Cooperation, fairness, trust, and resource coordination are cornerstones of modern civilization, yet their emergence remains inadequately explained by the persistent discrepancies between theoretical predictions and behavioral experiments. Part of this gap may arise from the imitation learning paradigm commonly used in prior theoretical models, which assumes individuals merely copy successful neighbors according to predetermined, fixed rules. This review examines recent advances in evolutionary game dynamics that employ reinforcement learning (RL) as an alternative paradigm. In RL, individuals learn through trial and error and introspectively refine their strategies based on environmental feedback. We begin by introducing key concepts in evolutionary game theory and the two learning paradigms, then synthesize progress in applying RL to elucidate cooperation, trust, fairness, optimal resource coordination, and ecological dynamics. Collectively, these studies indicate that RL offers a promising unified framework for understanding the diverse social and ecological phenomena observed in human and natural systems.

OBJECTIVE: To propose time-to-event estimators that help evaluate incident diagnostic coding and possible upcoding in Medicare as well as introduce an open-source software package that enables more reproducible methods development relevant to Medicare billing behavior. STUDY SETTING AND DESIGN: Observational analysis of simulated upcoding based on coding by insurers or providers that may be incentivized by Medicare Advantage risk adjustment. DATA SOURCES AND ANALYTIC SAMPLE: Two years of separately simulated incident health condition coding data for a Medicare Advantage population and a Traditional Medicare population where coding patterns are aligned with known practices in each program. PRINCIPAL FINDINGS: We propose several novel time-to-event estimators of incident coding intensity and possible upcoding in Medicare Advantage, including accounting for unreliable reporting. We demonstrate estimator performance in simulated data leveraging the National Institutes of Health's All of Us study and also develop an open source R package to simulate longitudinal realistic labeled upcoding data, which were not previously available for researchers. In simulations, our novel estimators recovered differences in upcoding within and across monitoring periods. Undercoding had a limited effect on our novel estimators while an existing estimator was more sensitive to undercoding. CONCLUSIONS: Our proposed estimators can help researchers and policymakers track new coding behaviors (e.g., as may be incentivized by risk adjustment formula updates) earlier and at scale while accounting for several real-world data considerations. Further, the R package we provide can be used to improve the development, accessibility, and reproducible evaluation of coding intensity and upcoding methodology.

By studying first a new decelerating sudden future singularity (SFS) universe we report finding a novel type of cosmological singularity which we dub a finite time pseudo-rip (FTPR) because unlike for a pseudo-rip, it happens in the finite future of the universe. In contrast to the new SFS model, where the expansion is decelerating before reaching the pressure singularity, the FTPR scenario is preceded by a super-accelerated phantom phase. Our claim is based on the thorough study of the energy conditions showing the violations of all of them for a FTPR, and only the dominant energy one for an SFS. Application of the so-called Raychaudhuri averaging shows that, alike within the requirement of geodesic completeness, these singularities are weak in the sense of this definition. We study the properties of the models including the behaviour of the cosmological horizons presented in the appropriate Penrose diagrams. Finally, we introduce a FTPR model containing standard radiation and dust fluids that can mimic the past expansion history of $Λ$CDM though facing a future pressure singularity.

Frequent tree mining asks us to enumerate tree patterns that occur frequently in a database of rooted trees. This problem is motivated by tree-structured data in bioinformatics, such as glycans and pseudoknot-free RNA secondary structures. A direct enumeration of all frequent trees is often highly redundant, because every subtree of a frequent tree is again frequent. Closed and maximal frequent trees are standard ways to reduce this redundancy, but their enumeration can still be computationally hard. In this paper, we study the effect of bounding the height of the input trees. This is a natural restriction for rooted trees, since the height is the depth of the hierarchy. We ask whether closed/maximal frequent tree mining remains hard when every input tree has a small height. Our results show that the answer depends sharply on the model. For rooted unordered trees of height at most 2, we give a polynomial-delay algorithm for enumerating closed frequent trees. On the other hand, for rooted ordered trees of height at most 2, we show that an output-polynomial time algorithm for enumerating closed frequent trees would imply an output-polynomial time algorithm for Dualization. For maximal frequent tree enumeration, we prove that no output-polynomial time algorithm exists unless P = NP already for rooted ordered trees of height at most 2 and for rooted unordered trees of height at most 3. Thus, even very small height bounds do not make the enumeration problems easy in general. At the same time, the unordered closed case of height at most 2 admits polynomial-delay enumeration. These results give a height-based classification of the complexity of closed and maximal frequent tree mining on shallow rooted trees.

While large language models (LLMs) are transforming engineering and technology through enhanced control capabilities and decision support, they are simultaneously evolving into complex dynamical systems whose behavior must be regulated. This duality highlights a reciprocal connection in which prompts support control system design while control theory helps shape prompts to achieve specific goals efficiently. In this study, we frame this emerging interconnection of LLM and control as a bidirectional continuum, from prompt design to system dynamics. First, we investigate how LLMs can advance the field of control in two distinct capacities: directly, by assisting in the design and synthesis of controllers, and indirectly, by augmenting research workflows. Second, we examine how control concepts help LLMs steer their trajectories away from undesired meanings, improving reachability and alignment via input optimization, parameter editing, and activation-level interventions. Third, we look into deeper integrations by treating LLMs as dynamic systems within a state-space framework, where their internal representations are closely linked to external control loops. Finally, we identify key challenges and outline future research directions to understand LLM behavior and develop interpretable and controllable LLMs that are as trustworthy and robust as their electromechanical counterparts, thereby ensuring they continue to support and safeguard society.

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties of the corresponding integral model of unitary Shimura varieties. As a direct application, we are able to improve the arithmetic inner product formula so that the places with local root number \((-1)\) are allowed to be ramified.

The theory of inter-spin-subband electric dipole spin resonance in transition metal dichalcogenide heterobilayers is proposed. Our symmetry analysis demonstrates that, in contrast to monolayers, the reduced symmetry of heterobilayers enables coupling between conduction band spin subbands by an electric field. We establish the optical selection rules for all six high-symmetry stacking configurations. The microscopic mechanism of the effect is identified as the spin-orbit coupling induced mixing of Bloch states from different conduction bands, which generates a non-zero momentum matrix element between the spin-split states. It also leads to the linear-in-wavevector spin-dependent terms in the effective Hamiltonian, i.e., the Rashba effect. Our estimates show that the rate of electric-dipole spin-flip transitions exceeds by far that of the magnetic-dipole transitions in transition metal dichalcogenide heterobilayers.

Gravitational wave (GW) astronomy opens a new venue to explore the universe. Future observatories such as LISA, the Laser Interferometer Space Antenna, are expected to observe previously undetectable fundamental physics effects in signals predicted by General Relativity (GR).One particularly interesting such signal is associated to the displacement memory effect, which corresponds to a permanent deformation of spacetime due to the passage of gravitational radiation. In this work, we explore the ability of LISA to observe and characterize this effect. In order to do this, we use state-of-the-art simulations of the LISA instrument, and we perform a Bayesian analysis to assess the detectability and establish general conditions to claim detection of the displacement memory effect from individual massive black hole binary (MBHB) merger events in LISA. We perform parameter estimation both to explore the impact of the displacement memory effect and to reconstruct its amplitude. We discuss the precision at which such a reconstruction can be obtained thus opening the way to tests of GR and alternative theories. To provide astrophysical context, we apply our analysis to black hole binary populations models and estimate the rates at which the displacement memory effect could be observed within the LISA planned lifetime.

These are the notes for the 4.5-hour course with the same title that I delivered in August 2025 at the Les Houches summer school ``Exact Solvability and Quantum Information''. In these notes I pedagogically introduce the setting of brickwork quantum circuits and show that it provides a useful framework to study non-equilibrium quantum many-body dynamics in the presence of local interactions. I first show that brickwork quantum circuits evolve quantum correlations in a way that is fundamentally similar to local Hamiltonians, and then present examples of brickwork quantum circuits where, surprisingly, one can compute exactly several relevant dynamical and spectral properties in the presence of non-trivial interactions.

We present a lattice QCD study of the chiral properties of $(2\!+\!1)$-flavor QCD in background magnetic fields at zero temperature with physical pion masses. Simulations are performed using the highly improved staggered quark action across four different lattice spacings to enable a controlled continuum extrapolation. We compute the renormalized chiral condensates together with pseudoscalar meson masses and decay constants for pions, kaons, and the fictitious $η^0_{s\bar{s}}$ pseudoscalar as functions of the magnetic-field strength $eB$ up to $eB\simeq1.2$ $\mathrm{GeV}^2$. The chiral condensates exhibit clear magnetic catalysis, increasing monotonically with the field strength. In the meson sector, neutral pseudoscalar masses decrease steadily with $eB$, whereas charged pseudoscalar masses display a nonmonotonic response: They rise at small fields, consistent with the lowest-Landau-level expectation, but then saturate and slightly decrease at larger fields, signaling sizable internal-structure effects. At the same time, neutral pseudoscalar decay constants are strongly enhanced by the magnetic field. To quantify deviations from chiral symmetry relations, we isolate the magnetic-field-induced shift in the Gell-Mann--Oakes--Renner corrections and find it to remain small for the neutral pion but to become sizable for the neutral kaon. To elucidate the origin of the magnetic response, we separately analyze the sea- and valence-quark contributions to both neutral and charged meson masses, finding that valence effects dominate at zero temperature. These results provide new insights into the interplay between QCD chiral symmetry breaking and strong magnetic fields.

We present results for the phase diagram of the parent compound LaCrAsO under electron doping using the matrix random-phase approximation. At low doping levels, the system stabilizes an antiferromagnetic state in which different Cr sublattices carry opposite spins, consistent with experimental observations. As the doping concentration increases, a stripe-type antiferromagnetic phase becomes favored. At even higher doping, the system repeats the two former magnetic states, but with incommensurate magnetic ordering vectors. The commensurate magnetic phases are associated with more localized electrons in the Cr $d_{3z^2-r^2}$ orbital, whereas the incommensurate phases are linked to the $d_{xy}$ orbital, whose stronger overlap favors itinerant-electron magnetism.

We study various types of consistency of honest decision trees and random forests in the regression setting. In contrast to related literature, our proofs are elementary and follow the classical arguments used for smoothing methods. Under mild regularity conditions on the regression function and data distribution, we establish weak and almost sure convergence of honest trees and honest forest averages to the true regression function, and moreover we obtain uniform convergence over compact covariate domains. The framework naturally accommodates ensemble variants based on subsampling and also a two-stage bootstrap sampling scheme. Our treatment synthesizes and simplifies existing analyses, in particular recovering several results as special cases. The elementary nature of the arguments clarifies the close relationship between data-adaptive partitioning and kernel-type methods, providing an accessible approach to understanding the asymptotic behavior of tree-based methods.

Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces a novel lower bound for asymmetric quantum hypothesis testing that is based on the Nussbaum-Szkoła mapping. The lower bound provides a unified recovery of converse results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bound arises from a single expression and enables, in some cases, the direct use of classical results. It is further demonstrated that the proposed bound provides accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight its improved tightness.

Stochastic renewal processes are ubiquitous across physics, biology, and the social sciences. Here, we show that continuous-time renewal dynamics can naturally produce a mixed discrete-continuous structure, with a macroscopic fraction of particles occupying a discrete state. For ensembles of continuous-time random walkers subject to Poissonian renewal resets, we develop an age-structured framework showing this discrete component corresponds to localization at the reset configuration. We next show that collective interactions can retain memory although all reset events are memoryless. Remarkably, the transition to collective memory is discontinuous, and we identify a discontinuous dynamical phase transition from weak collective bias, where the dynamics are stationary, to strong collective bias where the dynamics are nonstationary and display aging up to finite-size effects. We explicitly discuss ecological implications of our work, illustrating how continuous-time renewal dynamics shape macroscopic structure and collective organization with long-term memory.

Regulators and voluntary corporate sustainability efforts are increasingly adopting time-matching requirements (TMRs) for clean electricity procurement for large loads, such as data centers, and electricity-intensive fuel production, such as hydrogen. We use a stochastic capacity expansion model (CEM) framework to assess how inter-annual weather variability affects the cost, composition, and emissions of procurement-driven infrastructure to meet annual and hourly TMRs using the case study of a grid-connected hydrogen producer in Texas. Our approach, which relies on co-optimizing investments and hourly operations over nine weather scenarios, reveals that hourly TMR comes at a higher cost premium compared to annual TMR than previously estimated by single-scenario deterministic modeling, while emissions outcomes remain directionally consistent. Demand flexibility and partial hourly TMR (80-90%) lower the cost premium while preserving emissions benefits. We further examine how binding renewable portfolio standards (RPS) interact with TMR costs and emissions outcomes. When an RPS is applied to non-H2 electricity demand, annual TMR reduces emissions comparably to hourly TMR at a lower cost. Incorporating H2-related electricity demand directly into the RPS constraint, rather than imposing a separate TMR, achieves similar emissions outcomes at still lower cost, suggesting that TMR-based clean electricity procurement, particularly hourly matching, offers limited additional value in regions with stringent grid decarbonization policies.

We investigate the discovery potential of the $T_{bc}$ state with $J^P = 0^+$ in proton-proton ($pp$) collisions at LHCb at a center-of-mass energy of $\sqrt{s} = 13~\mathrm{TeV}$. The study focuses on the decay channel $T_{bc} \to B^- D^+$. A phenomenological approach is employed to construct the background model based on the associated production of $B$ and $D$ mesons, incorporating previously published LHCb results. Background processes are simulated using $\texttt{MadGraph5\_aMC@NLO}$ and $\texttt{Pythia8.3}$. We explore the parameter space of the $T_{bc}$ mass, width, production cross section, and the effective double-parton scattering cross section ($σ_{\mathrm{eff}}$) relevant for the $B D$ meson background. The integrated luminosity required for a $5σ$ discovery at LHCb is evaluated under various assumptions. In particular, we consider three representative $T_{bc}$ production cross section scenarios: an optimistic estimate of $103~\mathrm{nb}$, an intermediate value of $18~\mathrm{nb}$ obtained by scaling from the $T_{cc}^+$ production cross section, and a conservative lower bound of $0.3~\mathrm{nb}$. We find that a $5σ$ observation is achievable for a production cross section of $103~\mathrm{nb}$, which is expected to be within reach during Run~4. In contrast, the more realistic cross section estimate of $18~\mathrm{nb}$ requires the full Run~5 dataset ($300~\mathrm{fb}^{-1}$) under the most favorable parameter choices. For the conservative scenario, no significant signal would be observable even with $300~\mathrm{fb}^{-1}$. In addition, we estimate the minimum observable $σ(T_{bc}) \times BR(T_{bc} \to B^- D^+)$ for a $5σ$ discovery under different luminosity scenarios, providing guidance for future experimental searches at LHCb.

We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are illustrated with common examples of the theory of umbral calculus.

Gallium oxide (Ga2O3) is a wide-bandgap semiconductor with promising applications in high-power and high-frequency electronics. However, its complex polymorphic nature poses substantial challenges for fundamental studies, particularly in understanding phase-transformation behaviors under nonequilibrium conditions. Here, we develop a robust, accurate, and computationally efficient machine-learning interatomic potential (MLIP) for Ga2O3 based on the neuroevolution potential (NEP) framework combined with an energy-dependent weighting strategy. The resulting NEP potential demonstrates clear advantages over the state-of-the-art tabGAP potential with respect to both accuracy and computational efficiency. Furthermore, we introduce a physically process-oriented sampling strategy to systematically augment the training dataset, thereby enhancing the MLIP performance for targeted physical phenomena. As a representative application, a dedicated NEP potential is constructed for swift heavy-ion (SHI) irradiation simulations of \b{eta}-Ga2O3. The simulated results are in quantitative agreement with experimental observations and provide a consistent physical explanation for the reported experimental discrepancies regarding phase transformations in the ion track of \b{eta}-Ga2O3.

The 2D Euler system, which governs inviscid incompressible fluid flow, can admit infinitely many steady solutions in a given domain with slip boundary conditions. To select physical classical solutions, we investigate the vanishing viscosity limits of the steady Navier-Stokes system. The vanishing viscosity limits in periodic strips or bounded connected domains are completely characterized, even when strong boundary layers may appear. More precisely, we show that the only vanishing viscosity limits in a bounded connected domain are flows with constant vorticity. The significance of this result is that the approximating Navier-Stokes solutions are not required to have nested closed streamlines, an essential assumption in the century-old Prandtl-Batchelor theorem. For flows in an infinitely long strip, if the viscous velocity (but not the pressure) is periodic in the strip direction, we show that the only vanishing viscosity limits are constant flows, Couette flows, and Poiseuille flows. The proof relies on a delicate analysis of the streamlines for both viscous and inviscid flows, in which a key observation is that the set of chaotic streamlines for the Euler flow is null with respect to two-dimensional Lebesgue measure. The second result depends not only on the first but also on a powerful rigidity theorem that any non-shear steady classical Euler flow in a periodic strip must have closed streamlines, established via an analysis of streamlines and a novel total curvature estimate.

Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $q=p-1$ a formula for the variance of the digits of a period was given previously. This formula involves a Dedekind sum. We determine the variance in the case $q=(p-1)/2$. If $p\equiv 3$ mod 4 a Dedekind sum and the class number of $\mathbb Q(\sqrt{-p})$ occur in the respective formula. If $p\equiv 1$ mod 4, the formula may be much more complex since it involves linear combinations of (possibly many) products of two Bernoulli numbers attached to odd characters.

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the ètale cohomology of the finite level Drinfeld spaces.

We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable $C^*$-algebras. Along the way, we reprove some results of Kwaśniewski--Meyer on Fell bundle ideals.

The Fortuin-Kasteleyn-Ginibre (FKG) inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. This inequality has numerous applications and plays an integral role in the proof of various central limit theorems (CLTs), including recent work on ferromagnetic exponential random graph models (ERGMs) wherein a Hamiltonian tilt promotes the presence of small subgraphs like triangles. However, the FKG lattice condition fails to hold when confining a spin system to a particular phase in the low-temperature regime of parameters. Thus it is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the overall model (which is a mixture of phases) arise primarily from the global choice of phase. In this article, we show that the individual phases in ERGMs do indeed satisfy an approximate form of the FKG inequality internally. We use this to finish the proof of various CLTs within each individual phase in the phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini. We present the FKG inequality for ERGMs as a consequence of a more general result which holds under certain inputs related to metastable mixing; we expect this general result to be widely applicable, and we devote a section to spelling out the details of its application to a class of generalized higher-order ferromagnetic Curie-Weiss models where the necessary inputs are relatively transparent.

Negative states are an intrinsic property of relativistic quantum theory and related to anti-particles in the context of the Dirac sea concept. We show that negative states can dominantly contribute to the diffraction amplitude in the quantum dynamics of the two-photon Kapitza-Dirac effect. We draw our conclusion by investigating solutions from time-dependent perturbation theory, where the perturbative solutions are in match with numeric solutions of the relativistic quantum system and also with the numeric and analytic solutions from the relativistic equations of motion of a classical point-like electron in an external standing wave light field. While our numeric solutions assume a strong laser field, the analytic solutions indicate that negative state coupling remains dominant for arbitrary low field amplitudes, where in the single-photon case (Compton scattering) negative state coupling can be mathematically associated with the interaction of a virtual electron-positron pair in the context of a quantized theory in old-fashioned perturbation theory.

In this study, we explore temperature-dependent CPT violation during Big Bang Nucleosynthesis (BBN) through electron-positron mass asymmetries parametrized by $b_0(T) = αT^2$. The $T^2$ scaling naturally evades stringent laboratory bounds at zero temperature while allowing for significant CPT violation at MeV scales in the early universe \cite{ParticleDataGroup:2024cfk}. Using a modified version of the BBN code \faGithub \href{https://github.com/vallima/PRyMordial}{\,\texttt{PRyMordial}} with dynamically-solved chemical potentials and appropriate finite-mass corrections, we constrain electron-positron mass differences from observed abundances of Helium-4, Deuterium, and $N_{\rm eff}$. We find that $α$ must be greater than or approximately equal to $10^{-6}$ GeV$^{-1}$ for keV-scale mass differences at BBN. All three observables show no simultaneous $1σ$ overlap, though pairwise combinations allow for constrained regions of parameter space. We present three toy models demonstrating how $b_0(T) \propto T^2$ arises from field-theoretic mechanisms, including temperature-driven phase transitions. These results provide the most stringent constraints on early-universe CPT violation in this regime, probing parameter space inaccessible to laboratory experiments.

We analyze the mixing time of open quantum systems governed by the Lindblad master equation, showing that it is determined not only by the Liouvillian gap, but also by the trace-norm factor of each decaying Liouvillian eigenmode. By utilizing them as universal predictors of mixing time, we establish general conditions for the fast and rapid mixing, respectively. Specifically, we derive rapid mixing conditions for both the strong and weak dissipation regimes, formulated as sparsity constraints on the Hamiltonian and the local Lindblad operators. Our findings provide a general framework for calculating mixing time and offer a guide for designing dissipation to achieve desired mixing speeds, which has significant implications for efficient experimental state preparation.

The fragile quantum states of low-temperature quantum applications require protection from infrared radiation caused by higher-temperature stages or other sources. We propose a material system that can efficiently block radiation up to the optical range while transmitting photons at low gigahertz frequencies. It is based on the effect that incident photons are strongly scattered when their wavelength is comparable to the size of particles embedded in a weakly absorbing medium (Mie-scattering). The goal of this work is to tailor the absorption and transmission spectrum of an non-magnetic epoxy resin containing sapphire spheres by simulating its dependence on the size distribution. Additionally, we fabricate several material compositions, characterize them, as well as other materials, at optical, infrared, and gigahertz frequencies. In the infrared region (stop band) the attenuation of the Mie-scattering optimized material is high and comparable to that of other commonly used filter materials. At gigahertz frequencies (pass-band), the prototype filter exhibits a high transmission at millikelvin temperatures, with an insertion loss of less than $0.4\,$dB below $10\,$GHz.

The quantum-field-theoretic description for the U(1)-Goldstone boson of a scalar Bose-Einstein condensate with time-dependent contact interactions is developed beyond the acoustic approximation in accordance with Bogoliubov theory. The resulting effective action is mapped to a relativistic quantum field theory on a dispersive (or rainbow) cosmological spacetime which has a superluminal Corley-Jacobson dispersion relation. Time-dependent changes of the s-wave scattering length to quantum-simulate cosmological particle production are accompanied by a time-dependent healing length that can be interpreted as an analog Planck length in the comoving frame. Non-adiabatic transitions acquire a dispersive character, which is thoroughly discussed. The framework is applied to exponentially expanding or power-law contracting $(2+1)$-dimensional spacetimes which are known to produce scale-invariant cosmological power spectra. The sensitivity of these scenarios to the time-dependence of the Bogoliubov dispersion is investigated: We find a violation of scale-invariance via analytically trackable Transplanckian damping effects if the cut-off scale is not well separated from the horizon-crossing scale. In case of the exponential expansion, these damping effects remarkably settle and converge to another scale-invariant plateau in the far ultraviolet regime where non-adiabatic transitions are suppressed by the high dispersion. The developed framework enables quantitative access to more drastic analog cosmological scenarios with improved predictability in the ultraviolet regime that ultimately may lead to the observation of a scale-invariant cosmological power spectrum in the laboratory.

Ways in which people's opinions change are, without a doubt, subject to a rich tapestry of differing influences. Factors that affect how one arrives at an opinion reflect how they have been shaped by their environment throughout their lives, education, material status, what belief systems are they subscribed to, and what socio-economic minorities are they a part of. This already complex system is further expanded by the ever-changing nature of one's social network. It is therefore no surprise that many models have a tendency to perform best for the majority of the population and discriminating those people who are members of various marginalized groups . This bias and the study of how to counter it are subject to a rapidly developing field of Fairness in Social Network Analysis (SNA). The focus of this work is to look into how a state-of-the-art model discriminates certain minority groups and whether it is possible to reliably predict for whom it will perform worse. Moreover, is such prediction possible based solely on one's demographic or topological features? To this end, the NetSense dataset, together with a state-of-the-art CoDiNG model for opinion prediction have been employed. Our work explores how three classifier models (Demography-Based, Topology-Based, and Hybrid) perform when assessing for whom this algorithm will provide inaccurate predictions. Finally, through a comprehensive analysis of these experimental results, we identify four key patterns of algorithmic bias. Our findings suggest that no single paradigm provides the best results and that there is a real need for context-aware strategies in fairness-oriented social network analysis. We conclude that a multi-faceted approach, incorporating both individual attributes and network structures, is essential for reducing algorithmic bias and promoting inclusive decision-making.

A space $X$ is called a generalized Namioka space (g$\mathcal{N}$-space), if for every compact space $Y$ and every separately continuous function $f\colon X\times Y\rightarrow\mathbb{R}$, there exists at least one point $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Y$. We principally prove the following results: (1) If $X=\prod_{α\in A}X_α$ is non-meager such that each factor is a separable space or each factor is a pseudo-metric space, then $X$ is a g$\mathcal{N}$-space. (2) If $X$ is a separable space and $Y$ a pseudo-metric space such that $X\times Y$ is Baire (resp. non-meager), then $X\times Y$ is an $\mathcal{N}$-space (resp. a g$\mathcal{N}$-space). (3) If $X=\prod_{α\in A}X_α$ such that each factor is separable and $\prod_{α\in A^\prime}X_α$ is a non-meager space for each countable subset $A^\prime$ of $A$, then $X$ is a non-meager g$\mathcal{N}$-space. (4) If $X=\prod_{α\in A}X_α$ such that each factor has a countable $π$-base, then each tail set having the property of Baire in $X$ is either meager or residual. If $G$ is a g$\mathcal{N}$ right-topological group and $X$ a locally compact regular space, or, if $G$ is a separable first countable non-meager right-topological group and $X\times X$ a countably compact completely regular space, then any separately continuous action $G\curvearrowright X$ is jointly continuous.