We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$. Viewing these points as morphisms $\mathbb{P}^1 \to X$ of fixed degree, we obtain exact formulas for their number depending on the parity of $n$ and on the determinant of the quadratic form defining $X$, including secondary terms in some cases.

We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a local-product dichotomy. Over a finite commutative local principal ideal ring, the common centralizer of two noncommuting affine permutations is always abelian, so the pattern is impossible. Over a direct product of two commutative rings whose affine groups each contain a noncommuting pair, the same pattern is constructed by separating the two noncommuting families into different factors. More generally, over a finite commutative principal ideal ring, the pattern exists if and only if at least two local factors are not isomorphic to $\mathbb{F}_2$. Applied to residue rings, this yields an exact classification: $\mathrm{AGL}_1(\mathbb{Z} / n \mathbb{Z})$ contains the pattern if and only if at least two prime-power factors of $n$ exceed 2 . We also compare this phenomenon with the permutation-group setting, where the same pattern is easy to realize.

We demonstrate that strong-field ionization of atoms in circularly polarized laser fields generates a photoelectron spin texture with toroidal topology in momentum space. Using time-dependent Schrödinger equation simulations, spin-resolved classical-trajectory Monte Carlo calculations, and an extended spin-resolved strong-field approximation including intermediate excitation pathways, we show that the rotation angle of this spin torus provides access to attosecond relative time delays associated with photoelectron wave packets released by tunneling from the counter-rotating and co-rotating \(p\)-orbital channels. When intermediate-state dynamics become significant, the torus develops a clear splitting. These results establish photoelectron spin textures as a complementary source of dynamical information beyond conventional momentum spectroscopy, and identify spin polarization as a robust internal degree of freedom for self-referenced attosecond metrology.

The widespread use of digital devices has raised growing concerns about its impact on sustained attention at the population level. In this work, we propose a minimal dynamical framework to describe the collective evolution of attention under continuous exposure to screen-mediated environments. We introduce a macroscopic variable representing the average level of sustained attention and model its dynamics as the result of competing mechanisms: intrinsic cognitive recovery and degradation induced by digital stimulation. The digital environment is treated as an external control parameter that continuously perturbs the system, leading to a relaxational dynamics. The proposed mechanisms are consistent with empirical findings on attentional dynamics under digital exposure. We first analyze a linear formulation, which provides an analytically tractable baseline, and then extend the model by incorporating a nonlinear degradation term that captures amplification effects under high-intensity stimulation. We derive an explicit expression for the stationary state and show that the equilibrium attention level decreases monotonically with increasing exposure. An effective potential formulation is introduced, revealing that digital overstimulation progressively deforms the dynamical landscape, shifting the stable state toward regimes of reduced attention without generating multiple equilibria. Importantly, the model does not rely on social contagion or interaction-driven bistability, but instead describes a continuous displacement of the collective cognitive regime under environmental pressure. Our results suggest that the impact of digital technologies on attention may be understood as a gradual macroscopic effect emerging from persistent external stimulation, rather than as a transition between competing behavioral states.

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of $n$ locations in each time step $t=0,1,2,\dots$. Their goal is to minimize the expected time until they visit the same location and thus meet. Anderson and Weber [J. Appl. Prob., 1990] proposed a strategy that operates in rounds of $n-1$ steps: a player either remains in one location for $n-1$ steps or visits the other $n-1$ locations in random order; the choice between these two options is made with a probability that depends only on $n$. The strategy is known to be optimal for $n=2$ and $n=3$, and there is convincing evidence that it is not optimal for $n=4$. We show that it is not optimal for any $n\geq 4$, by constructing a strategy with a smaller expected meeting time.

Neural communication operates on both fast synaptic transmission and slower, diffusive extrasynaptic signaling, yet how these two modes jointly organize brain function remains unclear. Here, using the complete synaptic and neuropeptidergic connectomes of \emph{Caenorhabditis elegans}, we develop a unified multiplex framework linking anatomical wiring to functional communication. We infer structure-derived functional connectivity from the synaptic connectome using equilibrium principles from statistical physics, yielding a probabilistic map of information flow across all synaptic pathways, and compare this functional layer directly with the extrasynaptic connectome. This reveals a principled functional specialization across four communication regimes: (i) a topology-dependent layer that reinforces and stabilizes synaptic motor circuits, (ii) a topology-resilient modulatory layer supporting global regulation and behavioral state control, (iii) a purely extrasynaptic network sustaining survival and homeostasis, and (iv) a purely synaptic regime mediating rapid, low-latency sensorimotor processing. Together, these findings reveal that synaptic and extrasynaptic signaling form complementary architectures optimized for speed, modulation, robustness, and survival, and provide a general strategy for integrating structural and modulatory connectomes to understand how distinct communication modes cooperate to sustain coherent brain function.

Auxiliary Field Quantum Monte Carlo (AFQMC) has emerged as a powerful framework for treating strongly correlated electronic systems, offering a favorable balance between computational cost and accuracy. In this paper, we present a novel AFQMC method that uses the isometric tensor hypercontraction (ITHC) technique to diagonalize the two-body Coulomb interaction of molecular electronic Hamiltonians by introducing additional fictitious fermionic modes. Our method shows reduced theoretical complexity and better practical performance for both propagation and local energy evaluation compared to the standard AFQMC method. We demonstrate the efficacy of this approach by computing the ground-state energies of a linear $\ce{H10}$-chain and the benzene molecule. Our results show that the extended-basis AFQMC recovers many-body correlations with a precision comparable to that of high-level wavefunction methods such as Coupled Clusters (CC) or Density Matrix Renormalization Group (DMRG), while offering significantly improved scaling.

Introduction: Terbium-149 ($^{149}$Tb) is a promising radionuclide for targeted $α$ therapy that has a non-zero branching ratio (BR) for positron decay. However, its relatively low positron branching fraction and multiple prompt $γ$ emissions may challenge quantitative imaging. This study evaluates, for the first time, the imaging performance and quantitative accuracy of $^{149}$Tb using a clinical long axial field-of-view (LAFOV) PET/CT system. Methods: Quantitative accuracy of $^{149}$Tb was assessed with a NEMA IEC body phantom, which was filled with about 45 MBq $^{149}$Tb and a sphere-to-background ration of 10:1. The phantom was scanned for 20 min and shorter scan times and lower activities were simulated. Recovery coefficients, coefficient of variation, and lung residual error were evaluated for different reconstruction settings and compared to the EARL standard 2 for $^{18}$F. Results: High-quality PET images of $^{149}$Tb were obtained, even with a simulated total activity of 4.5 MBq. The 20 min and full activity scan yielded a mean recovery coefficient $RC_\textit{mean}$ of $0.55$, $0.69$, $0.73$, $0.76$, $0.79$, and $0.81$ for the six phantom spheres. Despite the low count statistics, the coefficient of variation stays mostly below $15\,\%$. Relative scatter correction combined with prompt $γ$ modeling provided robust quantification. Conclusion: $^{149}$Tb can be imaged using a commercial LAFOV PET/CT with a quantitative accuracy comparable to the EARL standard 2 for $^{18}$F. These findings demonstrate the feasibility of PET-based treatment verification and dosimetry for targeted $α$ therapy with $^{149}$Tb.

We investigate the asymptotic symmetries of General Relativity at spatial infinity within the first-order formalism described by the Holst action. Employing the covariant phase space method, we propose a set of relaxed boundary conditions for the co-tetrad and Lorentz connection that admit the full Bondi-Metzner-Sachs (BMS) group, including non-trivial supertranslations, which are typically eliminated in standard treatments. We demonstrate that the logarithmic divergences appearing in the symplectic structure can be removed by imposing specific, symmetry-preserving parity conditions on the asymptotic fields without suppressing the supertranslation sector. A detailed analysis of the conserved charges reveals that the Holst term contributes non-trivially to the charge variations due to the linear growth of Lorentz generators. We show that the naive surface integrals for the Holst charges exhibit linear divergences arising from the rotation of the background tetrad. These divergences are successfully regularized by supplementing the asymptotic symmetry generator with a compensating internal Lorentz gauge transformation defined to preserve the background structure. The resulting charges are manifestly finite and integrable. Crucially, we prove that while the Holst modification shifts the charges associated with Lorentz boosts and rotations, it leaves the supertranslation charges identically invariant. This framework provides a consistent derivation of the full BMS algebra at spatial infinity in terms of Ashtekar-Barbero variables, offering new insights into the role of the Immirzi parameter in classical and quantum gravity.

We study the syzygies of projections of elliptic normal curves. Let $C \subset \mathbb{P}^{d-1}$ be an elliptic normal curve of degree $d \ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green--Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q \in C^s \setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ for a general point $q$ in $\mathbb{P}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls.

We study the local synchronization of phases and frequencies for the Kuramoto model driven by rough noise. In particular, we prove exponential convergence towards synchronization and we give the explicit rate of convergence and quantify the size of the random basin of attraction. Furthermore, we show that the long time behavior of the system is determined by the evolution of phases' mean. Our result relies on the use of a Lyapunov function, capable of overriding the particular structure of the noise, taking in account only its intensity. Finally, we illustrate our analytical results and possible extensions with the help of numerical simulations.

In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.

We introduce an efficient stable algorithm for transforms associated with expansions in Hermite functions interpolated at Hermite polynomial roots. The Hermite transform matrix can be factorised into a diagonal component and an orthogonal matrix, leading to a form which allows both the forward and inverse Hermite transforms to be computed stably. Our novel algorithm computes this factorisation based on the eigendecomposition of the Jacobi matrix associated with Hermite functions. Through numerical experiments, we demonstrate the stability and efficiency gains of this novel method over prior work. Numerical experiments show that the new approach matches or improves on the accuracy of existing stabilized methods, is substantially faster in practice, and enables reliable use of large Hermite expansions in downstream PDE computations. We also provide an open-source implementation, together with reference implementations of previous methods, to facilitate adoption by the community.

Modern software systems rely heavily on Web APIs, yet creating meaningful and executable test scripts remains a largely manual, time-consuming, and error-prone task. In this paper, we present APITestGenie, a novel tool that leverages Large Language Models (LLMs), Retrieval-Augmented Generation (RAG), and prompt engineering to automatically generate API integration tests directly from business requirements and OpenAPI specifications. We evaluated APITestGenie on 10 real-world APIs, including 8 APIs comprising circa 1,000 live endpoints from an industrial partner in the automotive domain. The tool was able to generate syntactically and semantically valid test scripts for 89\% of the business requirements under test after at most three attempts. Notably, some generated tests revealed previously unknown defects in the APIs, including integration issues between endpoints. Statistical analysis identified API complexity and level of detail in business requirements as primary factors influencing success rates, with the level of detail in API documentation also affecting outcomes. Feedback from industry practitioners confirmed strong interest in adoption, substantially reducing the manual effort in writing acceptance tests, and improving the alignment between tests and business requirements.

This paper characterizes the capacity region of a two-user additive-multiplicative multiple access channel (AM-MAC) under heterogeneous input constraints. This model captures the fundamental limits of symbiotic radio, where an active primary transmitter (PT) conveys information via active transmission subject to an average power constraint, while a passive backscatter device (BD) modulates signals through backscattering under a peak amplitude constraint. Our main results are threefold. Firstly, we prove that the sum-rate capacity equals the PT's point-to-point capacity, achieved when the PT employs Gaussian signaling and the BD acts as a pure reflector to assist the PT's transmission. Secondly, to achieve the BD's maximum achievable rate, the PT must adopt a constant-envelope signaling strategy, while the optimal BD distribution exhibits a concentric-circle structure with a uniform phase. Thirdly, for the remaining boundary points, we establish that the optimal PT signal consists of a continuous uniform phase and a discrete amplitude, whereas the optimal BD distribution is fully discrete. Finally, numerical results are provided to characterized the capacity region by solving a specialized nonlinear optimization problem. To demonstrate the practical implications, we also characterize an baseline rate pair and evaluate the overall performance of the AM-MAC.

We solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields completely for 85 of the 112 possible types. We also determine for all 112 types the smallest field of existence. As an aside, we provide an example of a del Pezzo surface of degree 1 in characteristic 2 with more than one generalized Eckardt point.

Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$, close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$, $CS$, and $T$ gates.

The adoption of clean technologies (CTs) plays an important role in reducing carbon dioxide (CO$_2$) emissions. We study CT adoption in a large population of consumers with heterogeneous behavioral tendencies. We model the interaction among the agents as a multi-type mean-field game in which the agents choose between clean and polluting technology based products and may either behave as rationals (trading off price and moral incentives), herding agents (just follow the majority), or lethargic agents exhibiting inertia toward adopting the new technologies. We characterize equilibrium CT adoption levels using the recently introduced notion of $\boldsymbolα$-Rational Nash Equilibrium ($\boldsymbolα$-RNE) and its multi-type extension. We then identify a stable subset using the limits of a stochastic turn-by-turn behavioral dynamics. Our results highlight the role of population composition in determining CT adoption. In particular, widespread adoption requires either a sufficiently small price disadvantage for CTs or the presence of a sufficiently large herding population that can be influenced through social awareness programs. Surprisingly, we could prove that environmental damages do not provide sufficient incentives to increase CT adoption.

We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian $\boldsymbol{B}$, our algorithm determines the subgraph featuring Laplacian $\boldsymbol{B'}$ on the same vertex set, but $x$ out of $N$ inactive edges, minimizing the Frobenius distance $||\boldsymbol{B} - \boldsymbol{B'}||_\mathrm{F}^2$. We represent the $\binom{N}{x}$ graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up $\mathcal{O}(\sqrt{N^{x}/x!}N\log\log N)$ compared to $\mathcal{O}(N^{x+1}/x!)$ of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing $||\boldsymbol{B} -\boldsymbol{B'}||_\mathrm{F}^2$ from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.

I take a pragmatist perspective on quantum theory. This is not a view of the world described by quantum theory. In this view quantum theory itself does not describe the physical world, nor our observatons, experiences or opinions of it. Instead, the theory offers reliable advice on when to expect an event of one kind or another, and on how strongly to expect each possible outcome of that event. The actual outcome is a perspectival fact: a fact relative to a physical context of assessment. Measurement outcomes and quantum states are both perspectival. By noticing that each must be relativized to an appropriate physical context one can resolve the measurement problem and the problem of nonlocal action. But if the outcome of a quantum measurement is not an absolute fact, then why shoud the statistics of such outcomes give us any objective reason to accept quantum theory? One can describe extensions of the scenario of Wigner's friend in which a statement expressing the outcome of a quantum measurement would be true relative to one such context but not relative to another. However, physical conditions in our world prevent us from realizing such scenarios. Since the outcome of every actual quantum measurement is certified at what is essentially a single context of assessment, the outcome relative to that context is an objective fact in the only sense that matters for science. We should accept quantum theory because the statistics these outcomes display are just those it leads us to expect.

Entangled photon pair sources are a key enabling technology for quantum communication and networking, yet their deployment beyond laboratory environments is hindered by system-level complexity, limited operational stability, and insufficient industry compatibility. Here, we demonstrate a rack-based, mobile quantum light source architecture based on a semiconductor quantum dot emitter that directly addresses these challenges through modular system integration and automated operation. The source generates polarization-entangled photon pairs with an entanglement negativity 2n of up to $0.98(1)$, confirming near-maximal entanglement quality. In continuous, hands-off operation over a six-hour time window, the system achieves an average single-photon emission rate of $697(8)$ kHz and a maximum rate of $740(7)$ kHz, while maintaining 2n-value of more than $95$ $\%$. These results are enabled by the integration of optical excitation, collection, cryogenic operation, and control electronics within a standardized rack footprint, together with automated monitoring. By demonstrating simultaneously high entanglement quality, sustained brightness, and long-term operational stability in an industry-aligned system architecture, this work advances semiconductor quantum dot sources toward deployable entangled photon sources for applied quantum photonics.

In their fourth observing run, the LIGO--Virgo--KAGRA gravitational-wave observatories have found hundreds of new signals, but many are contaminated by non-Gaussian transient noise artefacts known as glitches. Left unaddressed, glitches can bias parameter inference and lead to misleading astrophysical conclusions. We present a series of case studies using GPBilby, a parameter estimation tool that employs a time-domain likelihood jointly modelling the astrophysical signal with a physical waveform and non-Gaussian noise with a Gaussian process. We first show that when the detector noise is Gaussian, GPBilby produces results consistent with those obtained with the standard Gaussian-noise likelihood, and then consider events affected by non-Gaussian features. For GW231123, the highest-mass binary black hole candidate observed to date, analyses using IMRPhenomXPHM reveal coherent residual structure that leads to measurable shifts in inferred source parameters. In contrast, analyses employing NRSur7dq4 show no significant excess residual power and remain consistent across likelihood choices. This demonstrates that waveform systematics and flexible noise modelling are intrinsically coupled, as the Gaussian process terms can partially absorb coherent waveform mismatches. For GW191109, we find that evidence for spin misalignment remains robust despite glitches in both LIGO detectors. For GW230630_070659, excluded from GWTC-4.0 owing to poor data quality, we find the data to be consistent with a BBH waveform model, with no additional residual power identified by the Gaussian process component. Overall, these results highlight how GPBilby can be used to perform glitch-robust inference and as a tool to understand waveform modelling systematics.

This paper introduces discrete differential form spaces over two-dimensional manifold meshes that feature enhanced subdivision-induced inter-element regularity compared to conventional finite element (FE) spaces. This increase in smoothness is achieved by pulling back refined subdivision basis functions along a hierarchy of increasingly fine meshes that are generated by a subdivision algorithm. We introduce a framework that casts several known instances of $k$-form subdivision schemes in the language of FE and derive conditions under which the resulting subdivision-induced hierarchy of FE function spaces satisfies a discrete de Rham complex. The paper further illustrates the enforcing of zero boundary conditions by discarding basis functions close to the mesh boundary and shows that this does not compromise the de Rham complex. To analyse our novel subdivision $k$-form spaces we solve the Maxwell eigenvalue problem to confirm the absence of spurious modes and to study the accuracy of the computed eigenvalues. Recovering accurately the expected analytic eigenvalue spectrum shows that our novel subdivision $k$-form spaces indeed preserve the de Rham complex, since this test case is known to be challenging for methods not preserving this structure. Further, we numerically investigate the approximation errors of these subdivision spaces for given analytic functions. The presented study shows that our method can be employed in two ways. Upon a suitable choice of parameters, the subdivision $k$-form spaces are up to $1.5$ orders of magnitude more accurate in the $L^2$ norm than conventional lowest-order FE spaces with the same number of degrees of freedom. Alternatively, for a given target accuracy, the number of required degrees of freedom can be significantly reduced, resulting in a speed-up by a factor of up to 6 for the discussed test cases.

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ at level $K$, where $K:Λ\timesΛ\to\mathbb Z$ is an even, integral, nondegenerate symmetric bilinear form, by exact zeta-regularized Gaussian evaluation of the formal quotient integral over connections modulo gauge. For closed $3$-manifolds, this yields a topological invariant; for manifolds with boundary, the relative functional integral produces the canonical boundary state. The resulting theory satisfies the required axioms of a $(2+1)$-dimensional TQFT.

Understanding axion production in the early Universe remains a pivotal challenge, given the axion as a compelling cold dark matter candidate. Conventional misalignment scenarios often overlook the possibility that a large initial axion velocity can fundamentally reshape the subsequent evolution of the axion field. In this letter, we provide a comprehensive analysis of how primordial magnetic fields impact the axion relic abundance. By accounting for the axion coupling to the Chern-Simons term of the hypercharge gauge field, the equation of motion of the axion is recast as a driven oscillator equation. This modification effectively shifts the onset of axion oscillations, leading to a significant reevaluation of the final relic abundance, a novel effect we term the axion helical misalignment mechanism. Furthermore, in the presence of primordial chiral asymmetries, the chiral magnetic effect (CME) emerges as a critical driver of axion dynamics. The interplay between the axion field and the CME not only profoundly influences the evolution of Standard Model chiral fermions but also provides a viable pathway for generating the observed baryon asymmetry of the Universe.

We investigate the dynamics of Bose-Einstein condensate droplets composed of $^{164}$Dy atoms formed in a double-well potential following removal of the interwell barrier. By solving the dipolar Gross-Pitaevskii equation, we determine phase diagrams of ground-state configurations as functions of the atom number confined in the double-well potential. For strong dipolar interactions, some of the lowest-energy configurations arise from spontaneous symmetry breaking of the droplet structure, which optimizes the interaction energy. We analyze the subsequent time evolution after removal of the central barrier, revealing both droplet oscillations and merger events leading to the formation of larger droplets. The oscillations are driven by the external potential and by the repulsive tails of the in-plane component of the dipolar interaction. Merger events occur when the initial excess energy is sufficient to overcome the interdroplet potential barrier. The oscillatory dynamics depend sensitively on the atom number, the strength of dipolar interactions, and the initial symmetry of the configuration. We find that both oscillations of individual droplets and atom leakage from the droplets, induced by close droplet-droplet encounters, contribute to the damping of the oscillations.

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences $(q_n)_{n\in\mathbb{N}}$, we show that the covering radius $\frac{1}{n}$ is sharp up to a constant: there exist $C>c>0$ such that, for Lebesgue-almost all $x$, the intervals of length $\frac{C}{n}$ cover $[0,1)$ infinitely often, while this fails for intervals of length $\frac{c}{n}$. Moreover, the lower bound holds for certain sub-lacunary rates and the results partially extend to all probability measures with sufficiently fast Fourier decay. As an application, we obtain a new bound for a variant of the inhomogeneous Littlewood-Cassels problem: for any badly approximable $α$ and $γ\in\mathbb{R}$, there exists a set of badly approximable $β$ of full Hausdorff dimension such that $ \|nα-γ\| \|nβ-δ\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $δ\in\mathbb{R}$. This improves upon previous works of Haynes-Jensen-Kristensen, Chow-Technau, and the third author, and is best possible when one restricts to best approximations of the first factor. Second, under certain arithmetic restrictions on $(q_n)_{n\in\mathbb{N}}$, we compute the almost-sure Hausdorff dimension of limsup sets generated by intervals of size $\frac{1}{n^ν}$ for $ν\geqslant 1$, centered at $\{q_n x\}$, and intersected with Ahlfors regular compact sets such as the middle-third Cantor set. In particular, our results apply to all real-valued lacunary sequences, to integer-valued polynomials, and to powers of primes. This substantially extends the work of Bugeaud and Durand, which applies only to certain super-lacunary integer-valued sequences.

Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric intersecting codes exhibit substantially different behavior. It was previously shown that a nondegenerate $[n,k,d]_{q^m/q}$ rank-metric intersecting code must satisfy $2k-1 \le n \le 2m-3$, and the tightness of the upper bound was left open. Using the geometric interpretation of rank-metric codes via $q$-systems, we prove that the dual subspace associated with a rank-metric intersecting code must satisfy strong evasiveness properties. This connection allows us to derive new restrictions on the parameters of such codes and to show that the bound $n=2m-3$ can be attained only when $k=3$ and $m\ge 6$. More generally, we show that $n \leq 2m-\lfloor(k+4)/2\rfloor$. Moreover, we obtain a geometric characterization of these extremal codes in terms of scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$. As a consequence, the existence problem for $[2m-3,3,d]_{q^m/q}$ rank-metric intersecting codes is reduced to the existence of scattered subspaces of dimension $m+3$. Using known constructions of maximum scattered subspaces, we derive existence results when $m$ is even. Finally, we prove that $[6,3,3]_{q^5/q}$ rank-metric intersecting codes do not exist for any prime power $q$, thus resolving an open problem posed by Bartoli et al. in 2025.

The tunability and twist engineering of van der Waals materials enable the emergence of electronic states not present in individual monolayers. Among them, monolayer 1T-TaS$_2$ is a well-known Mott insulating system, whose star-of-David charge density wave reconstruction realizes an emergent triangular lattice of local magnetic moments. Interestingly, in its bulk form, the insulating gap is not correlation-driven, but stems from interlayer coupling. Here, we exploit the stacking-dependent nature of the insulating gap to show that in twisted 1T-TaS$_2$ bilayers, the spatially dependent competition between many-body and single-particle gaps creates Mott-trivial mosaic superlattices, featuring regions with local magnetic moments and non-magnetic insulating regions. We further demonstrate the tunability of the mosaic correlated state with an interlayer bias, giving rise to controllable charge transfer and quenching of correlations. Our results establish twisted 1T-TaS$_2$ as a flexible platform to engineer mixed spatially modulated correlated insulating phases, arising from the moiré profile.

It is common when using cross-section or panel data to assign each observation to a cluster and allow for arbitrary patterns of heteroskedasticity and correlation within clusters. For regression models, there are many ways to make cluster-robust inferences. A number of different variance matrix estimators can be used. Hypothesis tests and confidence intervals can then be based on several alternative analytic or bootstrap distributions. Some methods typically perform much better than others, but no method yields reliable inferences in every case. Thus it can be hard to know which $P$ values and confidence intervals to trust. Nevertheless, by using a number of procedures to assess the reliability of various inferential methods for a specific model and dataset, we can often obtain results in which we may be reasonably confident.