Traditional inference on the coefficient in an instrumental variables regression does not retain size when the instrument set is weak. With constant treatment effects or one instrument, the Anderson and Rubin (1949) AR test, the Klieibergen (2002)-Moreira (2003) LM test, and the Moreira CLR test provide robust alternatives which retain validity. Under treatment effect heterogeneity, no valid inference procedure exists in the overidentified setting. This paper develops the TSLS likelihood ratio (TLR) statistic, for performing inference on the TSLS estimand. When combined with a two-step procedure in the spirit of Berger and Boos (1994), it retains uniform validity across both the weak- and strong-instrument regimes. The procedure retains power with small choices of first-step level, hence the test can be constructed to numerically coincide with the Wald test in the strong-instrument limit.

For a long time, additive quantizers, such as product quantization, have been considered the gold standard in terms of accuracy and efficiency. Recently, scalar quantization has re-emerged from the depths of history with a new wave of data-agnostic techniques. Inscribed in this general framework, we turn our attention to data-driven methods, showing that new highs in recall and speed can be achieved by reducing the number of dimensions while increasing the bitrate per dimension. Critically, this dimensionality reduction needs to be learned from data to be successful. We present ASH (Asymmetric Scalar Hashing), a data-driven encoder-decoder framework that applies dimensionality reduction to database vectors via a learned orthonormal projection, followed by scalar quantization, while keeping queries in their original form. This asymmetric design enables higher accuracy than the best additive and scalar quantizers at iso-compression, while admitting highly efficient similarity computations via SIMD operations. ASH has short learning and encoding times, making it attractive for real-world deployment. Extensive experiments on a variety of datasets demonstrate that ASH achieves state-of-the-art ANN recall and speeds across all compression regimes.

We prove global smoothness for smooth finite-energy axisymmetric solutions of the three-dimensional incompressible Navier--Stokes equations with arbitrary swirl. The proof is organized around the circulation \(Γ=ru^θ\), the lifted azimuthal vorticity ratio \(G=ω^θ/r\), and the axis-compatible circulation-gradient pair \[ Ξ=(A,W)=\left(\frac{Γ_r}{r},\frac{Γ_z}{r}\right). \] The principal near-axis difficulty is the source term \(\partial_z(F^2)\), where \(F=u^θ/r=Γ/r^2\), in the lifted \(G\)-equation. The first key observation is the exact identity \[ \partial_z(F^2)=\frac{2ΓW}{r^3}, \qquad dμ_5=r^3\,dr\,dz, \] which converts the source pairing into \(2\int GΓW\,drdz\). This term is controlled by an axis Hardy formula for \(Γ\), one-dimensional Sobolev estimates in the axial variable for radial energy densities, and the positive \(W/r\)-Hardy term in the \(Ξ\)-dissipation. The second key point is that the typed zero-output endpoint is no longer treated as an abstract bridge-profile problem. After all source, collar, macro, motion, projection, cascade, and backward-ancestor channels vanish, a small-threshold energy-seeding lemma gives \[ G\in L_t^\infty L^2(dμ_5)\cap L_t^2\dot H^1(dμ_5). \]

The Renormalization Group flow connects microscopic to macroscopic descriptions of a system and is therefore typically considered as an initial-value problem. Motivated by situations in which different couplings within a system of Renormalization Group equations are constrained at different scales, we instead consider boundary-value problems in Renormalization Group flows. We find that, unlike initial-value problems which provide $n$ conditions for $n$ couplings, boundary-value problems which provide $n$ conditions for $n$ couplings do not always have a unique solution. When the Jacobian matrix, i.e., the matrix of first derivatives of beta functions, has complex eigenvalues, boundary-value problems may be non-unique. We provide a diagnostic tool for non-uniqueness in systems with many couplings. We also provide two examples with potential relevance for physics, namely within the Standard Model as well as within the Einstein-Hilbert truncation of asymptotically safe quantum gravity.

One of S.T. Yau's problems asks the following: given a $3$-dimensional asymptotically flat manifold $M$ with non-negative scalar curvature and $L^2$-norm of the curvature tensor at most $1$, if the mass of $M$ is small, is there a bilipschitz diffeomorphism from $M$ to the flat Euclidean space $\mathbb{R}^3$? We provide a strong positive answer to this problem by using our previous work \cite{DS25}.

Recent experiments have provided evidence for excitonic insulator (EI) states in 1T HfTe2. In this work, we investigate EI states in monolayer, bilayer, trilayer, and bulk 1T HfTe2 using advanced meta generalized gradient approximation (meta GGA) calculations and a model Bethe-Salpeter equation (BSE) approach, together with structural and electronic symmetry breaking analyses. Our results show that both the monolayer and bilayer exhibit negative exciton energies, leading to the spontaneous formation of bound excitons and EI states, whereas the trilayer and bulk display positive exciton energies and do not support EI states. Structural symmetry-breaking calculations show very small in-plane displacements of the Hf atoms from their symmetric positions in the monolayer and multilayers, consistent with experimental observations. Interestingly, electronic symmetry-breaking calculations for the monolayer, performed using a symmetric structure and a hybrid functional, show a pronounced unfolded valence-band feature at the M point and no unfolded conduction-band states near the Fermi level at Gamma, in good agreement with experimental results. Overall, our findings support the existence of EI states in low dimensional 1T HfTe2. The methodology developed here can be readily extended to investigate EI behavior in other related quantum material systems.

When a solid support undergoes crystallization, the advancing amorphous-to-crystalline transformation front separates regions of distinct surface energy, creating a moving interfacial energy boundary. A supported nanoparticle straddling such a boundary experiences an asymmetric particle-substrate interfacial energy environment that constitutes a lateral thermodynamic driving force for migration. Here, using in situ transmission electron microscopy to track Pt nanoparticle motion statistically, paired with time-resolved diffraction and 4D-STEM analysis to characterize support crystallization, we demonstrate that propagating crystallization fronts in amorphous AlO$_x$ thin films actively drag supported Pt nanoparticles over long distances. Temporal correlation between the onsets of support crystallization and rapid particle migration, together with 4D-STEM virtual crystallinity maps, establishes that the front drives particle motion. Phase-field simulations confirm that particle-substrate interfacial energy contrast alone sustains particle drag, and identify curvature gradients along the particle surface as the mechanism by which the advancing front redistributes mass and displaces the particle. These results establish a general mechanism by which any propagating surface-energy boundary on a substrate can act as a deterministic driver of supported nanoparticle transport.

Understanding data change is critical towards understanding trends, normal vs. abnormal behaviours, recognizing patterns, and the causes of change. Existing database systems have limited support for change management, relying on statistics, triggers, and constraints. Data quality rules model sequential changes along a restricted set of attributes, quantify change among unordered tuples, and have limited ability to model the context under which attribute changes occur. In this paper, we introduce Change Rules (CRs) that quantify the sequential changes among ordered tuples in both the antecedent and consequent attributes. CRs aim to address the limitations of existing declarative dependencies to support trend analysis and causal relationships that trigger change among attributes. We propose CR-Miner, an automated algorithm for CR discovery that generates candidate change intervals in a level-wise manner. Experimental results show that CR-Miner achieves an average runtime improvement of 40-50% over existing baselines.

This paper presents a preliminary model for managing involuntary technical debt in agile environments by formulating, in an integrated way, the dynamics among backlog, debt, velocity, and economic value. The work distinguishes initiated but unfinished functional debt from a simple defect back log and from rework, interprets productivity degradation as technical-debt interest, and derives the naive maximum-remediation policy in order to show its limitations against an intertemporal value-based decision. On this basis, a dynamic policy uk is proposed to balance new development and remediation; a decreasing marginal-value structure is incorporated; and the model is extended to discrete, inhomogeneous items. Exploratory validation through sensitivity analysis and MonteCarlo simulation shows behavior consistent with the economic intuition of the model. Finally, the limits of the formulation are made explicit: its macroscopic nature, its dependence on organizationally stable parameters, its assumption of intertemporal rationality, and its requirement of weak coupling among stories.

How well turbulence mixes a scalar $θ$ is governed by the scalar dissipation rate $χ= 2D |\nablaθ|^2$, making scalar gradients central to turbulent mixing. We study the structure and amplification of these gradients for passive scalars driven by a uniform mean-gradient in isotropic turbulence, using DNS at grid resolutions up to $8192^3$. The $Re_λ$ spans $140-1000$, and $Sc\equivν/D$ spans $1-512$. We analyze joint statistical correlations of velocity and scalar gradients that underlie scalar-gradient amplification. Unconditional statistics reaffirm earlier observations that production of $χ$ is dominated by nonlinear amplification of scalar gradients by strain-rate. Scalar gradients preferentially align with the most compressive strain eigenvector and remain orthogonal to vorticity, with both trends virtually independent of $Re_λ$ and $Sc$. Conditional statistics reveal that this organization becomes dramatically enhanced in regions of intense scalar dissipation: scalar gradient becomes near-perfectly aligned with the most compressive eigendirection and orthogonal to other eigendirections and vorticity. This and visualizations suggest that intense scalar dissipation is organized in sheet-like structures formed in shear layers between vortex tubes, where intense strain also generally resides. However, the effective strain acting along intense scalar gradients is comparatively much weaker, indicating intense scalar dissipation arises primarily from optimal alignments rather than intense strain alone. Molecular diffusion arrests intense scalar-gradient events primarily by redistributing scalar-gradient variance away from intense structures. The contribution from imposed mean-gradient is negligible,but still imprints anisotropy directly onto smallest scales via the strain field. The statistics broadly become universal as $Sc$ and $Re_λ$ increases

The study of the large-scale distribution of galaxies provides essential information for testing the standard cosmological model, namely the $Λ$CDM paradigm. This scenario is based upon two foundations: General Relativity as the theory of gravity, and the Cosmological Principle, which states that the Universe is statistically homogeneous and isotropic on large scales -- so that we can measure distances and ages in the Universe assuming the FLRW metric. In this work, we perform a test of the Cosmological Principle by probing the angular homogeneity scale, $θ_H$, using the state-of-the-art observational data of Luminous Red Galaxies (LRGs) from the Dark Energy Spectroscopic Instrument Data Release 1 (DESI DR1). Our analysis is performed exclusively in two dimensions, across narrow redshift ranges inside a larger redshift sample of $0.4 < z < 1.1$, in two different surveyed regions of the sky (North and South Galactic Caps), as we want to minimize a priori dependences on an underlying cosmological model. We obtain that such a scale is indeed identified in all redshift ranges, and that they are consistent with mock simulations assuming the $Λ$CDM model. Moreover, our results are in great agreement with previous measurements using Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey Data Release 16 (SDSS-IV eBOSS DR16), as well as between the north and south galactic caps of the DESI DR1 survey. These findings help underpinning statistical isotropy and homogeneity of the Universe as a physically valid hypothesis in light of upcoming stage-IV redshift surveys, hence are consistent with one of the fundamental pillars of the standard cosmological model.

Unambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased). We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. In an independent and concurrent work, Buzet and Chailloux study similar fine-grained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.

Ultrasound Localization Microscopy (ULM) enables microscopic imaging of the cerebral microvasculature in vivo, but relies on a multi-stage processing pipeline in which acquisition settings and reconstruction processes strongly influence the final output. Existing public datasets remain sparse, restricting rigorous evaluation and slowing progress in algorithm development, including emerging machine-learning approaches, which by design require large quantities of data to be robust and reliable. We introduce \textbf{ULMShare}, an open-access dataset of 99 whole-brain transcranial ULM acquisitions from 61 healthy mice (36 females, 22 males, 3 unknown; mean age: $8.2 \pm 5.5$ weeks; mean weight: $17.7 \pm 4.2$ g), for a total of 30TB of raw data. The dataset spans three experimental procedures, multiple injection and anesthesia protocols, two ultrasound probes, and different imaging planes and orientations. Each acquisition includes raw ultrasonic data, detailed metadata, an illustrative reconstruction and the corresponding microbubble trajectories. Alongside the data, we report vascular saturation, Fourier Ring Correlation, and track-length statistics, plus expert visual gradings. ULMShare provides a broad, standardized and publicly available resource for method development, validation, and benchmarking. The full dataset is available on the Federated Research Data Repository and additional resources are hosted on the ULMShare Github repository.

We present PDE-Agents, a multi-agent ecosystem that automates the full lifecycle of partial differential equation (PDE) / finite element method (FEM) simulations through natural-language interaction. Three specialist large language model (LLM) agents (Simulation, Analytics, Database) are orchestrated via a LangGraph supervisor, with a local open-source LLM stack (Qwen3-Coder-Next, Llama 4 Scout) on dual NVIDIA RTX PRO 6000 GPUs. The architecture is model-agnostic, validated across two LLM generations. A GraphRAG knowledge base (Neo4j, 768-d vector embeddings) encodes curated material properties, known failure patterns, and prior run lineage. We report seven contributions: (i) a verification and validation (V&V) study confirming second-order spatial convergence (O(h^2)) on the heat-equation solver; (ii) a three-way ablation over 50 tasks with a frozen KG (KG On, KG Off, KG Smart), where KG Smart reaches 100% success and the highest output quality (physics 0.933 vs. 0.853 for KG Off; MPF 0.926 vs. 0.796); (iii) a novel-material experiment with three fictional materials known only to the KG, where KG Smart attains near-perfect material property fidelity (MPF = 1.00) versus 0.34 for the KG-free baseline; (iv) a failure analysis tracing KG On's three failures to budget exhaustion and timeout, establishing warm-start injection as the dominant reliability factor; (v) an adaptive framework selecting the optimal retrieval mode per task; (vi) production metrics from 1,369 runs (97.8% success, 57.6% first-try); and (vii) a 100-task KG growth experiment showing a difficulty-dependent gain, with hard-task MPF improving 8.8% while easy/novel tasks stay at ceiling. All code, models, and evaluation artifacts are released openly. Our findings show that integration pattern, not knowledge content, determines whether GraphRAG augmentation helps or hinders LLM agents.

We study orthogonal modular forms associated with moduli spaces of lattice-polarized K3 surfaces whose generic transcendental lattices are of the form $T = H \oplus H \oplus L(-1)$ where $L$ is a root lattice of type $A_n$ or $D_n$. In Picard numbers $10$ through $17$, we use explicit Jacobian elliptic fibrations to construct modular forms on type IV domains associated with orthogonal groups $\mathrm{O}^+(T)$. We show that the coefficients of suitable Weierstrass models naturally realize generators for the corresponding graded rings of orthogonal modular forms.

Platinum silicide (PtSi) thin films are promising for silicon-based superconducting quantum devices due to their compatibility with CMOS fabrication, air stability, and superconducting transition temperature near 1 K. We report a systematic study of PtSi phase formation, microstructure, and interface quality as a function of annealing temperature and duration, characterizing films using grazing-incidence X-ray diffraction, X-ray reflectivity, and electrical transport measurements. Phase-pure PtSi forms within minutes by rapid thermal processing at 600 °C and is stable under extended annealing, while 30 s anneals across 300-600 °C yield equivalent film quality with consistent microstructure and superconducting properties. X-ray reflectivity reveals that interfacial roughening is an intrinsic consequence of the Pt2Si-to-PtSi conversion step rather than a result of elevated temperature or prolonged annealing. These results establish a robust processing window for PtSi formation in silicon-based superconducting device fabrication flows.

The Behrens--Fisher problem concerns inference on the difference of two normal means when both variances are unknown and unequal. It is a classical example in which nuisance parameters prevent ordinary exact fixed-sample inference, and it has long served as a benchmark for the foundations of inference. We revisit it through the inferential model (IM) framework of Martin and Liu. After conditioning and regular marginalization, the exact association is two-dimensional, with one coordinate for the standardized mean contrast and one for the variance ratio. Their one-dimensional generalized marginal IM is then best understood as a cylindrical two-dimensional predictive random set: sharp in its mean-contrast projection, by Hsu's stochastic domination, and vacuous in the variance ratio. Our main result is a precise validity-first optimality: among prior-free procedures that retain exact, uniform, finite-sample validity, the IM interval is the shortest. We prove minimaxity and admissibility in the cylindrical class and, by a projection argument, extend this to rectangular and general two-dimensional predictive random sets. A companion tradeoff principle shows that any adaptive procedure can only redistribute interval width across variance-ratio regimes, never shorten it uniformly. A Monte Carlo study bears this out: Welch and the bootstrap under-cover, whereas the conservative fiducial does not dominate the IM interval, being shorter only where the latter over-covers and longer where validity binds.

We introduce two infinite families of 3-regular planar graphs. Both families are conceptual adversaries to the Pohl-Warnsdorf algorithm for finding Hamiltonians. We provide a closed form calculation of the number of Hamiltonians.

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection -- under suitable non-degeneracy assumptions -- belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=\frac{1-d}{2}$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. This result is consistent with the description of the filamentary visual patterns -- often described as ``scars'' or ``scarlets'' -- observed in numerical simulations of random waves since the work of Heller, O'Connor and Gehlen (1987), in the precise sense that our findings show that suitable observables of Berry's random wave exhibit fluctuations that, in the large-domain limit, become arbitrarily close to those generated by a Poisson line process. It also suggests that the emergence of linear scar-like patterns in random waves is a phenomenon that persists in any dimension. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the {\it Radon--Fourier coefficients} associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.

We construct and analyze an exactly divergence-free Scott-Vogelius finite element method for the surface Stokes problem. The proposed scheme simultaneously enforces the tangentiality and incompressibility constructs exactly and has the same number of unknowns as the two-dimensional Euclidean discretization. Our construction extends the surface finite element framework of [10,11] to Scott--Vogelius discretizations defined on curved Clough--Tocher triangulations. In contrast to previous isoparametric Scott--Vogelius methods based on macro-element constructions, the present approach defines the finite element spaces directly on the refined surface triangulation, leading to a substantially simpler and more practical implementation. We prove inf-sup stability of the method and derive optimal-order convergence in the isoparametric regime.

This work is motivated by the analysis of temporal difference algorithms, where stability can be guaranteed by bounding the eigenvalues of an associated matrix derived from a, typically non-reversible, Markov kernel. We generalize the existing sufficient conditions for stability and show that the associated eigenvalues can be bounded in terms of the Dirichlet spectral gap of the Markov kernel. We derive a collection of methods for showing that non-reversible Markov chains have positive spectral gaps. We show that if a Markov chain has positive absolute spectral gap, then it has a positive Dirichlet spectral gap. In the case of discrete-time linear Gaussian systems, we give explicit bounds for both Dirichlet and absolute spectral gaps. Additionally, we present an example of a Markov chain which is $V$-uniformly geometrically ergodic but has zero Dirichlet spectral gap.

Parker Solar Probe (PSP) observations in the young solar wind reveal new properties of both plasma particle velocity distributions (VDs) and associated electromagnetic (EM) wave fluctuations. The quasilinear (QL) kinetic theory of plasma wave instabilities has recently shown that new hammerhead (HH) proton distributions can be generated by the relaxation of proton beams through the instabilities of right-handed (RH) polarized waves. Such RH waves have indeed been reported in association with HH distributions. In this paper, new results from hybrid simulations of proton-beam-plasma systems with properties typical of those observed to excite EM-RH wave instabilities are presented. From the long-term evolution of these systems, it is found that beam relaxation is driven by instabilities and growing wave fluctuations, leading to HH-type features in the velocity distributions. The production of these features, as well as their prominence, depends on the magnetic power of the waves generated by the instabilities and, therefore, implicitly on the available free energy, quantified by the plasma beta parameter and the relative beam drift. The simulation results capture the self-consistent evolution of the instabilities and their nonlinear development. Linear theory, together with simulations, helps identify the nature of the unstable modes and the plasma conditions under which they arise. The good agreement with quasi-linear (QL) theory further indicates that it can serve as a computationally efficient complementary framework for interpreting the associated wave-particle interactions.

This paper presents SLRMentor, a conversational assistant designed to support both learning about the systematic literature review process and the execution of planning activities in software engineering. The tool offers general guidance on SLR methodology and supports key planning tasks, including search string construction and reasoning about inclusion and exclusion criteria, with explanations grounded in established SLR guidelines. A pilot validation with graduate students suggests that SLRMentor helps clarify the SLR process and planning decisions, lowers initial barriers for novice researchers, and supports learning while still requiring active methodological judgment.

Academic integrity in higher education is increasingly shaped by complex socio-technical environments marked by automated tools, evolving institutional practices, and heightened performance pressures. Within this context, large language models (LLMs) are becoming prevalent in software engineering education, further blurring boundaries around acceptable assistance and authorship. This study investigates how software engineering students describe their emotional experiences after using LLMs in ways they perceive as academically inappropriate. We conducted a cross-sectional survey with 116 undergraduate students. Results show emotionally heterogeneous responses. Indifference was most frequent, including among students who recognized risks to learning and academic standing. Guilt and anxiety were reported in relation to moral discomfort and concern about penalties. Relief and satisfaction were evident primarily in deadline-driven contexts and situations of unclear guidance.

The determination of nuclear charge radii is crucial for understanding the internal structure of nuclei and their fundamental interactions. A persistent discrepancy, not only in the measured proton charge radius but also in the light nucleus charge radius, between electron scattering and muonic spectroscopy has fueled ongoing debates in nuclear and particle physics. Using this discrepancy, we revisit the role of one of the proposed solutions, namely the use of Lorentz-boosted nuclear form factors to find a subtle connection between the boost and the cluster structure of nuclei. By applying two distinct relativistic formalisms, namely the Licht-Pagnamenta and Mitra-Kumari approaches, we systematically analyze corrections to the moments of the density distributions in hadrons and nuclei. Our results demonstrate that boosting the form factors from the Breit to the rest frame of the nucleus not only assists in reconciling the spectroscopic and scattering measurements but also provides a method to infer on the quark and nucleon cluster configurations within nuclei.

We determine the optimal parameters for thermal neutron-induced fission of $^{233,235}$U and $^{239,241}$Pu in the complete event fission model $\mathtt{FREYA}$. First, we revisit and, in most cases, improve the prior values determined for spontaneous fission using a genetic algorithm. Our optimization procedure is then applied to thermal neutron-induced fission, replacing the empirically-chosen parameters previously employed in $\mathtt{FREYA}$. Finally, the optimized parameter values are used to make predictions for spontaneous and thermal neutron-induced fission observables not included in the fits. This work represents the first step in a broader program to study neutron-induced fission as a function of incident neutron energy, to systematically improve the performance of $\mathtt{FREYA}$, capturing any energy-dependent trends of the physics-based $\mathtt{FREYA}$ parameters.

Deep learning offers a powerful approach to quantum many-body problems via neural network wavefunctions, but their optimization remains a severe bottleneck. Existing optimization methods, including natural gradient descent and stochastic reconfiguration, suffer from spectral gap-dependent convergence that limits their effectiveness on systems fraught with competing orders and nearly degenerate ground states, such as frustrated magnets and strongly correlated electron materials. Here, we introduce Projected Inverse Iteration (PII) by re-framing the ground-state search as an eigenvalue problem. PII achieves rapid, gap-insensitive convergence while preserving the favorable polynomial computational scaling of stochastic reconfiguration. Demonstrated on challenging two-dimensional spin systems, including the highly frustrated $J_1$-$J_2$ model, PII outperforms standard optimization techniques and presents a promising algorithmic strategy for discovering complex quantum states in the presence of small spectral gaps. More broadly, PII can be interpreted as a novel natural gradient method tailored for eigenvalue problems, opening up its application to related challenges within deep learning.

In this note, we establish an explicit upper bound for the dimension of the rational cohomology for a simple algebraic group over an algebraically closed field of prime characteristic.

We investigate motility-induced phase separation in a lattice gas of self-propelled particles with hard-core exclusion, where an internal director biases particle hopping along the lattice coordination directions while undergoing rotational diffusion, together with a thermal-like translational diffusion. Rather than employing stochastic simulations, we adopt a master-equation formalism within a general mean-field approximation. By linearizing the mean-field master equation around the homogeneous stationary state and applying Bloch's theorem, the stability analysis is reduced to a $z$-dimensional tight-binding eigenvalue problem. A perturbation expansion in the wavenumber near $\vk = 0$ then yields the spinodal surface in closed analytical form for six Bravais lattices: linear, square, hexagonal, simple cubic, body-centered cubic, and face-centered cubic. The influence of lattice geometry is shown to enter exclusively through a single coefficient $\mathcal{A}$ which we evaluate exactly for each case. We further show that translational diffusion smooths the interface between the dense and dilute phases. Finally, we determine the rotational probability currents associated with the inhomogeneous stationary states, a distinctive signature of the broken detailed balance underlying active-system dynamics.

Liquid-crystal spatial light modulators (SLMs) are widely used for programmable wavefront control but are typically operated as phase-only devices, limiting applications that require independent amplitude and phase shaping. Here we demonstrate full complex-field modulation using a single phase-only SLM by implementing two sequential modulation planes on different regions of the same device. The phase retardance introduced by the first SLM region is converted into amplitude modulation by a polarizer placed in the beam path, while the second region compensates the associated phase offset and imposes the required phase distribution. The field from the first region is imaged onto the second, enabling complex-field synthesis without a second modulator. We validate the approach by generating Bessel--Gaussian beams, helical-phase fields, and arbitrary focal-plane intensity patterns. This single-SLM platform provides a compact route to programmable complex wavefront engineering for structured illumination, holography, and electron--light interaction experiments.