AI for Science

The emergence of non-relativistic spin splitting (NRSS) in altermagnetic systems has introduced a new paradigm in antiferromagnets with vanishing net magnetization. Although sliding-induced valley-polarized phases have recently been demonstrated in two-dimensional altermagnets, the observed valley-polarized state represents only a partial manifestation of altermagnetism, and a comprehensive classification based on spinsplitting characteristics remains lacking. Here, using first-principles calculations, general stacking theory, and spin-Laue symmetry analysis, we propose a coupled quasialtermagnetic state representing a distinct subclass of altermagnetism, in which reversible type-IV NRSS is controlled through interlayer sliding. Accordingly, the sliding-induced phases are classified into two categories: altermagnetic and quasi-altermagnetic states. We establish a direct correspondence between reciprocal-space spin splitting and real-space switching between the two quasi-altermagnetic states. Importantly, the spin-polarized bands in these states remain spin split at Γ point even in the absence of spin-orbit coupling (SOC), distinguishing them within the proposed classification framework. To demonstrate the interplay between altermagnetic and quasi-altermagnetic states, we investigate the two-dimensional Lieb-lattice material Mn2WS4 and its Janus derivative Mn2WS2Se2, analysing how changes in the local environment influence the different magnetic phases. Importantly, the underlying mechanism is broadly applicable to a wide class of twodimensional square-lattice systems. We further investigate the effects of SOC, focusing on spin texture and transport signatures in coupled quasi-altermagnetic states.

Valley polarization in 2D TMDs is promising for low-power valleytronic and spin-valley information processing, but time-reversal symmetry in pristine nonmagnetic TMDs keeps the K+ and K- valleys degenerate, limiting device applications. In this work, we investigated the structural stability, electronic properties, and tunable valley polarization of V-alloyed MoTe2 monolayer, Mo0.75V0.25Te2, using first-principles density functional theory (DFT) calculations. Substitutional alloying of MoTe2 with V introduced magnetic exchange interaction, which, together with spin-orbit coupling (SOC), lifted the valley degeneracy at the unequal valleys. The alloyed structure was found to be energetically and dynamically stable due to the absence of imaginary phonon modes. In pristine MoTe2, SOC produced spin splittings of 34.0 meV and 218.9 meV in the conduction bands and valence bands, respectively, but no valley polarization was observed. In contrast, Mo0.75V0.25Te2 exhibited spontaneous valley polarization of 37.3 meV in the conduction band and 78.2 meV in the valence band. The valley polarization was further enhanced by external electric fields and biaxial strain. A transverse electric field along the crystal c axis produced the maximum valley splitting of 132.8 meV in the valence band, whereas biaxial tensile strain increased the valence band valley splitting up to 160.8 meV. The maximum conduction band valley splitting reached 54.4 meV under 2% biaxial compressive strain. These results demonstrated that V alloying, combined with electric-field and strain engineering, provides an effective strategy for achieving large and tunable valley polarization in MoTe2. Thus, Mo0.75V0.25Te2 can be considered a promising 2D platform for tunable valleytronic device applications, such as transistors and sensors.

The quantum spin Hall effect (QSHE) has attracted widespread attention due to its dissipationless transport, which is protected by non-trivial topological invariants and helical edge states. Because even weak magnetic disorder can destroy the stability of topological quantum states, current research on the QSHE has primarily focused on non-magnetic materials. In this work, we extend the research scope of the QSHE to altermagnets. We establish the relevant symmetry constraints and identify all magnetic point groups that can realize the altermagnetic QSHE. Symmetry analysis reveals that pronounced spin-valley locking or spin-valley-layer locking universally exists in these systems. The concerted interaction between band inversion and spin-valley locking collectively gives rise to the helical edge states. Using first-principles calculations and theoretical models, we demonstrate that monolayer Nb2SeTeO exhibits an altermagnetic QSHE characterized by spin-valley locking, while bilayer Hf3Se3Te2 manifests an altermagnetic QSHE featuring spin-valley-layer locking. This work clarifies the intrinsic symmetry correlation between altermagnetism and quantum spin Hall topological phases, providing a brand-new theoretical perspective and research platform for exploring magnetic topological systems and developing next-generation spintronic devices

Interlayer interactions in layered materials are often assumed to transfer from the bilayer to the bulk, but this assumption can fail when chemically active out-of-plane orbitals participate in bonding. We combine diffusion quantum Monte Carlo (DMC) and density functional theory (DFT) to determine how interlayer coupling evolves in arsenene multilayers. DMC shows that bulk gray arsenic is compact, whereas the corresponding few-layer structures remain at substantially larger interlayer separations despite sharing the same nominal A$_{1}$B$_{-1}$ adjacent-layer registry. Registry alone therefore does not determine the bonding regime; thickness and coordination reshape the interlayer interaction. Among the tested functionals, SCAN+rVV10 most closely reproduces DMC equilibrium separations and stacking energetics. Using the DMC-benchmarked SCAN+rVV10 calculations, we predict a thickness-driven stacking sequence from A$_{1}$A$_{1}$ to A$_{1}$B$_{1}$ and finally bulk-like A$_{1}$B$_{-1}$. The structural crossover coincides with a stacking-dependent DFT band-gap collapse driven by enhanced interlayer As p$_{z}$ hybridization.

Recently, layered corrected kagome metal CsCr3Sb5 have garnered significant attention attributed to its flat bands near the Fermi level (EF) and altermagnetic ground state [ Yi Liu et al., Nature 632, 1032 (2024)]. However, the van Hove singularities (vHSs) in bulk CsCr3Sb5 are far away from the EF, while an effective modulation of VHS toward the EF is essential for exploring intriguing electron transport properties. In this work, using first-principles calculations, we investigate electronic structures of two-dimensional (2D) Cr3Sb5 and CsCr3Sb5 monolayers which may be mechanically exfoliated from bulk materials. Notably, it is revealed that both flat bands and vHSs simultaneously reside in close proximity to the EF in Cr3Sb5 monolayer, signifying enhanced electronic correlations. Importantly, a tensile strain further shifts the incipient flat bands and vHSs of two monolayers simultaneously to the vicinity of the EF, suggesting strain tunable electronic correlations and concomitant quantum effects. Furthermore, altermagnetic ground state is also revealed due to retained mirror symmetry between two sublattices in these two monolayers. Thus, this work advances understanding and modulations of electronic properties of 2D CsCr3Sb5 monolayers, strengthening their great potential for exploring unconventional quantum phenomena and altermagnetism.

We theoretically investigate the optical spin-injection response in different stoichiometric configurations of graphane and fluorographene using density functional theory. Our goal is to determine which configuration yields the strongest degree of spin polarization. The results show that the fluorographene zigzag configuration yields the best degree of spin polarization response (${\cal DSP}^{\mathrm{z}}$), with 98\% spin polarized electrons at the band edge and over a wide range of excitation photon energies. In contrast, other graphane and fluorographene configurations achieve a ${\cal DSP}^{\mathrm{z}}$ of roughly 83--100\%, but only within a limited photon-excitation energy range. In structures with low spin-orbit coupling, the degree of spin polarization is close to 100\% over a wide range of photon energies. For higher spin-orbit coupling, this strong response appears, but only in a narrow photon energy region. Additionally, under the band-resolved decomposition scheme, the contributions of different band-to-band transitions to the ${\cal DSP}^{\mathrm{z}}$ spectrum are identified by summing only the selected valence and conduction bands. Our findings show that almost the entire ${\cal DSP}^{\mathrm{z}}$ spectrum of the fluorographene zigzag configuration comes from transitions that involve only the top valence band, which is a mixture of C--p and F--p states.

Predicting detailed atomistic structures of self-assembling systems remains a challenge for all-atom molecular dynamics simulations. Replica exchange with solute tempering (REST) has been used to study those systems by accelerating all monomers in a global and uniform manner. While such a global approach can in principle predict any morphology of the system, it has computational drawbacks such as inefficient replica traversal due to order-disorder transitions and the growing number of replicas with system size. To address these issues, here we propose an alternative, stepwise construction approach to modeling supramolecular polymers under the assumption of one-dimensional polymerization. Specifically, we generate polymer structures by adding new monomers one by one to the system and applying REST to the new monomers to find their optimal binding positions based on an energy-based scoring function. The monomer addition and enhanced sampling are repeated sequentially until a polymer of desired length is obtained. We test the above procedure using a model supramolecular polymer in explicit solvent, and show that it can generate a polymer structure with characteristic H-bonding patterns at reduced computational costs, while also improving the efficiency of replica traversal significantly. We thus expect that the sequential REST will be useful for modeling supramolecular polymers, particularly for cases where global REST simulations are too demanding computationally.

Rydberg atom simulators, in both analog and digital modes, have attracted significant recent interest due to their versatile geometric reconfigurability. In this work, leveraging this feature, we propose two complementary approaches, one for each mode, to characterize emergent fermions in critical quantum many-body systems. In the analog mode, we assemble the Rydberg atoms in a "developable" (namely, preserving local couplings) Möbius band geometry to realize antiperiodic boundary conditions, where fermionic states reside. Spectroscopic measurement in this sector then reveals universal energy ratios of the bosonic and fermionic states. In the digital mode, we carry out a fermionic version of Kibble-Zurek ramping with a quantum circuit, directly addressing the fermionic scaling form. Reconfigurability allows an exponential speed-up of this task, with an $O(\log L\log\log L)$ circuit-depth overhead. Our work establishes the Rydberg atom simulator as a uniquely powerful platform to attack the notoriously difficult issue of experimentally probing emergent fermions that are nonlocally defined in a bosonic system.

Quantum quenches drive edge states far from equilibrium, yet whether the memory of a topological initial state survives in a non-integrable, interacting system has remained largely unexplored. We study this question in the bond-alternating XXZ chain -- an interacting Su--Schrieffer--Heeger model hosting symmetry-protected topological edge modes with markedly enhanced boundary magnetization -- and analyze quenches across all combinations of single-particle and many-body initial and final Hamiltonians. The results organize by a single distinction as we rigorously establish in this work: whether the post-quench Hamiltonian is free or genuinely interacting. For a free post-quench Hamiltonian, the dynamics is solved exactly by a correlation-matrix approach; the boundary-mode return amplitude decays as $t^{-3/2}$, and initial interactions enter only through a dressed one-body density matrix. For a genuinely interacting post-quench Hamiltonian, finite-time stability bounds prove that away from local resonances the first-dimer magnetization remains stable on time windows growing as arbitrarily large powers of the inverse inter-dimer coupling. Matrix product state simulations across all four protocols show that interactions in the final Hamiltonian markedly extend finite-time boundary memory -- with local suppression near the isotropic $SU(2)$ point -- revealing a many-body protection mechanism in a non-integrable system where scrambling would otherwise wash out initial-state memory fast.

We present the exceptional phenomenon of complete absence of Anderson localization, and perfect transmission of particles, in a completely disordered diamond-dot chain. We analytically show a proof for the condition to observe this exceptional phenomenon, based on a transparent window emerging from a geometrical condition. We support our theoretical prediction by numerical simulations and direct experimental observation of the transmission probabilities of the light in a femtosecond laser-written diamond-dot photonic lattices. We additionally show that for a $π$ effective magnetic flux, extreme localization of the light in the same system may occur, independently on the specific geometry. Our results open up an excellent platform for controlling the transmission of energy from ballistic to zero transmission, in a completely disordered lattice system..

Foliated topological orders in three dimensions are layered systems in which anyons are free to move within a layer but cannot hop between them. A simple model with such a phase is a stack of decoupled two-dimensional electron gases in a strong magnetic field, each in the same fractional quantum Hall state. By focusing on the case of filling $ν=1/3$ of the lowest Landau level in each layer, we show that (i) the limit of decoupled Laughlin states is stable upon introducing interlayer interactions and (ii) the system can enter a spontaneously layer-trimerized foliated non-Abelian Fibonacci phase. We support our claims by numerical exact diagonalization of up to 10 layers as well as perturbative analytical calculations. Specifically, we show that the foliated Fibonacci phase exists in the 9-layer system with pseudopotential interactions within and between neighboring layers. We identify the phase via quasihole counting and by calculating the overlap with a model wave function which we derive from the associated conformal field theory. Our numerical results suggest the possibility of realizing these phases in layered van der Waals crystals in strong magnetic fields, as well as in multilayer heterostructures.

Tomography allows one to estimate the density matrix describing the state an ensemble of quantum systems are prepared in (for example, polarization tomography determines the polarization state of a beam of identically prepared photons). In general, it is not possible to uniquely decompose the density matrix into its pure state components. Agarwal et al. proposed a protocol which, for a mixture composed of any two pure states of a qubit (with arbitrary probabilities), allows an observer to infer not only the density matrix but the identity of those specific pure states and their weights - the additional requirement being that the qubits arrive in pairs, where both qubits in each pair are in the same state. We experimentally demonstrate this learning-from-pairs concept using photons in the polarization degree of freedom. We use tomography to measure a sequence of single photons and make use of their time-of-arrival information to 'pair up' the photons after the measurement. From here we are able to infer the photons' polarization states and their respective probabilities, and we demonstrate this for various different choices of polarization states and ratios. Finally, we investigate our ability to discriminate between two equal mixtures of distinct pairs of orthogonal polarization states. We find that on the order of approx. 10e4 photons is typically enough to achieve tomography fidelities of approximately 0.9999. This is sufficient to discriminate between two different preparations of the same mixed state, differing by angles of less than 5 degrees between the pure states used in the two preparations.

We investigate the propagation profile of positive solutions to \begin{equation*} u_t-du_{xx}=f(t,u) \mbox{ for } t>0,\ x\in(g(t),h(t)), \end{equation*} where $f(t,u)$ is monostable in $u$ and $T$-periodic in $t$, and the free boundaries $x=g(t), \ x=h(t)$ are determined by the Stefan condition $g'(t)=-μu_x(t, g(t)),\ h'(t)=-μu_x(t,h(t))$, coupled with $u(t, g(t))=u(t, h(t))=0$. For a special nonlinearity satisfying the strong KPP condition, the long-time behavior and asymptotic spreading speed of this problem were considered by Du, Guo and Peng \cite{DGP}. In this paper, by employing new techniques, we extend the results of \cite{DGP} to general monostable nonlinearities beyond the KPP framework and at the same time we obtain more precise description of the propagation profile: we prove the existence and uniqueness of a semi-wave and show that the spreading solution converges to this semi-wave as time goes to infinity.

For nearly a century, the transition state has been thought to lack an exact quantum counterpart: recrossing-free, one-way flux seems to require simultaneous knowledge of position and momentum. We show that this obstruction is illusory. The exact quantum flow contains a transition-state geometry: stable and unstable manifolds meeting in a unique bounded quantum transition-state trajectory that anchors a dividing surface carrying one-way quantum probability flux. The geometric framework of classical reaction dynamics survives in exact quantum mechanics, in a fundamentally quantum form.

Virtual accelerators and digital twins are increasingly essential tools for accelerator operations, controls development and verification, and model-based optimization. However, current implementations are often tightly coupled to specific simulation codes, facilities, and applications, resulting in fragmented, ad hoc solutions that are difficult to reuse or extend. To address this, we expand the LUME Python package to include standardized implementation and deployment of virtual accelerators and digital twins across heterogeneous simulation backends and control system interfaces. At the core of this change is the introduction of LUMEModel abstraction, which defines a fixed, simulator-agnostic API and a variable system that encodes metadata such as units and data types/validation. This design enables standardized interaction with physics-based simulators, surrogate models, and differentiable simulations, while supporting both Python-native workflows and IOC-based operation via EPICS using the lume-pva package. Facility- and simulator-specific details are encapsulated through extensible transformer layers, allowing consistent control-system semantics to be mapped onto diverse simulation engines. We describe the LUMEModel architecture, variable system, and package ecosystem, and present representative use cases including model interchangeability, staged and chained simulators, and continuous integration testing. This work will make implementing and using virtual accelerators easier and more flexible.

Rare $α$-particle-induced charge clusters appear in LSST images as compact, PSF-like sources with a median FWHM of $0.\!\!^{\prime\prime}95$ and median ellipticity consistent with zero, closely resembling unresolved astrophysical point sources. These events are detected in both dark and science exposures at a rate of approximately $10^{-12}\ \mathrm{pixel}^{-1}\ \mathrm{s}^{-1}$. Their collected charge and morphology are consistent with energy deposition from $\sim$5 MeV $α$-particles in silicon CCDs, and their spatial distribution across the focal plane suggests a localized material origin, plausibly associated with trace radioactive contamination in the cryostat aluminum. Despite their deceptive appearance, we demonstrate that a simple broadness statistic based on fourth-order moments cleanly separates these events from stellar PSFs, enabling efficient rejection in coadded images and real-time alert streams. Such charge clusters do not impose an intrinsic bright-end contamination floor for Rubin transient searches, as genuine fast astrophysical events would exhibit characteristically different morphological signatures.

An atomic Josephson junction realized with dipolar bosons in a double-well potential can be described by an extended Bose-Hubbard model in which dipolar interactions generate an effective on-site interaction and nearest-neighbor pair tunneling. Using mean-field theory and exact diagonalization, we investigate how this correlated process affects zero-temperature equilibrium and dynamical properties of the system. In equilibrium, we show that pair tunneling induces ground-state parity modulations and significantly reshapes the phase diagram, producing qualitative changes in the quantum phase transitions toward NOON and phase-NOON states, as well as quantitative shifts of the critical points. Out of equilibrium, we demonstrate that it modifies the conditions for macroscopic quantum self-trapping, and assess its impact by comparing mean-field and fully quantum evolution, including the emergence of dynamical quantum phase transitions.

A strange metal is an exotic state of correlated quantum matter; intensive efforts are ongoing to decipher its nature. Here we explore whether the quantum Fisher information (QFI), a concept from quantum metrology, can provide new insight. We use inelastic neutron scattering and quantum Monte Carlo simulations to study a Kondo destruction quantum critical point, where strange metallicity is associated with fluctuations beyond a Landau order parameter. We find that the QFI probed away from magnetic Bragg peaks, where the effect of magnetic ordering is minimized, increases strongly and without a characteristic scale as the strange metal forms with decreasing temperature, evidencing its unusual entanglement properties. Our work opens a new direction for studies across strange metal platforms.

Valley splitting in strained Si/SiGe quantum wells is a central parameter for silicon spin qubits and is commonly described with envelope-function and effective-mass theories. These models provide a computationally efficient continuum description and have been shown to agree well with atomistic approaches when the confinement potential is slowly varying on the lattice scale. In modern Si/SiGe heterostructures with atomically sharp interfaces and engineered Ge concentration profiles, however, the slowly varying potential approximation underlying conventional (local) envelope-function theory is challenged. We formulate an exact multi-valley envelope-function model by combining Burt-Foreman-type envelope-function theory, which does not rely on the assumption of a slowly varying potential, with a valley-sector decomposition of the Brillouin zone. This construction enforces band-limited envelopes, which satisfy a set of coupled integro-differential equations with a non-local potential energy operator. Using degenerate perturbation theory, we derive the intervalley coupling matrix element within this non-local model and prove that it is strictly invariant under global shifts of the confinement potential (choice of reference energy). We then show that the conventional local envelope model generically violates this invariance due to spectral leakage between valley sectors, leading to an unphysical energy-reference dependence of the intervalley coupling. The resulting ambiguity is quantified by numerical simulations of various engineered Si/SiGe heterostructures. Finally, we propose a simple spectrally filtered local approximation that restores the energy-reference invariance exactly and provides a good approximation to the exact non-local theory.

Characterization of the dynamics of an impurity immersed in a quantum medium is vital for fundamental understanding of matter as well as applications in modern day quantum technologies. The case of strong and long-ranged interactions is of particular importance here, as it opens the possibility to leverage quantum correlations in controlling the system properties. Here, we consider a charged impurity moving in a bosonic gas and study its properties out of equilibrium. We extract the stationary momentum of the ion at long times, which is nonzero due to the superfluid nature of the medium, and the effective mass which stems from dressing the impurity with the host atoms. The nonlinear evolution leads not only to emission of density waves, but also momentum transfer back to the ion, resulting in the possibility of oscillatory dynamics.

Photoluminescent color centers in diamond and hexagonal boron nitride (hBN) are powerful nanoscale solid-state quantum sensors that are explored in a plethora of quantum technologies. Methods for integrating them into macroscopic structures that improve their sensitivity and enable their large-scale deployment are highly sought after. Here, we demonstrate cavity-enhanced photoluminescence (PL) of fluorescent nanodiamonds (FNDs) and hBN nanoparticles (NPs) embedded in polymer-based thin-film optical cavities on the centimeter scale. The cavity resonances efficiently modulate the spectral PL peak position of nitrogen-vacancy (NV) centers in FNDs across the NV PL spectrum and lead to an up to 2.9-fold Purcell-enhancement of the NV PL decay rate. The brightness of hBN NPs increases by up to a factor of three and the PL decay rate is enhanced by up to 13-fold inside the cavities. Finally, we find a 4.8 times improved magnetic field sensitivity of 20 nm FNDs in thin-film cavities due to cavity-enhanced optically detected magnetic resonance contrast and PL brightness. Our study demonstrates a low-cost and scalable method for the fabrication of quantum sensor-doped thin-film cavities, which is an important step toward the development of advanced quantum sensing technologies.

This study investigates the dynamics of quantum information and computational resources using a tractable model of coupled harmonic oscillators. We precisely characterize the interplay between mutual information, synchronization, and circuit complexity, demonstrating that they serve as complementary yet distinct measures of quantum correlations. Our analysis reveals how coupling strength, detuning, and external magnetic fields modulate these quantities, with synchronization and mutual information exhibiting marked divergence in nonlinear regimes. By employing exact Gaussian methods, we compute the circuit depth required to prepare target states and connect increased fidelity to more regular dynamical behavior. Furthermore, we analyze single-ion transport in a harmonic trap, comparing sudden and adiabatic protocols. We introduce a nonadiabaticity metric to quantify the fidelity-complexity trade-off, showing that smooth control sequences significantly minimize operational errors by suppressing excitations. These results provide a refined understanding of quantum correlations and offer concrete principles for optimizing control strategies in quantum technologies.

The Richards equation, a nonlinear elliptic parabolic equation, is widely used to model infiltration in porous media. We develop a finite element method for solving the Richards equation by introducing a new bounded auxiliary variable to eliminate unbounded terms in the weak formulation of the method. This formulation is discretized using a semi-implicit scheme and the resulting nonlinear system is solved using Newton's method. Our approach eliminates the need of regularization techniques and offers advantages in handling both dry and fully saturated zones. In the proposed techniques, a non-overlapping Schwarz domain decomposition method is used for modeling infiltration in layered soils. We apply the proposed method to solve the Richards equation using the Havercamp and van Genuchten models for the capillary pressure. Numerical experiments are performed to validate the proposed approach, including tests such as modeling flows in fibrous sheets where the initial medium is totally dry, two cases with fully saturated and dry regions, and an infiltration problem in layered soils. The numerical results demonstrate the stability and accuracy of the proposed numerical method. The numerical solutions remain positive in the presence of totally dry zones. The numerical investigations clearly demonstrated the capability of the proposed method to effectively predict the dynamics of flows in unsaturated soils.

Electrons bound to the surface of superfluid helium have been proposed for scalable charge and spin-based quantum computing. However single electron quantum measurement in this system has remained elusive. Here we use a hybrid circuit quantum electrodynamic (cQED) device that comprises a quantum dot and a high-impedance superconducting resonator to demonstrate, for the first time, strong coupling between the resonator microwave field and the motional quantum state of the electron. We find a coupling strength between the electron motion and a resonator photon of $g/2π=118$ MHz, exceeding both the electron motional state decoherence and the resonator loss. These experiments open new avenues for investigating light-matter interaction at the single electron level, and are a key step towards measurement and control of electrons on helium-based spin qubits.

Gene regulatory networks govern cellular fate decisions through multistable dynamics. The genetic toggle switch is a canonical model of such behaviour; yet, the impact of cell division on its dynamics remains poorly understood. We derive analytical separatrices for a simplified Boolean toggle switch with and without division. We show that division can redirect trajectories with identical initial conditions to opposing stable states, and we define a region of disagreement where fate decisions are predicted incorrectly if division is neglected. Our results imply that division can fundamentally reshape fate boundaries in multistable regulatory networks.

We extend the theory of root clusters from perfect fields to general fields which are not necessarily perfect. We introduce the following notions for field extensions over any given base field and study their interesting properties: root cluster size, multicluster size and their generalizations root capacity, multiroot capacity; ascending index, ascending normal index and their generalizations intersection indicium, intersection normal indicium; compositum indicium and compositum normal indicium. We establish our results on the Inverse problems for these generalized notions over Hilbertian fields which generalizes our earlier results which were over number fields. In particular, we show over a given Hilbertian field, the existence of a polynomial for given degree, cluster size and multicluster size and existence of an extension for given root capacity and multiroot capacity with respect to that polynomial.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

Learning Bayesian network (BN) structure from sparse discrete data is hard: when each instance records only a few variables, most variable pairs lack the joint observations needed for reliable scoring, and data-only methods recover little structure. However, imperfect domain knowledge, expressible as a weighted directed knowledge graph (KG), is often available. We propose KG-SoftMAP, which encodes such a KG as a finite-strength, confidence-weighted edge prior and maximizes a MAP objective combining the BDeu score with a logit-form prior; the KG may be expert-curated or LLM-extracted. On synthetic benchmarks with known DAGs, KG-SoftMAP reaches Directed-F1 (DF1) $0.19$--$0.32$ at observation rate $ρ=0.05$ and DF1 $0.44$--$0.97$ at $ρ\geq0.2$, while every data-only learner tested stays near zero under the same sparse masks. Recovery tracks KG quality: controlled corruption degrades it smoothly, a zero-signal KG yields DF1 $0.00$, and a blindly LLM-extracted KG with imperfect precision and recall still drives substantial recovery. On three real sparse educational datasets, the learned BN acts as a concept-level posterior model: on SAF it matches logistic regression (LR) within $0.03$ F1_FAIL while providing an inspectable concept graph, calibrated Fail probabilities, and tractable posterior queries from partial observations.

Recently, Starr used asymptotic methods to disprove a conjecture by Heim--Neuhauser and Abdesselam about the log-concavity of the D'Arcais polynomials, without giving an explicit counterexample. We refine the asymptotics, to give the necessary estimates on convolutions of $σ_{-1}$, and identify the first counterexample at $λ= 65\,214\,507\,758\,400$. We also consider the asymptotic density of such counterexamples.

A finite group is said to be quartic-free if its order is not divisible by $p^4$ of any prime $p$. A finite group is called an $A$-group if all of its Sylow subgroups are abelian. Objective of this paper is to provide explicit structure of a quartic-free $A$-group. Further in the process of providing the explicit structure we also determine the derived length of a solvable quartic-free $A$-group.