We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $Ω(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. In this work, we give a simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu. We additionally discuss some additional applications of near-$X$-Ramanujan graphs to "on average" lossless expansion and rotating expanders.

Fluid dynamics spans phenomena from the Cheerios effect to cosmic evolution and has been called the 'queen mother' of science. Traditional modelling relies on numerical methods, including finite differences, volumes, and elements, that simulate flows across scales. Recent advances in machine learning have enabled data-driven fluid models, but these approaches are often complex and opaque. We introduce a transparent framework that links data-driven models directly to classical fluid-dynamics operators. A simple convolutional neural network (CNN) is trained on laminar-flow data to reproduce the exact behaviour of a finite-difference scheme, providing an interpretable bridge between numerical analysis and machine learning (ML). The CNN generalises across a wide range of unseen flow conditions and learns the forward-Euler three-point stencil, capturing principles such as consistency and symmetry with only three trainable weights. The approach extends beyond numerical data: the same architecture works when trained on analytical solutions and even molecular-dynamics trajectories. Its simplicity reveals when and why physics is or is not captured, offering insight into the limits and best practices of data-driven fluid modelling. Because it is grounded in finite-difference operators, the method naturally generalises to many structured-grid CFD problems, including turbulent, multiphase, and multiscale flows.

We present a terahertz time-domain spectroscopy (THz-TDS) system which accesses a broadband spectrum, efficiently covering the so-called "new THz gap" between 5 and 15 THz and extending beyond 25 THz. The system exploits nonlinear interactions within the organic crystal BNA (N-benzyl-2-methyl-4-nitroaniline) to generate and detect THz radiation upon excitation by a near-infrared (NIR) pulse centered at 1.03 $μ$m. To enable broadband THz spectral monitoring, the NIR pulse from a Yb-based solid-state laser undergoes spectral broadening in a gas-filled single-ring hollow-core photonic crystal fiber, followed by a pulse compression to achieve durations as short as 31 fs. This approach paves the way for broadband spectroscopy in hard-to-access THz regions using widely available near-infrared ultrafast sources.

The Future Circular Collider (FCC) stands at the forefront of the European Strategy for Particle Physics as the future flagship project at CERN. The H$\toττ$ decay, featuring a large branching ratio, clean identification in the FCC-ee environment, and the possibility to reconstruct polarization information, is an excellent channel to measure Higgs boson properties. This work shows the expected precision for the H$\toττ$ cross-section measurement at the FCC-ee in the ZH production mechanism at $\sqrt{s}=$240 GeV and $\sqrt{s}=$365 GeV, as well as via the vector boson fusion process at $\sqrt{s}=$365 GeV. Furthermore, we explore and evaluate a set of methods for reconstructing tau decays. These techniques are critical for unlocking the full physics potential of the FCC-ee and for improving the understanding of tau-related observables in both Standard Model measurements and New Physics searches. The results obtained significantly enhance the FCC-ee outlook in the H$\toττ$ channel, improving it by at least an order of magnitude compared to the current sensitivity of measurements' performance at the LHC.

Transonic buffet is a class of shock-wave/boundary-layer interaction known to exhibit self-sustained two-dimensional (2D) chordwise shock wave oscillations (Strouhal number St=0.05-0.1), and three-dimensional (3D) spanwise-modulated flow separation/reattachment (St=0.2-0.4). Due to computational cost, scale-resolving simulations of span-periodic configurations to date have been limited to narrow airfoils, insufficient to accommodate the 3D buffet cell instability reported in low-fidelity simulations and experiments. In this work, implicit large-eddy simulations (ILES) and modal analysis are performed on infinite swept wings up to AR=3. The sensitivity of the 2D and 3D modes to crossflow is detailed. Two flow conditions are examined, corresponding to minimally and largely separated mean flow at the shock location. For the minimally separated case, the shock dynamics remain essentially spanwise-uniform (quasi-2D), with only weak and intermittent separation cells confined to the trailing-edge region and exhibiting negligible interaction with the shock. In contrast, increased mean separation leads to the emergence of pronounced 3D buffet cells with a characteristic spanwise wavelength: 1-1.5c. SPOD reveals that a quasi-stationary low-frequency 3D separation mode previously identified on unswept wings (St=0.02) becomes a spanwise travelling mode as sweep is imposed, shifting monotonically to intermediate frequencies (St=0.06-0.35). The 2D shock mode is largely insensitive to sweep, whereas the frequency and energy content of the 3D mode increase with sweep while its wavelength remains unchanged. The results demonstrate that transonic buffet on infinite wings arises from the superposition of distinct but coexisting 2D shock motion and separation-driven 3D instabilities, with mean flow separation at the shock identified as a necessary condition for dominant 3D buffet dynamics to emerge.

Using (10087$\pm$44)$\times$10$^{6}$ $J/ψ$ events collected with the BESIII detector at the BEPCII collider at the center-of-mass energy of $\sqrt{s}=3.097~\rm{GeV}$, we report the first search for $η\toπ^0S\toπ^0χ\barχ$ with $S$ denotes an on-shell dark scalar boson and $χ$ an invisible dark matter particle. No significant signals are observed with $S$ mass ranging from 0 to 400 $\rm{MeV}/c^2$. The upper limits on the branching fractions and the new physics coupling strengths between $S$ and quarks are set to be $(1.8\sim5.5)\times10^{-5}$ and $(1.3\sim3.2)\times10^{-5}$ at the 90% confidence level, respectively. The constraints on the dark-matter-nucleon scattering cross section is improved by approximately 5 orders of magnitude over previous dark-matter-nucleon scattering experiments, providing unique insights into sub-GeV dark matter.

Engineering sublattice imbalance within the unit cell of bottom-up synthesized graphene nanoribbons (GNRs) represents a versatile tool for realizing custom-tailored quantum nanomaterials. The interaction between low-energy zero-modes (ZMs) not only contributes to frontier bands but can form the basis for magnetically ordered phases. Here, we present the bottom-up synthesis of a non-centrosymmetric GNR that places all ZMs on the majority sublattice sites. Scanning tunneling microscopy and spectroscopy reveal that strong electron-electron correlations, leading to the Stoner magnetic instability, drive the system into a ferromagnetically ordered insulat-ing ground state featuring a sizeable band gap of Eg ~ 1.2 eV. At higher temperatures, a chemical transformation induces an insulator-to-metal transition that quenches the ferromagnetic order. Tight-binding (TB), density functional theory, and GW calculations corroborate our experimental observations. This work showcases how control over molecular symmetry, sublattice polarization, and ZM hybridiza-tion in bottom-up synthesized nanographenes can open a path to the exploration of many-body physics in rationally designed quantum materials.

The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it also admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula. We provide a similar interpretation of the multiplicative inversion formula in terms of Chern numbers of the smooth theta divisors. In this relation we introduce a new formal group defined by the Catalan numbers and explain the topological meaning of the corresponding Hirzebruch genus. Finally, we discuss a related general problem of when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.

Research on the editorial boards of scholarly journals has predominantly relied on static, cross-sectional data, focusing on their composition or interlocking editorships at single points in time. To address this gap, a formal stock-flow framework is developed for analyzing the longitudinal dynamics of editorial boards. The model integrates three interconnected layers: journal demographics, the dynamics of editorial positions, and the dynamics of board members. This framework is applied to the Gatekeepers of Economics Longitudinal Database (GOELD), which contains annual snapshots of editorial boards for approximately 1,700 economics journals from 1866 to 2006 (by decade), plus the years 2012 and 2019. The period until 1946 was characterized by small-scale: few journals and compact editorial communities. The decade from 1946 to 1956 marked the shift toward a ''big science'' model, initiating an era of expansionary growth fueled primarily by the founding of new journals. The contemporary period (2006-2019) appears to represent a structural break, characterized by low flux and more stable and more closed editorial communities. The results shows that the proposed framework enables a dynamic, long-term analysis of how journals and their gatekeeping systems evolve, grow, and structure themselves.

We introduce the velocity coherence scale $R_v$, the scale at which the spherical volume average of the trace of the velocity correlation tensor transitions from scaling faster than the sphere radius to scaling more slowly. This corresponds to the radius at which the average motion of galaxies along their separation vectors transitions from correlated to anti-correlated. We derive a theoretical estimator for $R_v$ by defining the bulk in spheres $B_R$, a velocity-field analogue of the mean scale counts used in density-field correlation analyses. We show that, for a statistically homogeneous matter distribution, the logarithmic derivative of $B_R$ and the correlation dimension $D_2$ share the same asymptotic behaviour and therefore can be used to estimate the scale of transition to statistical homogeneity. Furthermore, we show that in standard $Λ$CDM cosmologies the velocity coherence scale is tightly connected to the matter-radiation equality scale $k_{eq}$, and that its value in comoving coordinates is redshift-independent. These results highlight the potential of $R_v$ both as a standard ruler and as a physically motivated scale characterising the onset of cosmic homogeneity. We present a proof of concept using measurements of the PV correlation functions from SDSS. We show that the main challenge in determining $R_v$ is the limited precision of PV measurements compared to density ones, as they typically rely on smaller samples with larger uncertainties that scale roughly linearly with survey depth. Fitting our theoretical estimators for $R_v$, we obtain $R_v \approx 132^{+29}_{-51}\,\mathrm{Mpc}/h$. Finally, we show that more precise determinations should be achievable with current and upcoming peculiar velocity surveys.

Solving quadratic equations over finite fields is a fundamental task in algebraic coding theory and serves as a key subroutine for computing the roots of cubic and quartic polynomials. Notably, any quadratic polynomial over binary extension fields can be transformed into the reduced form $x^2+x+c\in \mathbb{F}_{2^m}[x]$, for which existing formula-based methods rely on heavy exponentiation or case distinctions on $m$ (odd/even or powers of two), limiting uniformity and efficiency. This paper presents a unified, formula-based solution for all positive integers $m$ that uses only exclusive-OR operations (XORs). The approach leverages a Reed-Muller matrix characterization of evaluations and transforms the problem into computing a binary matrix-vector multiplication. The total cost is at most $m^2-2m+1$ XORs, and under parallelism, the latency is $\lceil \log_2 m\rceil$ XORs, making the method attractive for low-power, low-latency applications.

We define and study burst-covering codes. We provide some general bounds connecting the parameters of a code with its burst-covering radius. We then provide stronger bounds on the burst-covering radius of cyclic codes, by employing linear-feedback shift-register (LFSR) sequences. For the case of BCH codes we prove a new bound on pattern frequencies in LFSR sequences, which is of independent interest. Using this tool, we can bound the burst-covering radius of binary primitive BCH codes and Melas codes. We then present an efficient burst-covering algorithm for cyclic codes. Finally, we present a bound on the critical exponent of cyclic codes based on the burst-covering radius.

We revisit the quasinormal-mode/greybody factor correspondence for Kerr black holes in the eikonal limit and develop a systematic WKB-based formulation by recasting the radial Teukolsky equation into a Schrödinger-type equation with a short-range potential. Building on earlier studies of the correspondence in rotating backgrounds, we extend the analysis to gravitational perturbations and incorporate higher-order WKB corrections beyond the leading eikonal approximation. For gravitational perturbations, the predicted greybody factors are in good agreement with numerical results obtained from the generalized Sasaki-Nakamura equation, with increasing accuracy at large angular quantum number. We also identify the breakdown of the correspondence in the superradiant regime, where the WKB assumptions cease to be valid.

We study the mixing time of Glauber dynamics for Ising models in which the interaction matrix contains a single negative spectral outlier. This class includes the anti-ferromagnetic Curie-Weiss model, the anti-ferromagnetic Ising model on expander graphs, and the Sherrington-Kirkpatrick model with disorder of negative mean. Existing approaches to rapid mixing rely crucially on log-concavity or spectral width bounds and therefore can break down in the presence of a negative outlier. To address this difficulty, we develop a new covariance approximation method based on Gaussian approximation. This method is implemented via an iterative application of Stein's method to quadratic tilts of sums of bounded random variables, which may be of independent interest. The resulting analysis provides an operator-norm control of the full correlation structure under arbitrary external fields. Combined with the localization schemes of Eldan and Chen, these estimates lead to a modified logarithmic Sobolev inequality and near-optimal mixing time bounds in regimes where spectral width bounds fail. We complement these results by proving exponential lower bounds on the mixing time for low temperature anti-ferromagnetic Ising models on sparse random regular graphs and Erdös-Rényi graphs, based on the existence of gapped states as in the recent work of Sellke.

In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists a natural continuous perfect pairing \[ \mathrm{Br}_1(X)\times H_0(X,\mathbb{Z})_τ^{\wedge} \to \mathbb{Q}/\mathbb{Z}, \] where $\mathrm{Br}_1(X):=\ker(\mathrm{Br}(X)\to\mathrm{Br}(\overline{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z})_τ$ is the zeroth group of truncated homology $\mathrm{Hom}_{D(k_{\mathrm{sm}})}(τ_{\leq 1}Rϕ_*\mathbb{G}_{m,X},\mathbb{G}_{m,k})$, $ϕ$ is the structure morphism of $X$, and $(-)^{\wedge}$ is the profinite completion functor.

In this paper, we study continuous-state interacting multi-type branching processes with immigration (CIMBI processes), where inter-specific interactions -- whether competitive, cooperative, or of a mixed type -- are proportional to the product of their type-population masses. We establish sufficient conditions for the CIMBI process to never hit the boundary $\partial\mathbb{R}_{+}^{d}$ when starting from the interior of $\mathbb{R}_{+}^{d}$. Additionally, we present two results concerning boundary attainment. In the first, we consider the diffusion case and prove that when the constant immigration rate is small and diffusion noise is present in each direction, the CIMBI process will almost surely hit the boundary $\partial\mathbb{R}_{+}^{d}$. In the second result, under similar conditions on the constant immigration rate and diffusion noise, but with jumps of finite activity, we show that the CIMBI process hits the boundary $\partial\mathbb{R}_{+}^{d}$ with positive probability.

Higher-order interactions beyond pairwise relationships in large complex networks are often modeled as hypergraphs. Analyzing hypergraph properties such as triad counts is essential, as hypergraphs can reveal intricate group interaction patterns that conventional graphs fail to capture. In real-world scenarios, these networks are often large and dynamic, introducing significant computational challenges. Due to the absence of specialized software packages and data structures, the analysis of large dynamic hypergraphs remains largely unexplored. Motivated by this gap, we propose ESCHER, a GPU-centric parallel data structure for Efficient and Scalable Hypergraph Evolution Representation, designed to manage large scale hypergraph dynamics efficiently. We also design a hypergraph triad-count update framework that minimizes redundant computation while fully leveraging the capabilities of ESCHER for dynamic operations. We validate the efficacy of our approach across multiple categories of hypergraph triad counting, including hyperedge-based, incident-vertex-based, and temporal triads. Empirical results on both large real-world and synthetic datasets demonstrate that our proposed method outperforms existing state-of-the-art methods, achieving speedups of up to 104.5x, 473.7x, and 112.5x for hyperedge-based, incident-vertex-based, and temporal triad types, respectively.

We study the quantum Ising model on (2+1)-dimensional anti-de Sitter space using Matrix Product States (MPS) and Matrix Product Operators (MPOs). We explore the bulk phase diagram of the theory on regular tessellations of hyperbolic space with coordination number seven and find disordered and ordered phases separated by a phase transition. We find that the boundary-boundary spin correlation function exhibits power law scaling deep in the disordered phase of the Ising model consistent with holography. At the critical point, we find the boundary entanglement entropy scales logarithmically with subsystem size but away from this, we see a linear scaling. In comparison, the full system exhibits a volume law scaling, which is expected in chaotic and/or highly connected systems. We also measure Out of time Ordered Correlators (OTOCs) to explore the scrambling behavior of the theory.

The $R^2$ and the single-field-like regime of $R^2$-Higgs inflation are disfavored by the observed high spectral index $n_s$ from the combined cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) measurements at the $\sim2σ$ level. The addition of a dimension-six $R^3$ term in the action helps alleviate this tension. We show that the parameter space accounting for the observed high $n_s$ also induces rapid Goldstone and Higgs preheating. The preheating, especially from Goldstone modes, helps match the CMB and inflationary scales, which in turn supports the observed $n_s$.

Predicting when a chaotic trajectory will switch between the lobes of the Lorenz attractor is a long-standing challenge in nonlinear dynamics. This work shows that algebraic conservation laws, constructed by augmenting phase space with history-accumulating auxiliary variables, provide a deterministic solution. Systematic enumeration identifies eighteen valid invariants in three classes, each tied to a nullcline of the Lorenz flow, while six candidates fail, proving that the dynamics constrains which conservation laws are admissible. One class generates sharp spikes synchronized with lobe-switching events, achieving $99.2\%$ sensitivity with $0.3\%$ false-positive rate ($\mathrm{AUC} = 0.9995$) as a continuous Poincaré section analogue. The spike amplitude predicts switching latency via $Δt = t_{\min} + C\mathcal{A}^{-n}$ with $R^2 > 0.95$ across all parameter combinations tested. At canonical parameters $(σ, ρ, β) = (10, 28, 8/3)$, $n = 2.14 \pm 0.17$ with $R^2 = 0.93$ for individual events; the exponent increases with $β$ and decreases with $ρ$, while the $σ$-dependence is non-monotonic. The latency distribution reveals a topological gap of width $Δt_{\mathrm{gap}} \approx 0.68 \pm 0.01$ for $ρ$ sufficiently above the onset of chaos, explained by the Shilnikov passage map. Under stochastic perturbations, lobe-sensitive invariants are ${\sim}\,10^3$ times more robust than their smooth counterparts. In the Rayleigh-Bénard convection context, the auxiliary variables correspond to integrated heat-flux anomalies. Conservation is verified to $O(10^{-36})$.

We consider heavy neutral bosons in the 3-3-1 model with axionlike particles (331ALP), including the Higgs boson and the $Z^\prime$ boson which are outside the standard model (SM). Based on gluon-gluon fusion at the LHC, we investigate the signals of cross-sections in the parameter space region satisfying the current experimental limits of lepton flavor violating decay, including processes involving both charged leptons and Higgs boson, and provide predictions of $m_{h_2}\geq 600 ~\mathrm{GeV}$. A new gauge boson, labeled as $Z^{\prime}$, is predicted $m_{Z^{\prime}}\geq 5.1 ~\mathrm{TeV}$ based on the search for high-mass dilepton resonances at ATLAS and CMS. We consider the stability of odd-$Z_2$ particles, with $Z_2$ is assumed a residual symmetry after spontaneous symmetry breaking stages, to point out dark matter candidates in the model. Investigating the relic density of dark matter within experimentally permissible limits, we established a relationship between the mass of dark matter and the breaking scale of axion.

The 3-fold cyclic quotient singularity denoted $\tfrac{1}{7}(1,2,4)$ admits a crepant resolution X with three exceptional Hirzebruch surfaces intersecting pairwise along curves. We show that the derived category D(X) carries a faithful action of a quiver braid group, where the relevant quiver is a 3-cycle encoding the intersection data.

Catastrophe risk has long been recognized to pose a serious threat to the insurance sector. Catastrophe risk pooling offers an effective way to diversify losses arising from catastrophic events. In this paper, we investigate a structure of catastrophe risk pool and optimize it so that participants can attain the maximum diversification benefit from joining the pool. Determining the practical optimal pool entails solving a high-dimensional optimization problem, for which analytical solutions are typically unavailable and numerical methods can be computationally intensive and potentially unreliable. To address this challenge, we evaluate the diversification benefit in the limit and use it to derive an asymptotically optimal pool which approximates the practical optimal pool. Through simulation studies, we show that the asymptotically optimal pool provides an accurate and reliable approximation to the practical optimal pool. We also conduct an empirical analysis using data from the U.S. National Flood Insurance Program to illustrate how the framework can be applied in practice.

Conventional robust H2/H-infinity control minimizes the worst-case performance, often leading to a conservative design driven by very rare parametric configurations. To reduce this conservatism while taking advantage of the stochastic properties of Monte Carlo sampling and its compatibility with parallel computing, we introduce an alternative paradigm that optimizes the controller with respect to a stochastic criterion, namely the conditional value at risk. We present the problem formulation and discuss several open challenges toward a general synthesis framework. The potential of this approach is illustrated on a mechanical system, where it significantly improves overall performance by tolerating some degradation in very rare worst-case scenarios.

Research in explorable uncertainty addresses combinatorial optimization problems where there is partial information about the values of numeric input parameters, and exact values of these parameters can be determined by performing costly queries. The goal is to design an adaptive query strategy that minimizes the query cost incurred in computing an optimal solution. Solving such problems generally requires that we be able to solve the associated verification problem: given the answers to all queries in advance, find a minimum-cost set of queries that certifies an optimal solution to the combinatorial optimization problem. We present a polynomial-time algorithm for verifying a minimum-weight basis of a matroid, where each weight lies in a given uncertainty area. These areas may be finite sets, real intervals, or unions of open and closed intervals, strictly generalizing previous work by Erlebach and Hoffman which only handled the special case of open intervals. Our algorithm introduces new techniques to address the resulting challenges. Verification problems are of particular importance in the area of explorable uncertainty, as the structural insights and techniques used to solve the verification problem often heavily influence work on the corresponding online problem and its stochastic variant. In our case, we use structural results from the verification problem to give a best-possible algorithm for a promise variant of the corresponding adaptive online problem. Finally, we show that our algorithms can be applied to two learning-augmented variants of the minimum-weight basis problem under explorable uncertainty.

Quantum illumination represents one of the most interesting examples of quantum technologies. On the one hand, it can find significant applications; on the other hand, it is one of the few quantum protocols robust against noise and losses. Here we present a short summary of the history of this quantum protocol.

We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $σ,μ$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_σ(μ)$ satisfying $σ\boxplus μ= σ\triangleright \mathbb{A}_σ(μ)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_σ(μ)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.

We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia's coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of $\ell^1(\mathbb R)$, to bounding two key probabilities measuring proximity to the boundary of the subset in question. We apply this strategy to obtain an asymptotic formula for counting integers in $[1, x]$ that have a divisor in an interval $(y, z)$ in the regime $z/y \to \infty$ as $x \to \infty$.

The topological dynamics of the horocyclic flow $h_{\mathbb{R}}$ on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular, on such a surface, the flow $h_{\mathbb{R}}$ is minimal, or the minimal sets are the periodic orbits. When the surface is geometrically infinite, the situation is more complex, and the presence of possible non-closed and non-dense orbits, called irregular orbits, complicates the description of minimal sets. In this text, we will show that such an orbit is recurrent, or its closure is non-$h_{\mathbb{R}}$ minimal. This would allow us to almost complete the description of $h_{\mathbb{R}}$-minimal sets.

Dynamic community detection concerns inferring how community memberships evolve over time, including the emergence, persistence, merging, and dissolution of groups in temporal networks. We propose a Bayesian nonparametric model for time-evolving sparse networks, which captures power-law degree distributions and dynamically overlapping communities. The model is constructed from vectors of completely random measures coupled through a latent Markov process governing the evolution of node affiliations. This construction provides a flexible and interpretable approach to model dynamic communities, naturally generalizing existing overlapping block models to the sparse and scale-free regimes. We establish asymptotic results characterizing sparsity and degree heterogeneity over time, and develop an approximate inference procedure for recovering time-varying community trajectories. Applications to synthetic and real-world dynamic networks show that the model accurately uncovers evolving community structure and yields interpretable temporal patterns.