Emerging machine learning interatomic potentials (MLIPs) offer a promising solution for large-scale accurate material simulations, but stringent tests related to the description of vibrational dynamics in molecular crystals remain scarce. Here, we develop a general MLIP by leveraging the graph neural network-based MACE architecture and active-learning strategies to accurately capture vibrational dynamics across a range of polyacene-based molecular crystals, namely naphthalene, anthracene, tetracene and pentacene. Through careful error propagation, we show that these potentials are accurate and enable the study of anharmonic vibrational features, vibrational lifetimes, and vibrational coupling. In particular, we investigate large-scale host-guest systems based on these molecular crystals, showing the capacity of molecular-dynamics-based techniques to explain and quantify vibrational coupling between host and guest nuclear motion. Our results establish a framework for understanding vibrational signatures in large-scale complex molecular systems and thus represent an important step for engineering vibrational interactions in molecular environments.

We consider the class of Hopfield models of associative memory with activation function $F$ and state space $\{-1,1\}^N$, where each vertex of the cube describes a configuration of $N$ binary neurons. $M$ randomly chosen configurations, called patterns, are stored using an energy function designed to make them local minima. If they are, which is known to depend on how $M$ scales with $N$, then they can be retrieved using a dynamics that decreases the energy. However, storing the patterns in the energy function also creates unintended local minima, and thus false memories. Although this has been known since the earliest work on the subject, it has only been supported by numerical simulations and non-rigorous calculations, except in elementary cases. Our results are twofold. For a generic function $F$, we explicitly construct a set of configurations, called mixed memories, whose properties are intended to characterise the local minima of the energy function. For three prominent models, namely the classical, the dense and the modern Hopfield models, obtained for quadratic, polynomial and exponential functions $F$ respectively, we give conditions on the growth rate of $M$ which guarantee that, as $N$ diverges, mixed memories are fixed points of the retrieval dynamics and thus exact minima of the energy. We conjecture that in this regime, all local minima are mixed memories.

We present a method for constructing numerical schemes with up to 3rd strong convergence order for solution of a class of stochastic differential equations, including equations of the Langevin type. The construction proceeds in two stages. In the first stage one approximates the stochastic equation by a differential equation with smooth coefficients randomly sampled at each time step. In the second stage the resulting regular equation is solved with the conventional operator-splitting techniques. This separation renders the approach flexible, allowing one to freely combine the numerical techniques most suitable to the problem at hand. The approach applies to ordinary and partial stochastic differential equations. In the latter case, it naturally gives rise to pseudo-spectral algorithms. We numerically test the strong convergence of several schemes obtained with this method in mechanical examples. Application to partial differential equations is illustrated by real-time simulations of a scalar field with quartic self-interaction coupled to a heat bath. The simulations accurately reproduce the thermodynamic properties of the field and are used to explore dynamics of thermal false vacuum decay in the case of negative quartic coupling.

We develop protocols for generating loss-tolerant quantum tree-codes; these are designed to safeguard information against qubit losses, with wide applications in quantum communications. Contrary to previous proposals, our method enables top-to-bottom fast encoding and decoding, thereby reducing losses due to the lagging and photon-reordering at the repeater stations. At the hardware level, we show how to achieve this with a single quantum emitter equipped with a static feedback mechanism, which we leverage to engineer entangling gates between a fed-back qubit and multiple emitted qubits in parallel. In addition, analyzing typical patterns within the error-correction decoding graphs, we find optimizations of the structure of tree-codes, which enable improved performance by also reducing the code size; these are based on the introduction of asymmetries in the code, which mimic the intrinsic adaptiveness of the recovery procedure. We show numerically that these improvements together significantly enhance the loss-correction performance. Specifically, focusing on quantum repeater protocols, we show that our fast recovery scheme (decoding-encoding) allows for improved repeater rates with smaller photon numbers per code.

This work investigates the allowed parameter spaces of a simplified dark matter (DM) model characterized by a spin-0 mediator with masses in the low to intermediate range ($ < $ 10 GeV). We systematically divide the parameter space into various mass regions of the mediator and constrain the model parameters using a diverse set of observables, including flavour-changing charged and neutral current processes such as rare and semi-leptonic decays of pseudoscalar mesons (B and K), electroweak precision observables, alongside data from fixed-target experiments. Additionally, we explore the model's capability to explain recent Belle-II data on invisible B-meson decays. Our study includes a detailed examination of DM properties and the constraints from Big Bang nucleosynthesis. We present bounds on model parameters through individual and simultaneous analyses of the available inputs and highlight their implications for understanding DM phenomenology. Furthermore, we obtain bounds on the couplings of the possible gauge-invariant dimension-5 operators, leading to the possible interactions between the spin-0 mediator and the SM gauge bosons and fermions. This study comprehensively investigates the constraints and theoretical implications associated with low-mass spin-0 mediator DM models.

Ingham studied two types of convolution sums of the divisor function, the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their asymptotic formulas as $N \to \infty$. There have been numerous works extending Ingham's result on the shifted convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result of Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for certain integers $M \le N$. This involves careful choice of an applicable range of $M$. We also study the convolution sum $\sum_{n < M} f(n) g(N-n)$ for certain arithmetic functions $f$ and $g$ with absolutely convergent Ramanujan expansions, which in turn leads us to a well-established prediction of Ramanujan.

Moiré superlattices (MSLs) in van der Waals (vdW) heterostructures have demonstrated their incredible power in driving emergent electronic phenomena, some of which are reminiscent of those usually only observed in bulk strongly correlated quantum materials. With the recent discovery of van der Waals $f$-electron materials, the design of novel MSLs of intrinsic strong correlation is now within the reach. Here we study the novel electron phases of two-dimensional heavy-fermion MSL with increasingly diluted f-electron local moments. By applying dynamical mean field theory (DMFT) with numerical renormalization group (NRG) as an impurity solver, we demonstrate the appearance of a new energy scale and a re-entrant Kondo breakdown in connection with the emergence of a flat band in the system. We further compare our numerical findings with predictions derived from the Lieb-Mattis theorem and show the necessity of the new energy scale to consistently reconcile the predictions with the conventional single-impurity limit for exceedingly large unit cells.

We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,α}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,α}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.

The continuous functional calculus is perhaps the most fundamental construction in the theory of operator algebras, especially $C^{*}$-algebras. Here we document our formalization of the continuous functional calculus in Lean, which constitutes the first such formalization in any proof assistant. Our implementation is already merged into Lean's mathematical library, Mathlib. We provide a brief introduction to the mathematical theory for those unfamiliar with the subject, and then highlight the design decisions in our formalization which proved to be important for usability. Our exposition is aimed at a general mathematical audience and provides a glimpse into the world of formalization by laying bare the discovery process.

Algebraic theories with dependency between sorts form the structural core of Martin-Löf type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical structures have been introduced to model them (contextual categories, categories with families, display map categories, etc.) Comparisons of these models are scattered throughout the literature, and a detailed, big-picture analysis of their relationships has been lacking. We aim to provide a clear and comprehensive overview of the relationships between as many such models as possible. Specifically, we take *comprehension categories* as a unifying language and show how almost all established notions of model embed as sub-2-categories (usually full) of the 2-category of comprehension categories.

A fluid, with broken time-reversal symmetry, would exhibit odd transport coefficients, such as odd viscosity, thermal conductivity and diffusion coefficient, which may fundamentally alter the fluid properties and significantly influence the structure and dynamics of immersed objects. Here, we develop an efficient coarse-grained simulation approach for the odd fluid, that captures all essential features of real odd fluids. Based on microscopic kinetic theory, we analytically derive the transport coefficients of the mesoscale odd fluid. Furthermore, we validate our approach by performing both simulations and theoretical calculations to explore the intricate transport phenomena of the odd fluid under various external drivings. Our work thus paves the way for studying anomalous transport in odd fluids and for large-scale simulations of odd complex fluids.

In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition.

The period of oscillation of a simple pendulum ($T = 2π\sqrt{l/g}$) is a familiar formula to most first-year physics students. However, deriving this expression from first principles requires linearizing the equation of motion under the small-angle approximation and solving the resulting differential equation. From our point of view, this method may seem obscure to students in the early stages of learning calculus and lacking in physical insight. Therefore, we propose an alternative approach to the derivation of this formula that relies on geometry, algebra, and physical intuition. Our method follows the foundational idea of integral calculus, replacing the circular path of the pendulum with a successive collection of infinitesimal inclined planes and summing the travel times along each plane as the number of planes becomes very large. Remarkably, evaluating the limit of this sum relies solely on geometric reasoning, making the approach accessible to any student, even those not yet familiar with differential equations or integration techniques.

We propose a test for the identification of causal effects in mediation and dynamic treatment models that is based on two sets of observed variables, namely covariates to be controlled for and suspected instruments, building on the test by Huber and Kueck (2022) for single treatment models. We consider models with a sequential assignment of a treatment and a mediator to assess the direct treatment effect (net of the mediator), the indirect treatment effect (via the mediator), or the joint effect of both treatment and mediator. We establish testable conditions for identifying such effects in observational data. These conditions jointly imply (1) the exogeneity of the treatment and the mediator conditional on covariates and (2) the validity of distinct instruments for the treatment and the mediator, meaning that the instruments do not directly affect the outcome (other than through the treatment or mediator) and are unconfounded given the covariates. Our framework extends to post-treatment sample selection or attrition problems when replacing the mediator by a selection indicator for observing the outcome, enabling joint testing of the selectivity of treatment and attrition. We propose a machine learning-based test to control for covariates in a data-driven manner and analyze its finite sample performance in a simulation study. Additionally, we apply our method to Slovak labor market data and find that our testable implications are not rejected for a sequence of training programs typically considered in dynamic treatment evaluations.

Dense bacterial suspensions exhibit turbulent behaviour called ``bacterial turbulence''. The behavior of the bulk unconstrained bacterial turbulence is described well by the Toner-Tu-Swift-Hohenberg (TTSH) equation for the velocity field. However, it remains unclear how we should treat boundary conditions on bacterial turbulence in contact with some boundaries (e.g. solid walls). To be more specific, although the importance of the ``edge current'', the flow along the boundary, has been demonstrated in several experimental studies on confined bacterial suspensions, previous numerical studies based on the TTSH equation employ non-slip boundary conditions and do not seem to describe properly the behavior of bacteria near the boundaries. In this study, we impose a slip boundary condition on the TTSH equation to describe the bacterial motion at boundaries. We develop a method to implement the slip boundary condition. Using this method, we have successfully produced edge current and discovered that the direction of the edge current temporally oscillates. The oscillation can be attributable to the advection term in the TTSH equation. Our work demonstrates that boundary conditions could play an important role in the collective dynamics of active systems.

This paper studies an infinite-horizon optimal control problem for a pendulum with quadratic control cost via its associated Hamiltonian system. The problem is strongly degenerate, as the linearization at the upright equilibrium has purely imaginary eigenvalues with multiplicity, making standard linear analysis inconclusive. Using Lie series and Chetaev's instability theorem, we show that the equilibrium is weakly unstable in a non-hyperbolic sense. We further identify a family of periodic orbits forming an invariant cylinder, whose monodromy matrix exhibits a maximally degenerate Floquet structure consisting of a single Jordan block at the unit multiplier. This structure implies slow, non-exponential dynamics and provides a mechanism for long-time transitions with arbitrarily small control effort. Although derived for a pendulum, this phenomenon arises from degeneracy in the optimal control formulation and may occur more broadly, suggesting the possible nonexistence of optimal control despite stabilizability.

We investigate phenomenologically the viability of fuzzy dark matter (FDM). We do this by confronting the predictions of the model, in particular, the formation of a solitonic core at the center of dark matter halos, with a homogeneous and robust sample of high-resolution rotation curves from the "LITTLE THINGS in 3D" catalog. This comprises a collection of isolated, dark matter dominated dwarf-irregular galaxies that provides an optimal benchmark for cosmological studies. We use a statistical framework based on Markov chain Monte Carlo techniques that allows us to extract relevant parameters such as the axion mass, the mass of the solitonic core, the mass of the dark matter halo and its concentration parameter with a rather loose set of priors except for the implementation of a core-halo relation that is predicted by simulations. The results of the fits are used to perform various diagnostics on the predictions of the model. FDM provides an excellent fit to the rotation curves of the "LITTLE THINGS in 3D" catalog, with axion masses determined from different galaxies clustering around $m_a\approx2\times10^{-23}$ eV. However, we find two major problems in our analysis. First, the data follow scaling relations of the properties of the core which are not consistent with the predictions of the soliton. This problem is particularly acute in the core radius - mass relation with a tension that, at face value, has a significance $\gtrsim5σ$. The second problem is related to the strong suppression of the linear power spectrum that is predicted by FDM for the axion mass preferred by the data. This can be constrained very conservatively by the galaxy counts in our sample, which leads to a tension exceeding again $5σ$. We estimate the effects of baryons in our analysis and discuss whether they could alleviate the tensions of the model with observations.

Minimal linear codes play an important role in coding theory and cryptography, particularly in the construction of secret sharing schemes. In this paper, we investigate the structure and construction of two-dimensional minimal linear codes over the finite rings $\mathbb{Z}_{p^n}$. We provide an explicit construction of a family of two-dimensional linear codes generated by a structured $2\times m$ matrix over $\mathbb{Z}_{p^n}$ and prove that these codes are minimal whenever the generator matrix contains all $p^n+p^{n-1}$ essential types of column vectors. We further show that this condition is necessary: removing any of these column types destroys the resulting code's minimality. As a consequence, we establish a lower bound on the length of two-dimensional minimal linear codes over $\mathbb{Z}_{p^n}$. Several examples are presented to illustrate the construction and to verify the theoretical results. We also demonstrate that the proposed construction cannot be extended in a straightforward manner to rings of the form $\mathbb{Z}_{p^n q^l}$. Finally, we apply our results to the design of secret sharing schemes derived from minimal linear codes over $\mathbb{Z}_{p^n}$ and analyze the corresponding access structures. Our study highlights structural differences between minimal codes defined over finite rings and those over finite fields, revealing new perspectives for coding-theoretic constructions in cryptographic applications.

The coherence characteristics of a tin-vacancy color center (SnV) in diamond are investigated through optical means, including linewidth broadening effects and coherent population trapping (CPT) between the ground state orbital levels. Spectral analysis is required as due to the large spin-orbit splitting of the orbital ground states, thermalization between the ground states occurs at rates that are impractical to measure directly in the time domain. First, by implementing a temperature-dependent linewidth broadening measurement, including the challenging-to-measure D transition, phononic coupling coefficients are determined. These measurements are performed on an emitter with a lifetime-limited linewidth and atom-like properties, making the measurement representative for high-quality SnVs. Next, a CPT-type experiment is carried out to independently analyze thermal decoherence processes at 4 K. The spectral information is transformed into its conjugate variable time, providing picosecond resolution and revealing an orbital depolarization timescale of ${\sim30{\rm~ps}}$. Consequences of the investigated dynamics are then used to estimate spin dephasing times limited by thermal effects.

Recent advancements in quantum technology threaten the cryptographic foundations of Accounting Information Systems (AIS), necessitating a transition to quantum-safe standards. This paper investigates why quantum standards fall within the purview of accounting by framing them as essential institutional governance mechanisms that ensure the integrity, auditability, and legitimacy of data. Utilizing neo-institutional theory, the study analyzes how coercive, normative, and mimetic pressures drive the adoption of these standards across jurisdictions. Through a structured documentary analysis of major standard-setting bodies, the research identifies significant divergence between U.S. and EU/European approaches: U.S. standards emphasize market-driven innovation and pragmatic legitimacy, while EU and Pan-European standards prioritize regulatory harmonization and societal privacy objectives. The findings suggest that while these standards are currently voluntary, their inconsistent implementation creates risks of decoupling and fragmented assurance practices, challenging the global comparability of AIS security controls.

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and Füredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

The theoretical physicist and mathematician Aristophanes Dimakis passed away on July 8, 2021, at the age of 68, in Athens, Greece. We briefly review his life, career and scientific achievements.

As a generalization of irrational rotations and a dual case of higher-dimensional Kronecker sequences, we study the discrepancy of sequences of linear forms on the circle. Given irrationals $α_1,\dots,α_d$, consider the set of $N^d$ points $\{k_1α_1+\cdots+k_dα_d \mod 1 : 1\le k_j\le N\}$. We prove that for a full-measure set of vectors $(α_1,\dots,α_d)\in\mathbb{R}^d$, the maximal discrepancy of these points relative to intervals in $[0,1)$ has the optimal principal order $(\log N)^d$, up to powers of $\log\log N$. This result provides a nearly sharp dual analogue, in the setting of linear forms, to Beck's celebrated theorem on multidimensional Kronecker sequences (Ann. of Math., 1994). The proof combines Fourier analysis, metric multiplicative Diophantine estimates, and a duality argument which reduces certain lattice-counting errors to Beck's discrepancy theorem.

We analyse the rack structure of conjugacy classes in simple Suzuki and Ree groups and determine which classes are kthulhu. Combining this results with abelian rack techniques, we show that the only finite-dimensional complex pointed Hopf algebras over the simple Suzuki and Ree groups are their group algebras.

The fractional quantum anomalous Hall (FQAH) effect occurs in moiré superlattices in both twisted bilayer MoTe$_2$ and rhombohedral $n$-layer graphene aligned to hexagonal boron nitride (R$n$G/hBN) as a novel quantum phase driven by intertwined electron correlation and topology. Although several fractional states in the Jain sequence have been identified, the $1/3$ state, the most robust and fundamental state in conventional fractional quantum Hall (FQH) systems, was missing in either FQAH system. Determining whether it exists would have a major impact on understanding the mechanism of FQAH, especially in the theoretically still-debated R$n$G/hBN system. Here we report the FQAH effect at moiré filling factor $ν= 1/3$ in R$5$G/hBN moiré superlattice devices, through a combination of quantum capacitance and transport measurements. By tuning the displacement field, we observed a topological phase transition from a $1/3$ fractional Chern insulator (FCI) to a trivial charge density wave state. With the inclusion of the $1/3$ state, the FQAH states in R$5$G/hBN now exhibit a surprising level of particle-hole symmetry about half-filling, closely resembling the behavior of FQH states in the lowest Landau level. Additionally, we perform compressibility and transport measurements at a filling of one electron per moiré unit cell, $ν=1$, and also for $ν\lesssim 1$, where previous transport measurements displayed the extended quantum anomalous Hall (EQAH) effect. While our transport measurements show no change between the integer quantum anomalous Hall state (IQAH) and the EQAH region, compressibility measurements reveal a distinct transition from a gapped IQAH state to a gapless and highly compressible EQAH state. Our direct thermodynamic characterization of the rich phase diagram paves the way to engineering of anyon braiding and non-Abelian quasiparticles at zero magnetic field.

Let $pp (n)$ denote the number of ways to write $n$ as a sum of primes. In this paper, we show that $$ \log pp (n) \sim 2π\sqrt{\frac{n}{3\log n}}\;. $$ While sharper estimates are already known, they rely on highly involved and lengthy proofs. In sharp contrast, our approach uses a short, elementary recipe that easily adapts to yield similar asymptotic estimates for several related, extensively studied problems.

In this paper, we give a one-pass quantum streaming algorithm for Max-$k$SAT that uses $\operatorname{polylog}(n)$ space and achieves a $0.7172$-approximation on instances with $n$ variables. In contrast, prior work by Chou, Golovnev, and Velusamy (FOCS 2020) implies that achieving an approximation ratio better than $\sqrt{2}/2 \approx 0.7071$ for Max-$k$SAT requires $Ω(\sqrt{n})$ space for any classical streaming algorithm. Therefore, it yields an exponential quantum space advantage for Max-$k$SAT in the streaming setting. We further give a one-pass quantum streaming algorithm for Max-2OR that uses $\operatorname{polylog}(n)$ space and achieves a $0.7425$-approximation on instances with $n$ variables. Combining with the known results, it gives a complete classification of quantum space advantages for all Boolean Max-2CSPs.

Governments worldwide are increasingly regulating digital platforms to reduce online harms, particularly those affecting children. However, access restrictions can alter user behaviour and introduce new privacy and security risks. The UK Online Safety Act (OSA), passed in October 2023, illustrates this trend: it extends age-assurance and safety requirements to social media, search, and pornography services, and rolled out in phases. Ofcom's illegal content enforcement duties came into force in March 2025, and mandatory age verification for adult content took effect in July 2025. This phased rollout enables real-time observation of behavioural responses to regulation. To address this, we analyse Reddit discourse across VPN and UK Politics communities and conduct a privacy-policy risk analysis of 69 unique VPN services. We find that each of these three milestones produced significant stepwise increases in VPN-related discussion on Reddit: among UK-based users, posts and comments explicitly about VPN use in a regulatory or privacy context rose by +100%, +217%, and +415% respectively. UK Politics communities showed even larger effects, with OSA-related political discourse rising by +213%, +545%, and +464%, respectively, among UK-based users. UK VPN search interest on Google rose by +89% at the age-verification deadline. Users primarily framed this response around privacy, surveillance, and distrust of age-verification intermediaries rather than simple access-seeking. Demand increased across low, medium, and high-risk VPNs, but the proportional distribution remained broadly stable. These findings suggest that online safety regulation can create secondary privacy costs even when it does not disproportionately shift attention toward higher-risk providers.

Analog In-Memory Computing (AIMC) accelerators execute matrix-vector multiplications directly within memory arrays, reducing data movement and improving DNN inference efficiency. Their limited effective precision motivates heterogeneous architectures that combine analog compute tiles with digital processing units. This letter classifies existing methods for partitioning DNN workloads across these resources by mapping granularity, optimization strategy, and model support, and distills them into a unified four-stage workflow. To demonstrate the workflow on a model class not yet addressed by existing methods, we apply its first two stages to GPT-2, producing the first AIMC-specific precision sensitivity profile for a decoder-only transformer. Sensitivity is dominated by 4 of 49 projections, with the first decoder block's attention output dominating by an order of magnitude. This suggests that projection-level mapping and selective digital execution of early-block and output-facing projections are important for reliable decoder-transformer deployment on AIMC hardware.