We consider a four dimensional $SU(3)$ lattice gauge theory with 12 staggered fermions having identical masses and an unimproved action. Using sets of plaquette distributions for various inverse bare couplings $β$, we reconstruct the density of states with the Ferrenberg -Swendsen method and calculate the zeros of the partition in the complex $β$ plane with bare quark masses $m_q$ = 0.02, 0.06, 0.08 and 0.1 for hypercubes of linear size $L$= 4, 6, 8, 10, and 12. Our hypothesis is that there is a line of first order transitions in the $(m_q,β)$ plane ending at a second order phase transition. We expect this transition to be in the 4D Ising, mean field, universality class. We fit the $L$ dependence of the zeros with the lowest imaginary part using two ($y = bL^{-d}$) and three ($y = a + bL^{-d}$) parameter fits. For $m_q$ = 0.02 the results provide strong support for a first order phase transition ($d=3.98(6)$, and $a$ statistically compatible with 0). The results also indicate, with less statistical significance for $m_q=0.06$, that the three other masses are above the critical value $m_q^c$. In addition, we suggest that the infinite volume gap for the lowest zero $a$, can be represented as $a\simeq A(m_q-m_q^c)^{B}$ with $m_q^c\sim 0.05$ and $B\sim 1$. Given that there are only three data points with significant error bars, it is difficult to rule out the mean field value $B=3/2$. Combining this result with spectroscopic results by Jin and Mawhinney, indicates that the gap with real axis (Lee-Yang edge) scales roughly like $m_σ^2$, where $m_σ$ is the mass of the $0^{++}$ scalar which is also the lowest excitation.

We study the gauge invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group SU(2). Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-abelian fields through spherical surfaces centered at origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at the Euclidean time $x^4 = 0$ and radius $r=1$, which here corresponds to the size $λ$ of those solutions, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $θ$-vacuum.

Let $P$ and $P'$ be standard parabolic subgroups of $\mathrm{SL}_n(\mathbb{C})$ whose Levi subgroups are conjugate. We construct isomorphisms $Φ(P,P'):\overline{T^*(\mathrm{SL}_n/[P,P])}^{\mathrm{aff}}\rightarrow\overline{T^*(\mathrm{SL}_n/[P',P'])}^{\mathrm{aff}}$ between the affinizations of the cotangent bundles of the Braverman--Kazhdan spaces $\mathrm{SL}_n/[P,P]$ and $\mathrm{SL}_n/[P',P']$, and we show that they satisfy $\mathrm{SL}_n\times L^{\mathrm{ab}}$-equivariance and braid relations. These isomorphisms are the quasi-classical limits of Braverman--Kazhdan normalized intertwining operators. Our construction proceeds by defining reflection functors for quiver varieties with an $\mathrm{SL}$-gauge group, thereby generalizing Wang's construction of the Gelfand--Graev action on $\overline{T^*(\mathrm{SL}_n/U)}^{\mathrm{aff}}$ via quiver varieties in the case $P=B$.

The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as $T_K/D \approx 10^{-8}$ in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.

Contrary to naive expectations, poles used to define hadron resonances rigorously in the physical world may not evolve continuously to become degenerate in the SU(3)$_F$ and chiral limits of QCD. Instead, other shadow poles, usually ignored, may be the ones that degenerate and characterize the resonances in these limits. This feature is general, and we illustrate it first with the simple and familiar light-vector mesons, followed by the much-discussed light-scalar case. Their shadow poles and their degeneracy are found using the QCD low-energy effective theory unitarized to one loop.

We present a new version of the Sartre event generator for exclusive diffraction at small xP in the colour dipole model, for $ep$ and $e$A scattering at the EIC and the LHeC as well as ultra-peripheral $pp$, $p$A, and AA collisions at RHIC and the LHC. Sartre stores the first and second moment of the interaction amplitudes in lookup tables which are then used for efficient event generation. There are many possible combinations of processes in Sartre, with different initial state nuclear targets and different final state vector mesons or real photons. We also want to implement different versions of the dipole model: with and without non-linear saturation effects, with and without nucleon hotspot substructure. A long-standing bottleneck for simulating all possible exclusive processes has been the production of lookup tables, which take a few CPU-year for each combination, necessitating the use of computing farms. The calculation also involves integrations of rapidly fluctuating integrands, which may cause numerical glitches which takes much effort to smoothen out. In this paper we present a solution to these issues, by presenting a new numerical calculation which improves the efficiency in the table production by 3-4 orders of magnitudes. This enables us to produce lookup tables for any process that we may be interested in, in a few hours. The new calculation does also not exhibit numerical glitches. We provide novel predictions for the EIC and the LHeC using the new version of Sartre.

We investigate whether relativistic massless classical particles may emerge as the classical limit of massless quanta. To address this question independently of any specific dynamics, environment, or pointer basis, we develop an axiomatic and purely kinematical framework for the coarse-graining approach. In this formulation, a candidate classical phase space is taken as the outcome space of a POVM subject only to minimal classicality and covariance under the relevant spacetime symmetry group. Applying this framework to the Poincaré group, we prove a no-go theorem for massless particles: the covariance requirement is incompatible with the operational conditions for classicality. The theorem leaves open field-like limits of massless quanta, for example the emergence of electromagnetic or gravitational fields, while ruling out classical massless particles, such as classical photons or gravitons.

We adapt the recent theory for unique solvability of perturbed saddle-point problems in Banach spaces to the case of parameter-independent stability bounds. This also constitutes an extension to the Brezzi--Braess theory for parameter-robust stability of perturbed saddle-point problems from the Hilbert to the Banach setting. We apply the new abstract result to the mixed formulation of the advection-diffusion-reaction equation and tackle also its coupling with the Darcy--Forchheimer equations. The complete system is shown to be stable irrespective of the model parameters of permeability, Forchheimer coefficient, and reaction modulation. We discretise the problem with mixed finite element methods and show convergence in appropriately parameter-weighted norms. We also design operator-based preconditioners for the full system, utilising weighted norms that provide robustness with respect to model parameters.

Electrostatic parametric instabilities in the inhomogeneous near-SOL tokamak plasma, driven by the combined action of a high-harmonic fast wave (HHFW) with a frequency near the 30th ion-cyclotron (IC) harmonic and electron and ion temperature gradients, are investigated numerically. The results indicate the parametric decay of the HHFW into a HHIC (Bernstein) wave and HHIC quasimode. The instability is found to exist within a finite wavelength range of the HHFW. The development of the parametric HHIC quasimode decay instability leads to the onset of parametric turbulence accompanied by anisotropic ion heating, with the ion heating rate across the magnetic field significantly exceeding that along the magnetic field.

The tensor train (TT) model is widely used to approximate high-dimensional tensors, enabling efficient handling of data that may exceed available memory. TT helps address the curse of dimensionality in applications such as system identification and dynamic programming. In some applications, TT is known as a ``matrix product state" (MPS). This paper introduces algorithms that facilitate the summation, Hadamard (elementwise) product, and matrix--vector product of matrices and vectors (tensors) represented in the tensor train (TT) format. The last product is also known under the acronym MPO--MPS. The proposed algorithms achieve an improved tradeoff between computational efficiency and accuracy compared to state-of-the-art methods.

John E. Wetzel conjectured that the 30-60-90 triangle T obtained by placing a square of side 1/3 on the hypotenuse covers every unit arc in the plane. We give a computer-assisted proof of this conjecture with independently checkable interval certificates. The proof reduces a hypothetical noncovered arc to a finite family of 599 closed second-order cone models, covering all representative and raw tail-order branches, and certifies a polygonal-chain lower bound greater than one in every model by interval validation of stored dual certificates. Since every certified lower endpoint exceeds 1.0048, the homothetic copy T/1.0048 still covers every unit arc. Its area is 0.260956..., below the area pi/12 approx. 0.261799 of the 30-degree unit sector, a certified area improvement over the sector cover within this convex Wetzel-cover setting.

Numerous transition-metal oxides have low-temperature antiferromagnetic (AFM) states and high-temperature paramagnetic (PM) phases, where the AFM state is usually insulating while the PM phase can be either insulating or metallic. Without involving strong correlation, we use symmetry-broken density-functional theory (DFT) to obtain the PM phases of insulating NaFeO3 vs the recently discovered metallic NaOsO3. We develop the understanding of insulating and metallic behaviors in paramagnetic oxides by analyzing the interactions between magnetic and positional symmetry breaking: The insulating gap is governed by the competition between the spin disorder that induces a distribution of different magnitudes of local magnetic moments and the polymorphous distribution of off-center atomic displacements. NaFeO3, on the other hand, has large positional displacement with small spin-disorder-induced moments distribution, leading to insulating PM phase, whereas NaOsO3 has a pronounced spin-disorder-induced moments distribution that forces the PM phase to become metallic. Our work identifies this symmetry-breaking competition as a general framework to bridge seemingly disparate metal-insulator behaviors in transition-metal oxides paramagnets without invoking strong correlation.

Magnetic Navigation (MagNav) has emerged as a vital alternative Positioning, Navigation, and Timing (PNT) solution, leveraging Earth's magnetic field for robust navigation in GPS/GNSS-degraded or denied environments. Despite its potential, the successful deployment of MagNav is currently hindered by the lack of standardized, high-fidelity geomagnetic reference maps. Existing datasets, primarily designed for geological exploration or academic research, do not meet the distinct operational requirements of navigation systems regarding spatial resolution, error quantification, and global accessibility. This paper initiates a community-focused dialogue on future geophysical data requirements for MagNav, grounded in extensive real-world flight trials. We distinguish between two primary use cases with divergent data needs: Operational MagNav, which requires globally consistent, queryable, and uncertainty-aware datasets for field deployment, and MagNav R&D, which demands comprehensive access to raw survey data to foster innovation. We provide a prioritized set of recommendations for future data requirements, including the development of cohesive, merged datasets, the inclusion of localized 3D uncertainty estimates, and the expansion of the World Magnetic Model (WMM) core field model to spherical harmonic degree 13 to improve consistency. Finally, we emphasize the strategic necessity of designated test ranges to validate these requirements and ensure the operational robustness of MagNav infrastructure.

The thermal Sunyaev--Zel'dovich (tSZ) power spectrum is a sensitive probe of cosmology and cluster astrophysics, but its statistics are non-Gaussian because the signal receives a significant contribution from rare, massive, low-redshift galaxy clusters. As a result, a Gaussian likelihood fails to describe the statistics of its power spectrum on large scales. We use simulation-based inference (SBI) to test the accuracy of the standard Gaussian power-spectrum likelihood for a \textit{Planck}-like tSZ analysis. Using halo-based simulations of full-sky Compton-$y$ maps, we train neural posterior and likelihood estimators and compare the resulting constraints with those from a Gaussian likelihood assumption. Using only multipoles $\ell < 1000$, we find that the Gaussian likelihood assumption gives unbiased cosmological constraints, while the SBI-based inference shows a mild broadening of the posterior distributions for the amplitudes of residual foregrounds. This suggests that the Gaussian likelihood assumption is sufficiently accurate for cosmological inference for a \textit{Planck}-like tSZ analysis, while SBI provides a useful validation tool to model non-Gaussian likelihoods beyond analytic approximations.

Prolonged unemployment carries serious economic, health, and wellbeing costs. With federal support, most U.S. states now operate a Reemployment Services and Eligibility Assessment (RESEA) program to help Unemployment Insurance (UI) claimants return to work faster. We report results from a large (N = 23,549) preregistered randomized controlled trial (RCT) evaluating Rhode Island's RESEA program from February 2022 to September 2023. We estimate that selection into the program increased annualized wages by \$1,153, increased reemployment by 1.5 percentage points, and reduced UI duration by nearly two weeks. The vast majority of these wage and reemployment effects appeared within two quarters of claimants' first pay dates and persisted through at least the following year, and we estimate that each dollar spent on the program saved the state \$2.64. Using causal forests, a machine learning technique for estimating heterogeneous treatment effects (HTE), we also conduct an exploratory analysis to investigate if there are differential effects of selection into the RESEA program. We find that all participants experienced positive wage benefits from RESEA selection, with particularly large effects for older and lower-income workers. Finally, we improve upon prior RESEA evaluations by explicitly controlling for the week of treatment assignment -- a methodological refinement absent from several existing RCTs of job-training programs that is important to eliminate confounding bias. We also discuss ways to harvest precision gains from baseline covariate adjustment without introducing large-sample bias.

Machine learning (ML) and Artificial Intelligence (AI) components are increasingly embedded in software products, yet undergraduate software engineering (SE) curricula rarely provide systematic preparation for building, testing, deploying, and maintaining AI/ML-based software systems. This paper aims to provide evidence-based guidance for integrating AI/MLrelevant content into core SE education. We compile and define a structured inventory of topics relevant to SE practice in AI/MLbased software, then map these topics against required courses in a set of representative SE curricula to identify coverage gaps. To assess educational priorities and feasibility, we survey SE instructors on topic importance and integration constraints. Based on the crosswalk between topic definitions, curriculum coverage, and instructor prioritization, we derive a guideline that recommends where and how high-priority topics can be embedded within existing SE courses.

Over time, GitHub has introduced different strategies for sharing reusable code artifacts. In addition to fork-based reuse, template repositories provide a distinct feature for generating new projects from scaffolding. Although this feature has been available since 2019, little is known about the domains it supports, its maintenance characteristics, or the practices that guide practitioners for effective template design. To address this gap, we conduct a large-scale empirical study of GitHub template repositories across the five most used programming languages. First, we mine and categorize templates to analyze the domains they serve, exploring the LLM-as-a-judge strategy. Next, we explore the reliability of templates by evaluating the associations between repository characteristics and activity, and quality-related issues (e.g., code smells, vulnerabilities, and security hotspots) through statistical analysis. Finally, we qualitatively analyze a representative subset of templates to derive practical guidelines and recurring pitfalls for template design and management. Our results show that Web Development is the predominant domain across ecosystems, while maintenance and quality issues vary by programming language. We further find that high-quality templates tend to adopt established software engineering practices, while providing comprehensive documentation and clear guidance for use. Overall, our findings offer empirical insights and actionable guidance to support practitioners in designing and adopting high-quality template repositories.

Geostatistics aims to infer a spatially continuous phenomenon from observations collected at a finite number of locations, frequently measured with error. Whenever there is stochastic dependence between the spatial and sampling processes, preferential sampling occurs. Ignoring this problem drives to incorrect and biased estimates and, therefore, recognizing it is quite important, but not always simple to execute and understand. In this work, a test for assessing preferential sampling, simple and easy to implement, is presented, overcoming the previous concerns. It is based on the dependence between the number of sampled points and the values of the corresponding measures. The performance of the proposed test id assessed through a large simulation study, which consideres different levels of preferentiability, relation with a covariate, different sample sizes and different test procedure conditions. The results are quite encouraging, with high levels of correct preferential sampling detections, further confirmed by the test application to already known real data sets of lead concentrations in moss samples and red and blue shrimp capture data.

Picture naming requires the transformation of visual object information into a spoken lexical response through perceptual, semantic, lexical, and articulatory processes. This study asked whether semantic-category information is recoverable from high-density EEG during overt picture naming. Sixteen native French-speaking participants performed a picture-naming task using line drawings. Picture labels were embedded with a multilingual text-embedding model and organized into nine interpretable semantic categories, providing a data-driven semantic target space for neural decoding. EEG activity was represented channel-wise using a pre-trained single-channel EEG encoder over an early post-stimulus window, a later naming-related window, and their combination. Nine-class decoding showed above-chance semantic-category discrimination in all temporal representations. Balanced accuracy increased from 0.562 in the early window to 0.610 in the naming-related window, and reached 0.781 when both windows were combined, with a maximum Macro-F1 of 0.784. Class-level F1 scores showed consistent gains across semantic categories, and sensor-level decoding maps indicated spatially distributed category information. These findings suggest that semantic-category structure is reflected in EEG activity during overt picture naming and that early and naming-related temporal windows provide complementary information. The results support the use of modern neural decoding methods as tools for investigating lexical-semantic processing in spoken language production.

Tensor networks provide a powerful framework for efficiently representing high-dimensional data and many-body quantum states. Endowing tensor networks with a Riemannian manifold structure provides a natural setting for numerical optimization and analysis. A central feature of tensor networks is their gauge freedom, whose characterisation (captured by so-called fundamental theorems) underlies both their intrinsic structure and the design of numerical algorithms. In this work, we study the interaction between the Riemannian manifold structure and the gauge freedom for several families of tensor networks. Using group actions and Riemannian submersions, we establish a Riemannian fundamental theorem for the tensor network families studied.

Interdiction problems arise in a number of application areas, including global security, supply chains, and critical infrastructure protection - the goal is inhibit the movement of goods, people or information. An area of particular interest is counter-narcotics, where nodes or edges in a network are placed under surveillance or blocked to minimize the flow of illicit drugs from source to the destination. A fundamental challenge in this narco-traffic interdiction is data scarcity: available datasets are limited by the very nature of the problem and provide only partial and uncertain views of trafficking networks. Thus, developing robust interdiction methods that take this inherent lack of information is critical. In this paper we initiate the study of network flow interdiction problems under network uncertainty. First, using a limited real-world dataset, we generate an ensemble of plausible network realizations representing alternative trafficking scenarios. The method combines simulations with mathematical programming techniques to generate network ensembles that are consistent with the observed data. Second, we formulate the robust network flow interdiction problem and develop an integer linear program to solve the problem. We evaluate the optimal interdiction strategy and obtain the residual flows over the scenarios. Our analysis reveals that even modest budgets can yield significant flow reductions. However, optimal solutions vary substantially across scenarios, motivating the need for robust solutions. We show that the robust strategy achieves near-optimal performance across all near-real world realizations while remaining stable under structural uncertainty. This simulation-driven approach provides a principled basis for policy analysis and supports maximizing the return on interdiction investments in uncertain, data-limited environments.

We study the data-to-solution value map for quasistatic nonlinear elasticity in the linearized-kinematics regime, allowing both generalized loads and variable Dirichlet data. Under standard direct-method hypotheses, the negative minimum potential energy is finite and locally Lipschitz, convex in the load variable, and concave in the Dirichlet datum. Its supporting slopes, and its first variations at differentiability points, are the equilibrium displacement and the Dirichlet reaction. This convex--concave structure leads to a mechanics-preserving saddle minmax architecture in which displacement atoms generate load slopes, reaction atoms generate Dirichlet slopes, and the coupling coefficients are the corresponding trace-reaction pairings. Manufactured samples are produced by prescribing displacement--reaction pairs and computing the associated ambient data and exact value labels. The resulting architecture directly approximates the negative minimum-potential-energy value map and provides mechanical subgradient readouts. Immersed representations and cell-center quadrature make the construction implementable on background grids and geometry-rich domains. We prove uniform convergence on compact data sets with respect to atom enrichment and quadrature refinement, and illustrate the method on elementary examples.

Concurrent programming is a core component of Computer Science curricula, yet remains notoriously difficult for students to master due to its inherent complexity and the nondeterministic nature of concurrency bugs such as deadlocks and race conditions. In this work, we present ParaView, an educational tool designed to help students understand, debug, and correct concurrency issues in parallel programs written in C/C++. ParaView provides transparent execution recording and visualization to make parallel execution observable and comprehensible. We evaluated ParaView through a series of debugging and implementation tasks, with 17 students participating. Results showed a significant improvement in debugging and implementation successes compared to previous course iterations. A student survey confirmed that most participants found ParaView helpful. To further support learning outside the classroom, we explored using Large Language Models (LLMs) to analyze concurrency bugs and suggest fixes. While LLMs were highly effective in identifying bugs and explaining execution traces, the correctness of their bug fixes varied, especially for more complex synchronization patterns. Our findings suggest that recording-visualization tools like ParaView, complemented by artificial intelligence (AI), can improve teaching and learning of concurrent programming.

The paper is dedicated to calculate the number of semi-fast convergent representations of a set of subsums. The formula for specific irreductible sequences is obtained. We study the Central Cantor sets and consider complete sequences of natural numbers.

Agent-based modelling is gaining recognition as a powerful approach for simulating complex cellular pathways, owing to its ability to reproduce emergent biological behaviours without requiring extensive kinetic parameterisation. In this article, we present a GPU-accelerated agent-based simulator specifically designed to model and analyse signalling pathways involved in cancer progression, and to evaluate therapeutic interventions. Our approach leverages the computing capabilities of FLAME GPU 2, a GPU-accelerated agent-based modelling framework, to efficiently manage simulations involving millions of molecules interacting within a three-dimensional environment. Each molecule is represented as an autonomous agent with defined physical properties, capable of binding, releasing reaction products, migrating between compartments, and interacting based on spatial proximity. An intuitive graphical interface supports model construction, parameter setup, and real-time modification of treatment strategies. As the primary focus of this paper, we validate the simulator on the MAPK/ERK cascade affected by the BRAFV600E mutation, demonstrating that it accurately reproduces dose-response trends observed in clinical data and outperforms both deterministic models and our prior agent-based implementations. A second case study extends the approach to nuclear signalling by reproducing the dynamics of cFos expression and phosphorylation. This demonstrates the simulator's ability to capture compartmentalised regulation, reproducing transient mRNA responses and protein accumulation, including the effect of an unresolved negative transcriptional regulator. Together, these results show that GPU-accelerated ABM can faithfully replicate both drug response and emergent gene expression dynamics, providing a scalable and biologically grounded computational tool for supporting precision oncology.

Problem C7 from the 2006 IMO Shortlist gives $A-B=V-1$ for a generic convex polyhedron $P\subset \mathbb R^3$, where $A$ counts antipodal vertex pairs and $B$ counts antipodal edge-midpoint pairs. We study arbitrary convex polyhedra through the defect $δ(P)=V(P)-1-A(P)+B(P)$. To $P$ we associate an antipodal square complex $X(P)$ and prove $H_0(X(P);\mathbb Z)\cong\mathbb Z$, $H_1(X(P);\mathbb Z)\cong\mathbb Z/2$, and $H_2(X(P);\mathbb Z)\cong\mathbb Z^{δ(P)}$. In particular $δ(P)=β_2(X(P);\mathbb Q)\ge 0$, equivalently $A(P)-B(P)\le V(P)-1$. We also give an exact local formula for $δ(P)$ on the projective normal fan: it is the sum over exact opposite face pairs $\{F,G\}$ of $e(F)e(G)-(v(F)-1)(v(G)-1)$, equivalently in dimension three it is supported only on edge-facet and facet-facet exact pairs. This yields a facet-opposite formula, a zero-defect criterion, extremal bounds, and a spherical normal-graph profile. We further determine the integral lattice generated by square boundaries, obtaining the even-cycle lattice in the antipodal graph and Smith factors $1,\ldots,1,2$. Finally, we study the ordered representation space $\mathcal R(P)=\{(x,y)\in P\times P:x-y\in\partial(P-P)\}$ in all dimensions and show that it equivariantly deformation retracts onto $\partial(P-P)$, with unordered quotient homotopy equivalent to $\mathbb{RP}^{d-1}$.

We study ideals in a polynomial ring with partially linear (virtual) resolutions. We establish an effective bound beyond which their powers coincide with a power of the maximal ideal, proving a slightly weaker version of the Eisenbud-Huneke-Ulrich conjecture for a more general class of ideals. We also obtain an upper bound on their Castelnuovo-Mumford regularity, extending Macaulay's bound and a theorem of Eisenbud-Huneke-Ulrich. We introduce higher syzygy bundles, which generalize the classical Green-Lazarsfeld syzygy bundle. The key ingredient is the relationship between the syzygies of partially linear ideals and the sheaf cohomology of higher syzygy bundles.

Mid-infrared (mid-IR) spectroscopy enables label-free molecular detection and is widely employed in biomedical, environmental, and chemical sensing; however, its broader deployment remains limited by bulky instrumentation and insufficient analytical sensitivity. Photonic metasurfaces supporting strong mid-IR resonances provide a promising route toward compact on-chip spectrometers and enhanced molecular sensing, yet their practical implementation is often constrained by limited spectral coverage and fabrication complexity. Here, we present a scalable, resist-free stencil lithography approach for fabricating arrays of Ge pillar metasurfaces on CaF2 substrates that support polarization-insensitive Mie resonances dominated by electric dipole modes. By varying the geometric design parameters during fabrication, we engineered metasurface arrays with discrete resonances spanning the 950-1700 cm^-1 molecular fingerprint region. Furthermore, leveraging the thermo-optic response of Ge, we achieved dynamic tuning of these discrete resonances, demonstrating continuous and reversible resonance shifts of approximately 36 cm^-1 per metasurface over 300-500 K, corresponding to a tuning rate of 0.18 cm^-1/K. The thermally induced spectral sweeping enables multiplexed detection of poly(methyl methacrylate) vibrational modes and continuous reconstruction of its absorbance spectrum across 1100-1215 cm^-1. These results establish a scalable dielectric metasurface platform with spectrally reconfigurable mid - IR modes for molecular sensing across the fingerprint region and compact infrared sensor technologies.

Advances in experimental techniques have opened new routes for harnessing non-equilibrium dynamics in mesoscopic quantum systems. In this context, we study the impact of on-site interactions on the transport properties of a continuous quantum thermal machine composed of two coupled oscillators connected to two thermal reservoirs. In the weak system-reservoir coupling regime, where a long-standing debate concerns which reduced description should be preferred, we first show that the Redfield master equation (RME) provides an accurate and unifying framework that interpolates between two well-known limits: the \textit{local} and \textit{global} master equations. By relying on the Hierarchy of Pure States (HOPS), a numerically exact stochastic method, we then explore the full parameter space and show that interactions can be leveraged to tune the efficiency of the thermal machine at high temperatures (while leaving it essentially unchanged at low temperatures), induce non-reciprocal transport under asymmetric reservoir couplings, and generate steady-state entanglement within the junction. We derive expressions for system-bath correlators, such as heat and particle currents, consistently across different frameworks. Our work features on-site interactions to enhance the versatility of quantum thermodynamic junctions and clarifies the role of non-Markovianity and non-linearities in quantum transport.

Raman and infrared anomalies associated with H-bond symmetrization in $δ$-AlOOH, including mode softening and linewidth broadening at 5-10 GPa, occur at significantly lower pressures than predicted by static harmonic theory. To resolve this discrepancy, we combine harmonic phonon calculations with strongly constrained and appropriately normed (SCAN)-based deep-potential molecular dynamics and phonon quasiparticle analysis at 300 K. This framework extracts temperature- and pressure-dependent frequencies and lifetimes from long-time trajectories, capturing the branch reorganization and rapid linewidth growth characteristic of the disordering regime. Incorporating quasiparticle renormalization and directional longitudinal-optical-transverse-optical (LO-TO) splitting further yields near-quantitative agreement with the ambient-pressure OH-stretching Raman multiplet. These results identify finite-temperature dynamical effects and the progressive loss of spectral coherence as the origin of the spectroscopic signatures of H-bond symmetrization.