Efficient sparse array reconfigurability is essential for cognitive sensing in dynamic radio frequency environments, where rapid interference variations require both adaptability and stability. This work presents a framework for designing sparse arrays optimized over broad angular sectors, enabling near-optimal beamforming that maximizes the signal-to-interference-plus-noise ratio (SINR) across a range of interferer angles. Full data correlation matrices are computed for candidate configurations, and an angular-sector-based class reduction strategy is applied to merge adjacent sectors dominated by the same configuration, resulting in 56 representative classes. Controlled up- and down-sampling produce four dataset variants involving, high and low sample count, balanced and unbalanced datasets, to systematically evaluate the effects of dataset size and class distribution on neural network performance. A lightweight convolutional neural network (CNN) and a deeper ResNet 50 architecture are trained and evaluated using these datasets. Results demonstrate high classification accuracy, with ResNet 50 achieving up to 97.3%, while SINR deviations remain below 1% for most classes and below 5% even for challenging interference angles near broadside. The proposed approach enables robust sparse array selection, maintains strong SINR performance, reduces unnecessary reconfigurations, and provides an effective framework for real-time cognitive sensing and adaptive interference mitigation.

This paper investigates the use of convolutional neural networks (CNNs) for learning sparse array configurations that achieve near-optimal beamforming under varying source and interference angles. Unlike conventional or convex optimization based algorithms, the proposed deep learning approach enables rapid reconfiguration of sparse arrays in highly dynamic propagation environments. The paper considers a single desired source and a single interference signal at arbitrary angles, analyzing scenarios with both fixed and varying desired source directions. To avoid retraining for each possible source angle, an array pre-steering strategy is introduced, whereby the network is trained only at broadside, while test inputs are pre-steered to align with the broadside direction. To account for practical imperfections, the effect of pre-steering errors is examined, and a robust error-augmented training is adopted. The approach systematically incorporates small, structured pre-steering perturbations during training, enabling the network to maintain high classification accuracy and maximize the signal-to-interference-plus-noise ratio (SINR) even under angular uncertainty. The results demonstrate that the proposed method achieves over 90% test accuracy across wide ranges of source and interference angles, highlighting its potential for real-time, robust sparse array configuration in dynamic environments.

This work conceives two affine precoding based system models, common precoding with joint channel estimation (CP-JCE) and user-specific precoding for decoupled channel estimation (USPDCE). Considering a dual-wideband effected partially connected architecture, we rigorously model the terahertz (THz) multiple input multiple output (MIMO) channel for each subarray corresponding to each user by incorporating the absorption, reflection, and freespace losses. Next, to address the significant bandwidth overhead associated with conventional pilot-based channel estimation, we employ superimposed pilots. Building on this, we formulate a structured sparse channel model and develop a variational Bayesian inference algorithm that jointly estimates the channel coefficients and learns the underlying sparsity structure through hyperparameter inference, thereby enabling robust and high-precision superimposed pilotbased channel estimation under severe model uncertainty. Lastly, we compare our results for both systems and provide a trade-off analysis between them.

This work conceives SEMIKHORN, a semisupervised channel charting (CC) framework for mmWave localization, which leverages t-SNEkhorn, a doubly stochastic variant of t-distributed Stochastic Neighbor Embedding (t-SNE) that utilizes entropic optimal transport to construct pairwise similarities. Unlike standard t-SNE, which normalizes affinities independently for each data point, t-SNEkhorn generates globally balanced similarities ensuring consistent neighborhood representation. We consider wireless networks with distributed base stations (BSs) equipped with multiple antennas, where each BS constructs a local dissimilarity matrix from the channel state information (CSI). These local dissimilarity matrices are then fused to obtain a single global dissimilarity matrix, which is processed through manifold learning to embed users onto a geometric map. The performance is evaluated in a simulated outdoor environment, and Bayesian optimization is employed on the framework hyperparameters to minimize the mean localization error (MLE). Experimental results demonstrate that the proposed framework achieves an MLE of 6.86% in a circular vicinity of radius 100m, requiring less than 15% of labeled CSI samples.

The comparison of two binary images is formulated in terms of mathematical morphology. A new operator, the dilated symmetric difference, is introduced. It is shown that the dilated symmetric difference effectively detects differences between binary images, provided that the residual alignment error is within specified bounds.

In this paper, we study simultaneous hypothesis testing for multivariate Gaussian means under arbitrary covariance dependence. Building on the Maximum Residual Down (MRD) procedure of Cohen et al. (2009), we investigate a new calibration strategy based on the generalized step-down critical constants of Gavrilov et al. (2009). The resulting procedure retains the covariance-adaptive residualization mechanism of MRD while replacing the original model-dependent threshold specification with a simple stagewise calibration rule. Since the proposed procedure belongs to the class of monotone residual-based step-down procedures studied by Ghosh and Chakrabarti (2026), its admissibility follows directly from their theory. We also derive alternative representations of the MRD residual statistics that express all active residuals through a single active precision matrix, substantially reducing computational complexity. Simulation studies across a broad range of dependence structures show that the proposed methodology often achieves a lower normalized misclassification risk than several widely used marginal testing procedures. Under several structured dependence models, the procedure also exhibits strong signal-recovery behavior, attaining false discovery rates near the nominal level, extremely small false non-discovery rates, powers approaching one, and average numbers of rejections close to the expected number of true signals. These findings provide empirical evidence that covariance-adaptive residualization and stagewise calibration may interact in a highly favorable manner for dependent multiple testing.

We study community detection on Markovian random networks outside of the Stochastic Block Model (SBM) framework. Specifically, we consider a random network growth process which generates $K$ separate preferential attachment trees and connects them with Erdős--Rényi edges, so that each tree represents a community and each node inherits the label of the tree to which it belongs. This model is able to produce many features of real world networks that are improbable under SBM, such as power law degree distribution and the existence of chains and hubs. Given only the final graph, without any knowledge of the growth process, we seek to recover the unobserved community membership of the nodes. We first prove that it is impossible for any algorithm to consistently recover the community label of all the nodes. However, we design algorithms which are provably able to recover the community labels of subsets of central nodes, for several different notions of node centrality such as arrival time or degree. Our procedure consists of two stages where, in the first stage, we classify high degree nodes and then, in the second stage, extend the community assignments to the remaining vertices. Numerical experiments and a real data application on a coauthorship network demonstrate the effectiveness of our proposed approach.

Knowledge about optimal designs for standard addition seems to be scattered among literature and is also, at least partially, only available in mathematical literature that is not quickly accessible for readers not skilled in the field of design optimality theory. Therefore, the idea for this work was to summarize what is already available in analytical literature and to apply the respective results from optimality theory, where needed, to the special case of standard addition. It is shown, for measurement errors that are non-decreasing, e.g., are constant or increase linearly or quadratically with increasing analyte concentration, that the optimal design in the case of a linear response is a two-point design irrespective of the particular behavior of measurement error variance. In addition, it is demonstrated that the optimal allocation of measurements depends on the concrete setting, which means that the optimal distribution of measurements may deviate significantly from a 50:50 ratio. It is also investigated how the range, i.e., the largest added concentration influences the result. Last but not least, also the question of applying weighted regression is discussed and it is shown, that, in contrast to designs using more than two spiked concentrations, no weighting is necessary to achieve optimal results, when a two-point design is used. While the focus lies on the precision of the concentration estimate also the implications for the bias are investigated.

Relative-shift regression provides a principled framework for modeling compositional covariates by quantifying how the response changes when mass is reallocated from one component to another. Yet many emerging compositional data problems extend beyond this classical setting, involving high-dimensional predictors and regression effects that vary across latent subpopulations. This complexity poses a dual challenge unmet by existing methods: recovering latent cluster structure while simultaneously achieving dimension reduction within each cluster. We propose a Bayesian heterogeneous relative-shift regression model that jointly learns latent clusters and parsimonious effect structures. Methodologically, we combine a projection-based shrinkage prior on identifiable contrasts, which induces exact coefficient ties within mixture components, with a mixture of finite mixtures prior that infers the number of clusters. Computationally, we develop a scalable hybrid MCMC algorithm that embeds a deterministic surrogate collapse operator within NUTS. Theoretically, we establish posterior consistency for both the latent partition and cluster-specific effect structures. Simulations confirm accurate recovery and strong predictive performance, and applications to cross-country macroeconomic data and spatial transcriptomics demonstrate the method's interpretability and practical utility.

High-stakes computerized adaptive tests (CATs) require a continuous supply of calibrated items, yet traditional item piloting is slow, expensive, and operationally hazardous. We introduce the S2A3 framework -- Soft Scoring (S2) and Adaptive Adaptive Administration (A3) -- which unifies item calibration and test administration into a single online process. Thompson sampling enhances item selection by drawing provisional parameters from each item's posterior distribution and selecting the item maximizing expected Fisher information, naturally routing uncertain items to informative test-takers while maintaining measurement precision. Soft scoring integrates over parameter uncertainty so that incompletely calibrated items exert appropriately attenuated influence on ability estimates. A stochastic variant of Sympson-Hetter exposure control balances measurement efficiency against bank security via a tunable temperature parameter and item-specific weights. We validate S2A3 on Yes/No Vocabulary and Vocabulary-in-Context tasks from the Duolingo English Test, demonstrating rapid item calibration and preserved scoring reliability even when cold-start items constitute a significant fraction of the active pool.

This chapter explores ways to reduce the dimensionality of the data while preserving key information relevant to the analysis of multivariate extreme values.

Missing outcomes in randomized controlled trials are often handled by multiple imputation (MI). Rubin's rules are routinely used to estimate standard errors but can fail to provide valid standard error estimates for some commonly used procedures, such as reference-based imputation. We propose a one-step alternative by explicitly targeting the treatment effect implied by a given imputation model and constructing an efficient one-step estimator for that treatment effect via its influence function. Unlike Rubin's rules, this approach yields asymptotically valid inference. Moreover, the proposed method circumvents the stochastic component and computational burden of MI. We illustrate the approach with examples spanning a range of imputation models, including reference-based imputation and intercurrent-event-dependent imputation.

Generalised Bayesian inference tempers the likelihood by a learning rate $η$ to mitigate model misspecification, and the choice of $η$ is consequential. Zafar and Nicholls (2024) proposed selecting $η$ by a posterior predictive check (PPC): one chooses the smallest $η$ at which a log-likelihood PPC $p$-value is not rejected. An exact, finite-sample analysis of this selector on the Gaussian linear model is given. With known variance and a flat prior, the PPC $p$-value equals $P(χ^2_n > \mathrm{RSS}/σ_0^2)$ for every $η$, so the selector is $η$-invariant; under variance misspecification it is two-sided non-identifying. With unknown variance and the reference prior, the $p$-value depends only on $(n,d,η)$ and not on the realised data or the data-generating process. Consequently the selector's output is fixed before any data are seen, typically collapsing to the smallest grid value, which over-tempers and inflates predictive intervals relative to held-out selection. The phenomenon is a pivotality property specific to the Gaussian scale--location family and the reference prior; it disappears under informative priors. These results delineate the selector's scope, identify a canonical class on which it cannot identify the learning rate, and motivate a cheap, data-free pre-screening diagnostic.

The kappa statistic is the most widely used measure of inter-rater agreement for categorical data. Despite its popularity, applied researchers often encounter two major hurdles: (i) determining the sample size required to achieve a desired level of agreement with given power, and (ii) computing appropriate kappa coefficients with proper interpretation. Existing R packages such as irr and kappaSize provide these functionalities but require programming skills and lack an integrated, user-friendly interface. We present CATEKAPPA, an R package that bridges this gap by combining sample size planning (via kappaSize) and agreement analysis (via irr) into a single Shiny-based web application. The package supports Cohen's kappa for two raters, Fleiss' kappa for three or more raters, and Light's kappa, and provides automatic interpretation using the Landis & Koch scale. Users can either launch an interactive graphical interface or use command-line functions for scripting. The package is freely available on CRAN.

Background and objective: Prediction stability is increasingly recognised as important for reliable clinical prediction model development, but the effect of continuous predictor modelling choices is unclear. This study examined how approaches to modelling continuous predictors influence prediction stability. Methods: We used a real clinical dataset of 19,418 emergency department patients to create five sample size scenarios ranging from 437 to 8,739 patients. Six methods were compared: dichotomisation at the median (DIC), tertile categorisation (TER), linear terms (LIN), quadratic terms (QUA), multivariable fractional polynomials (MFP), and extreme gradient boosting (XGB). Prediction stability was evaluated using a bootstrap-based framework. Optimism-corrected AUC and calibration were estimated through internal validation. A method was considered stable when at least 90% of individual predictions had a mean absolute prediction error (MAPE) <=5%. Results: Stability increased with sample size and varied by method. At n = 437, no method met the stability criterion; LIN was the most stable, followed by DIC. At n = 874, DIC and LIN achieved stable predictions with similar calibration, although DIC had lower AUC. At n = 1,748, QUA achieved stability, whereas MFP and XGB did not. At n = 3,496 and n = 8,739, all methods achieved stability. LIN, QUA, MFP, and XGB generally had higher AUCs than DIC and TER, while XGB showed the highest AUC but persistent miscalibration. Conclusion: Continuous predictor modelling methods appeared to influence prediction stability. LIN achieved stable predictions from the base sample size onwards, whereas QUA, MFP, and XGB required larger samples. Although XGB showed high discrimination, calibration concerns persisted. These findings suggest that, in smaller datasets, simpler approaches, particularly LIN, may provide more stable predictions.

As European bidding zones are highly interconnected by physical transmission lines, spatial influences propagate across neighboring nodes through a network. It is reflected in the day-ahead electricity prices across European bidding zones, as the auction algorithm also uses information beyond each bidding zone's geographic boundary. To capture how this interconnection affects the electricity prices in neighboring bidding zones, we have used a metric graph to map the spatial coverage of information using a well-defined neighborhood measure. We propose the Networked Spatio-Temporal Model (NSTM), which maps irregular spatial nodes into an ordered network, enabling the systematic incorporation of neighborhood information. We implement the NSTM across 39 bidding zones covering the majority of European electricity markets in a high-resolution, streaming-forecasting setup. The model uses autoregressive, cross-hour, and seasonal effects, along with fuel and emission prices and day-ahead forecasts of fundamentals, as interconnected information to predict the day-ahead prices for each bidding zone. A Europe-wide study presented in this paper shows that the NSTM consistently outperforms traditional island-based pure local models. This paper provides a framework that demonstrates the critical role the networked structure plays in propagating information across interconnected markets and its vast implications for day-ahead electricity price forecasting.

Driven by four applications of Plackett-Burman (PB) designs, this paper proposes a causal inference framework based on potential outcomes. First, we define the causal effects of the PB designs under finite populations. The Neymanian estimator of causal effects is then obtained, including the estimated variance and covariance. Furthermore, we conduct a sharp null-hypothesis test and construct the Fisherian interval using an algorithm. Finally, the proposed methods are illustrated through these applications.

We study the problem of testing whether two conditional distributions are equal using generative models. The proposed method learns a conditional generator from each sample and uses it to create responses at covariate values observed in the other sample, allowing generated and observed responses to be compared directly. By aligning covariates through cross-generation, the approach avoids conditional density-ratio estimation and local smoothing over high-dimensional covariates. The population version of this construction yields a conditional discrepancy that characterizes equality of the two conditional distributions under suitable overlap conditions, while the sample version leads to a test statistic defined as the supremum of an RKHS-indexed empirical process with multiplier bootstrap calibration. A computationally efficient algorithm for evaluating the statistic and its bootstrap analogue is developed based on alternating maximization and the kernel trick. Theoretically, we derive the limiting distribution of the test statistic under both the null and alternative hypotheses, prove bootstrap validity and consistency of the resulting test, and show that the proposed procedure attains a double-robustness property with respect to conditional generator estimation errors. Simulations and real data applications suggest that the proposed method performs well for multivariate responses and high-dimensional covariates.

DeepKriging-style models, such as Spatio-Temporal DeepKriging, improve scalability through basis-function embeddings and stochastic gradient learning; however, fixed regular-grid spatial bases remain inefficient under highly non-uniform sampling patterns, often over-allocating capacity to sparse regions while under-resolving dense clusters. To address this limitation, we propose a practical extension of DeepKriging for reliable spatio-temporal distributional forecasting, incorporating cluster-adaptive spatial bases - whose centers and scales are initialized from {the spatial sampling density} - to better capture heterogeneous spatial sampling, together with cluster-aware conformal calibration that determines prediction-interval widths within spatial clusters (with a global fallback when calibration samples are insufficient). The resulting calibration pipeline explicitly targets spatial heterogeneity and local miscalibration, and experiments, including simulation studies and PM$_{2.5}$ data analysis, demonstrate substantially improved coverage accuracy and tail reliability under clustered observation patterns compared with a global conformal baseline.

With the increasing availability of ranking data, there has been a growing demand for appropriate unsupervised rank-based inferential frameworks capable of handling high-dimensional datasets and providing uncertainty quantification for all estimates. Rank-based methods have also seen a growing popularity in -omics pipelines, as ranking continuous measurements provides a robust means of handling non-normally distributed data. The Bayesian Mallows model (BMM) has emerged as a promising choice because of its adaptability to various types of ranking data and its flexible framework, integrating cluster-wise rank aggregation with inference at the individual level. However, the scalability of BMM to ultra-high-dimensional settings, such as -omics analyses, has remained limited. The present paper addresses this issue by introducing the first rank-based model generalizing BMM to jointly handle clustering and variable selection, namely the lower-dimensional Bayesian Mallows Model Mixture (lowBM3). The proposed method provides a novel Bayesian framework that simultaneously handles heterogeneity in the sample, unsupervised parameter estimation, and model selection in a scalable manner for ultra-high-dimensional data. Additionally, a companion postprocessing framework is introduced to provide posterior summaries of the discrete posterior distributions of both the consensus ranking and the variable selector. Simulation studies are performed to assess the performance of the method. The usefulness of the method is also shown in an application to signature discovery for cancer genomics, where RNA-seq bulk gene expression data obtained from breast cancer patients are clustered genome-wide.

Highly reliable products are often tested under accelerated conditions to provoke failures within a feasible timeframe. For products whose service life involves repeated alternation between two stress levels, such as automotive air-conditioners, batteries, and aerospace components, cyclic-stress accelerated life testing (CyALT) provides a more realistic loading profile than conventional accelerated tests. In practice, failures are often recorded only at scheduled inspection times, leading to interval-censored counts rather than exact lifetimes. Moreover, traditional maximum likelihood estimation is sensitive to data contamination, which is a genuine concern in small-sample industrial experiments. This paper develops robust inferential procedures for CyALT models with lognormal lifetimes under interval monitoring. Robust estimators are obtained by minimizing a weighted density power divergence (WDPD), leading to the weighted minimum density power divergence estimator (WMDPDE). We establish the asymptotic distribution of the WMDPDE, derive influence function expressions to characterize the robustness, and present asymptotic and bootstrap confidence intervals for important lifetime characteristics. A simulation study confirms that the WMDPDE provides substantial protection against outliers while retaining high efficiency under clean data. The methodology is illustrated through the analysis of an air-conditioner reliability dataset, demonstrating the practical advantages of robust inference in the CyALT framework.

Forecasting systems are commonly refreshed at every review period, and that refresh usually bundles two distinct operations: estimating parameters and selecting the model form. Recent evidence suggests the second operation is often unnecessary, since intermediate updating strategies can hold forecast accuracy roughly fixed while cutting computational cost and forecast instability. This technical note takes up the complementary question. Once a system has adopted a reduced-update policy, when should it interrupt that policy and re-specify the model form? We define specification debt as the evidence accumulated against the deployed model form, and we use it to build a cost-sensitive trigger for re-specification. In a closed discrete model space the trigger reduces to a threshold on the negative log posterior probability of the deployed specification. In open production settings the same decision rule can be run with predictive score gaps, stacking weights, or calibrated monitoring diagnostics. Fixed update frequencies turn out to be a special case of the rule, recovered when evidence against the deployed form accumulates at a constant rate. We illustrate the idea on 500 monthly M4 series, comparing full updating, fixed model-form update frequencies, parameter-only updating, and capped adaptive score-triggered updating, and within the finite ETS grid we also compute information-criterion analogues of specification debt from AIC and BIC weights over the candidate forms. In that illustration the best capped adaptive policy is comparable to full updating in accuracy, runs in about 28 percent of full-update computational time, lowers forecast instability, and behaves like a fixed schedule with a small number of evidence-based exceptions.

In various application domains, there is a certain `null cell', inside a multinomial setup, where observations are recorded for the other cells, but where one cannot count the number of occurrences for the null cell. I develop inference theory for assessing such unknown numbers, counting the uncounted, in situations where counts are available for the other cells, via parametric modelling. The methods are used to estimate the number of persons killed in Guatemala during the Genocidio guatemalteco years 1978--1995. There are three carefully curated lists of killed people, where the information can be mapped to a Venn diagram with $2^3=8$ cells. Summing over the seven observed cells, $R=\hbox{47,803}$ killed individuals can be identified, but how big is $N_{0,0,0}$, and hence $N=N_{0,0,0}+R$?

This paper presents an open-economy macroeconomic equilibrium model for Proof-of-Stake (PoS) networks with fee-burn mechanics (EIP-1559) that formalizes the strategic interplay between a Kelly-optimizing rational institutional investor and a utility-driven retail consumer. We analyze network dynamics across two behavioral regimes. In The Unbounded Accumulation Model, the consumer purely accumulates tokens, creating an exclusive buy-side pressure that interacts with institutional portfolio rebalancing to fuel an ever-expanding speculative bubble and generate compounding excess returns for investors. Conversely, in The Utility-Consumption Model, the consumer dynamically buys and sells tokens to balance crypto wealth against real-world fiat consumption. Within this framework, we derive an explicit steady-state equilibrium price for ETH, demonstrating how token valuation anchors to a stable fundamental baseline that scales directly with network adoption while completely dissolving the institutional yield premium. Our numerical simulations show that while exogenous traditional finance (TradFi) shocks propagate through portfolio rebalancing to drive high token price volatility, network inflation remains highly stable. Furthermore, we prove that network security is insulated from institutional monopoly by counter-cyclical consumer behavior. Our findings reveal that institutional excess wealth creation in PoS ecosystems is not native to the staking protocol itself, but is strictly driven by the leveraged extraction of the retail consumer's continuous demand for transactional utility.

Museums can serve as policy tools when their content is purposefully curated. We designed a framed field experiment at the Santa Maria della Scala museum in Siena that leveraged the site's historical role offering care and hospitality.Student visitors randomly assigned to a tour emphasizing this function later donated more to an NGO supporting refugee than those who followed a standard artistic itinerary, with effects concentrated among female participants. These results show that thematically targeted museum experiences can measurably boost charitable behavior toward vulnerable groups, underscoring the untapped potential of cultural institutions in behavioral public policy.

Decomposing impact as the difference between realized and counterfactual returns and requiring both to be diffusive yields an identity that restricts admissible impact scaling at the level of individual participants. This constraint implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive -- a diagnostic that many propagator and latent liquidity models fail to satisfy.

This article introduces an unconventional model for price processes in finance that emerges from the classical Heston model under Mechkov's fast-reversion limit. This new fast-excursion Heston model exhibits instantaneous (i.e. fast) excursions through an interval of prices at each time, which are invisible to vanilla options but critical for hitting probabilities and continuously monitored exotics. Theoretically, the model provides a rare example of a non-degenerate limit of stochastic volatility models that escapes the Skorokhod topologies. This leads us to a class of interval-valued processes which exist as lifts of subordinated Levy processes, through the concept of selections in the theory of random closed sets. On the practical side, we show how the model can be simulated using price-time parametric representations, and utilise a purpose-built classical Heston simulation scheme in order to visualise convergence. Finally we demonstrate how this model raises hitting probabilities for barrier options considerably (of order 10% for one-month EURUSD options), due to taking excursion risk into account.

This paper develops a binary-gamble framework for characterizing risk sensitivity and loss aversion in Cumulative Prospect Theory (CPT). The proposed probabilistic risk-sensitivity metric is defined as a probability-threshold ratio that determines acceptance and preference thresholds in choice problems involving either a certain outcome and a binary gamble or two binary gambles. We show how standard notions of symmetric and non-symmetric bet aversion can be recovered within this framework, and we compare the resulting threshold-based conditions with utility premia, probability premia, and Arrow--Pratt curvature measures. The analysis clarifies when these criteria coincide and when they diverge, particularly for increasing aversion conditions, binary gambles with unequal probability distributions, and settings involving probability weighting functions. We also identify technical restrictions that arise when CPT-utility functions are used to represent loss aversion at the reference point. The resulting framework provides a decision-theoretic interpretation of risk sensitivity that is directly tied to probability thresholds and complements existing premium-based approaches.

In the demographic framework, mortality payoff function describes the cost of an interaction and fertility payoff function describes its reward. So while mortality cost depends on opponent's strategy, fertility reward can be affected by the density-dependent juvenile recruitment survival. This motivates an analysis of the eco-evolutionary dynamics of the classical Hawk-Dove game. It is shown that the stable and unstable equilibria (determined by the intersections of frequency and density nullclines) are connected by heteroclinic orbits, which attract nearby trajectories. The resulting bundle of trajectories leads to the discovery of the so-called subnullcines (manifolds placed between frequency and density nullcline) before they converge to the stable rest point. The initial isolated system is then extended by adding environmental seasonality (periodic background mortality), which acts as an external factor. This leads to complex cycling behavior and the subnullclines act as barriers to the propagation of the perturbation (resilience/resistance threshold). Thus, in a way, this paper completes, yet extends, previous works on the eco-evolutionary dynamics of games with demographic payoffs.

Brains routinely generate highly flexible and complex behaviors on a relatively stable structure and limited resources. A key mechanism underlying this ability is compositionality, which allows the brain to efficiently decompose complex tasks into simpler, reusable primitives. While network modularity has often been linked to compositionality in biological and artificial networks, a rigorous mathematical characterization of this relationship in nonlinear networks is still lacking. In this work, we formally investigate how structural modularity supports functional compositionality in inhibition-dominated threshold-linear networks (TLNs). We introduce a novel class of modular network assembly called low-rank gluings, where component subnetworks with arbitrary internal connectivity are connected via specific low-rank couplings. We prove that the global fixed points of these networks are constrained to be combinations of the local fixed points of their constituent modules. For a more structured subclass, called rank-1 gluings, we provide a complete characterization that determines which combinations of local fixed points yield global ones. We apply these results to graph-based networks, extending fixed point decomposition rules from combinatorial threshold-linear networks (CTLNs) to the more flexible family of generalized CTLNs (gCTLNs), thereby proving that these structural rules are more robust than initially posited. Finally, we demonstrate that these gluing rules provide a mathematically tractable recipe for engineering compositional dynamics, enabling the construction of networks with a combinatorially large repertoire of predictable attractors that can be understood from simpler component motifs, ranging from compositions of fixed points to compositional limit cycles.