are described in terms of a three-color analogue of the positive roots of type \( A \), which leads to descriptions of their spines and congruence lattices. We also obtain several enumerative results for the extra slow Tamari lattices and their associated structures. Finally, we derive new structural and enumerative results for the slow Tamari lattices.

Optomechanical systems provide a versatile platform for precision measurements and investigations of fundamental physics, where bringing macroscopic resonators into the quantum regime is a widely pursued goal. Achieving such quantum behavior of solid-state mechanical resonators at room temperature would greatly broaden their applications by removing the need for cryogenic environments. Reaching this goal requires efficient cooling of mechanical motion, among various laser cooling methods, dynamical backaction cooling (DBC) is widely utilized in experiments but fundamentally limited when operating in the sideband-unresolved regime. Coherent feedback cooling (CFC) can overcome this limitation, while avoiding state collapse and the electronic restrictions inherent to measurement-based feedback. Here, we experimentally demonstrate CFC using an ultracoherent density phononic crystal membrane. By combining CFC with strong DBC in a relatively narrow cavity, we achieve a phonon occupation reduction from $5.5\times10^{6}$ to $166\pm7$, corresponding to a cooling factor of $3.3\times10^{4}$ at room temperature, even with current experimental limitations. Our results show the potential of CFC for approaching the ground state of high-$Q$ membranes at room temperature.

Topological phononics extends the foundational concepts of topological condensed matter physics to the realm of lattice vibrations and classical mechanical waves, unlocking robust, defect-immune states and phenomena beyond the reach of conventional phononic engineering. This review provides a unified, systematic framework for understanding topological phonons across natural and artificial systems, spanning solid-state materials, acoustic/mechanical metamaterials, and non-Hermitian platforms. We cover the core theoretical principles -- from Berry curvature and symmetry-protected topological invariants to bulk-boundary correspondence -- alongside experimental advances in probing topological phonon states via inelastic scattering and momentum-resolved techniques for solid-state phonons as well as pump-probe measurements in acoustic/mechanical metamaterials. Key topics include Weyl/Dirac/nodal-line phonons in crystalline solids, symmetry-engineered topological phases in metamaterials, non-Hermitian effects (exceptional points, skin effect), and emergent directions such as Floquet engineering, synthetic dimensions, and real-space topological textures (skyrmions, merons). We also highlight technological applications in robust waveguides, on-chip surface-acoustic-wave devices, and acoustofluidics, while outlining future challenges and opportunities in quantum phononics, nonlinear topological phenomena, and interdisciplinary integration with photonics and electronics. This review serves as a comprehensive guide across physics, materials science, and engineering, bridging fundamental theory with cutting-edge experiments and innovations in topological phononics.

We revisit the instability properties of the recovery of the absorption coefficient for the radiative transfer equation in the diffusive regime. To this end, we develop a rather robust framework building on [Koch-Rüland-Salo, 2021] which allows us to deal with nonlinear critical stability transition phenomena. In particular, this permits us to consider rather general geometries based on the identification of compression properties of the forward operator. Given the albedo operator as the measurement data, we show that in the regime of vanishing Knudsen number there is a transition from Hölder to logarithmic stability in the inverse problem for the radiative transfer equation. As a central ingredient, we rely on suitable a priori estimates for the radiative transfer equation which we deduce by building on the strategy from [Demattè-Velázquez, 2025].

We study single-loop gradient-flow dynamics for nested optimization, where the outer variable evolves while auxiliary variables track the inner solution map. While existing analyses typically rely on problem- and condition-specific Lyapunov constructions, we propose, to our knowledge, the first unified Lyapunov template for continuous-time analysis that covers minimax, bilevel via a lifted penalty formulation, and min--min--max. Our proof is modular, built from reusable lemmas that yield a unified characterization of time-scale separation. This characterization bridges regimes from strong convexity/concavity to mere convexity through an error-bound condition, and produces explicit closed-form thresholds that avoid the coupled ratio conditions common in discrete-time analyses. We further compare the penalty dynamics with the ideal hyper-gradient flow, derive a finite-time tracking bound, and discuss an Euler one-step analogue; hypercleaning diagnostics show that the predicted relative time-scale regions remain visible under stable forward-Euler discretization.

Graph algorithms are central to large-scale applications such as navigation systems, social networks, and data analysis platforms. This thesis studies two important challenges in such systems: robustness to failures and fairness in clustering outcomes. In the first part, we investigate fault-tolerant reachability preservers in directed graphs. We present the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers that remain resilient to two edge or vertex failures, achieving a sparse construction of size $O(n^{4/3}|\mathcal{P}|^{1/3})$. In the second part, we study fair clustering algorithms that ensure balanced representation of protected groups. We develop approximation algorithms for fair consensus clustering and introduce the framework of closest fair clustering, establishing hardness results and efficient algorithms for multi-group settings. Building on this framework, we obtain improved guarantees for fair correlation clustering and design the first streaming algorithm for fair consensus clustering using only logarithmic memory. Together, these results contribute toward the design of graph algorithms that are both robust and socially responsible.

We consider the asymptotics of sums of the form $$ \frac1{F_n^σ} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(πm/F_n)}\right|^σ} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(πF_{n-1}m/F_n)}\right|^σ} $$ where $(F_n)_{n \in \mathbb N} = (1, 1, 2, 3, 5, 8, 13, \dots)$ are the Fibonacci numbers. Such sums appear, for example, in the context of discrepancy theory and numerical integration methods reformulated as energy minimization problems. We show that for parameters $σ> 1$ and a large class of functions $f$ the above sum behaves asymptotically like $$ C n + D + O\left((1-\varepsilon)^{n}\right) $$ for some constants $C$ and $D$. These constants can be given via infinite series connected to the Dedekind zeta function over the algebraic number field $\mathbb Q(\sqrt5)$. In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that $$ \sum_{m=1}^{F_n-1} \frac1{\sin(πm/F_n)^2} \frac1{\sin(πF_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$