Papers

We construct magnetohydrostatic equilibria in rotationally symmetric toroidal domains whose magnetic lines define an $(m,n)$ Seifert fibration for arbitrary coprime integers $m,n$.

Many modern learning approaches are still struggling with spatial reasoning tasks, i.e. they lack the ability to utilize geometric information of perceived entities and their spatial relation to each other to solve problems. We introduce a novel Adaptive Neural Cellular Automata (aNCA) architecture which uses deformable convolutions to dynamically adapt the perceptive field and iteratively reason over 2D spatial relations on grid-like data structures (e.g. images). Empirical results on public benchmarks show state of the art comprehensible results with high generalization abilities for solving image based puzzles like Sudoku or finding the shortest path in a maze.

We study the random geometric graph formed by connecting $n$ independent uniform points on $\mathbb S^{d-1}$ whenever their inner product exceeds a threshold chosen to give edge density $p$. We establish the conjectured dimensionality threshold for indistinguishability from the Erdös-Rényi graph $G(n,p)$ throughout the sparse regime with polynomially growing average degree. Specifically, for every fixed $\delta>0$, if $p\to0$, $p\ge n^{-1+\delta}$, and $d\gg (np\log(1/p))^3$, then the Kullback-Leibler (KL) divergence between the two distributions vanishes. Our proof builds on the posterior analysis framework introduced in previous work of the authors, combining a truncated second moment analysis with Fourier and spherical harmonic expansions. As in prior works, we reduce the KL bound to controlling a truncated $\chi^2$-divergence $\Delta$ for the neighborhood of the last vertex conditional on the graph induced by the preceding vertices. We then bound $\Delta$ using a Fourier expansion of the conditional likelihood ratio for this neighborhood, where the high-frequency contribution is made negligible by truncation to a typical-degree event, while a spherical harmonic expansion approximates the low-frequency contribution in terms of posterior moments of bounded-degree harmonic polynomials. The key observation is that these moments and $\Delta$ satisfy a coupled system of cavity recurrences, yielding a self-bounding inequality that implies the desired KL bound.

We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every $n\geq3$. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, its exact minimum is the affine interpolation from one half to one. This proves the bound conjectured for six or more parties. The result holds for arbitrary binary measurements and density operators on tensor products of finite- or infinite-dimensional local spaces. We also determine the exact minimum for qubits, which is strictly higher at every interior score. One qutrit and qubits elsewhere attain the unrestricted minimum with measurements fixed as the score varies and, at interior scores, realize the smallest product of ranks of the local reduced states among states attaining that minimum. For $n\geq4$, the attaining state with this rank product is unique at each interior score, up to local isometries and party permutations. On one qutrit and qubits elsewhere, for $n\geq4$ and scores a fixed fraction away from both endpoints, a sufficiently small excess of extractability above the minimum controls trace-norm distance to the attaining state of the same score, up to local unitaries. The dependence is of square-root order, with optimal exponent and explicit constants depending only on that fraction, uniformly in $n$. Equality and stability concern actual extractability; identical Bell statistics can accompany different extractabilities.

We study the dimensional reduction of non-Abelian Yang-Mills theory from five to four dimensions. In this framework, Hadamard descent is formulated as a restriction to gauge-equivalence classes that are invariant, up to gauge transformations, under translations along the reduced direction. The central result is that, prior to gauge fixing, the adjoint scalar of the reduced theory is not the connection component $A_u$ itself, but the gauge-covariant combination $\widetilde{\Phi}=\epsilon-\iota_YA$, which follows naturally from the covariant Cartan identity. The usual identification with $A_u$ is recovered, up to the sign convention adopted here, in the $u$-independent gauge, while global holonomy data remain additional information to be specified separately. We show that the reduced theory takes the form of a Yang–Mills–Higgs theory with the Higgs field in the adjoint representation. As an example, we focus on the $U(2)$ gauge theory and analyze the Higgs mechanism, together with the associated symmetry breaking pattern, when the reduced theory is expanded around the ground state of the parent theory. We then include fermions and show that, in the $u$-independent gauge, the extra-dimensional component of the gauge connection generates a pseudoscalar Yukawa coupling which, on the vacuum configuration, can be converted into a fermion mass term by a chiral field redefinition, while the gauge interaction remains vectorial.

Execution feedback lets coding agents revise programs and learn from their own corrections. A correction's learning weight should reflect both the transitions supported by its executions and the amount of evidence behind that support. We introduce Effective-Evidence Self-Distillation (EESD), which represents these quantities separately. Normalized execution relevance determines relative transition support and an effective pseudo-count mass; a Dirichlet posterior then produces an uncertainty-penalized weight for KL-anchored correction learning. Under a symmetric prior, changing mass preserves category ordering, and effective mass yields a supervised coefficient bounded by its matched fixed-mass counterpart. Across four model-domain history sweeps, increasing visible observations from one to eight reduces future-outcome NLL by 55.0-59.3%. At eight observations, effective mass achieves lower NLL than fixed mass in all four comparisons. In the primary matched DeepSeek/RunBugRun study, argmax predictions agree on all 3,000 examples, with the largest NLL gain under concentrated relevance. After one correction-learning round, DeepSeek/CodeARC all-tests Pass@1 increases from 15.0% to 20.4%, with a paired 95% source-bootstrap interval of [+2.8, +8.0] percentage points. The twelve-setting downstream evaluation establishes the model-domain scope of this update. These results show how separating evidence support from evidence mass changes probability estimation and correction learning in coding agents.

LLMs are increasingly deployed as autonomous agents in social environments, making it critical to study their ability to faithfully simulate human interactions. Central to this is grounding agents in realistic user personas, yet existing datasets rely on fictional personas and are limited to a handful of languages, lacking the empirical grounding necessary to evaluate behavioral fidelity across diverse populations. We introduce Wiki-Talkie, a multilingual dataset of real-world conversations from Wikipedia Talk pages across five languages spanning two language families: Germanic (German, English) and Romance (Spanish, French, Italian), paired with personas derived from real user communities and encompassing sociodemographic attributes, self-descriptions, and behaviorally grounded interaction traits. Using Wiki-Talkie, we evaluate agent interactional behavior on a next-turn generation task across various persona conditioning strategies. Our evaluation assesses whether agents collectively reproduce the distributional behavioral patterns observed in human discussions. Results show that user's comment history exemplifying interaction behavior consistently outperforms explicit persona information. In addition, models systematically underproduce negative or extreme sentiments, while over producing references and suggestions, revealing biases toward agreeableness and positivity. Crucially, these patterns hold robustly across languages, with small cross-lingual differences.

We extend the dynamical vertex approximation (D$\Gamma$A), in its most commonly used ladder-version, to systems with retarded electronic interactions. Such an extended scheme, which we name "dynamical-$U$ D$\Gamma$A", allows us to study strongly interacting electrons coupled to bosons including non-local spatial correlations on all length scales on top of the purely local ones captured by dynamical mean-field theory. We demonstrate the applicability of the method in the specific case of the coupling to a single dispersionless phononic mode, by studying the Hubbard-Holstein model on the square and cubic lattices, focusing on regimes of sizable Hubbard interactions where strong electronic correlations compete with phonon-mediated interactions. As illustrative examples, we consider the antiferromagnetic phase in the three-dimensional model and the d-wave superconductor in two dimensions, analyzing the effect of the electron-phonon coupling, including its frequency dependence, on both instabilities.

In this paper, we first introduce fiberwise supramenability for locally compact Hausdorff étale groupoids with compact unit space. Then, for such a minimal ample groupoid $\mathcal{G}$, we show that if $\mathcal{G}$ is fiberwise supramenable or topologically amenable, then the clopen type semigroup $S(\mathcal{G})$ is almost unperforated. Consequently, if $\mathcal{G}$ is $\sigma$-compact, then it has groupoid strict comparison. We then present several applications, including Matui's AH conjecture and pure infiniteness for groupoids and groupoid $C^*$-algebras.

Paired translation between quasiperiodic physiological waveforms (i.e., recovering a target oscillatory signal from the source) is central to the interpretation of cardiovascular signals derived from wearables placed at different body locations. This source-to-target mapping in these problems carries inherent geometric structure: the phase wraps around the cycle and must be treated as a circular variable, the amplitude remains strictly positive, and the beat-to-beat alignment can drift unpredictably across cycles and subjects. While deep neural networks have been used for phase estimation and complex-valued signal modeling, prior work does not explicitly learn phase transport between paired signals. Consequently, neither endpoint-supervised regression nor the standard affine path used in flow matching accounts for this phase–amplitude structure. We introduce cylindrical geodesic flow matching for paired cardiovascular waveform translation. We show that the standard affine path used in flow matching distorts intermediate amplitude and instantaneous frequency when interpolating between quasiperiodic signals; replacing it with a closed-form geodesic on the phase–amplitude cylinder eliminates these artifacts and converts each training pair into dense, geometry-consistent velocity supervision. On zero-shot photoplethysmography and limited-support seismocardiography adaptation benchmarks, our method consistently outperforms interpolation baselines and matches or exceeds direct supervised prediction, reducing Hilbert Transform, $L_2$, and Dynamic Time Warping distance by up to ${\sim}15\%$ over the strongest competing baseline. These results suggest that bridge geometry is a critical inductive bias for flow matching on oscillatory signal translation.

Gaussian steering sets a sharp threshold for creating Wigner negativity at a distance. A measurement by Bob can make Alice's part of a shared Gaussian state Wigner negative only if Alice can steer him with Gaussian measurements. We ask whether multiple copies, ancillas, or catalysts can lift this requirement when Alice is restricted to Gaussian operations. We introduce the steered negativity, a multiplicative monotone whose logarithm equals the Gaussian steerability for Gaussian states. It bounds the negativity that Bob can herald and rules out activation by copies, by ancillas of Bob, and by catalysts without remote negativity of their own. For a strictly unsteerable Gaussian state and a single round of feed-forward to a Gaussian unitary of Alice, we further prove that even a Wigner-negative or steerable catalyst cannot help if it is returned exactly. Gaussian steering thus remains a barrier to remote Wigner negativity beyond the single-copy setting.

We investigate decentralized gradient descent (DGD) with emphasis on efficient communication and computation resource utilization under budget constraints. As a first step toward the broader communication-computation allocation problem, we consider and analyze a multi-consensus decentralized gradient descent (mcDGD) scheme, where the number of consensus rounds and the stepsize are allowed to vary across iterations. Building on a unified analytical framework for DGD, we derive finite-time convergence bounds that explicitly characterize the interaction between consensus quality and optimization dynamics. Our analysis requires only convexity of the local objective functions while assuming smoothness and strong convexity of the global objective. The resulting bounds enable a principled consensus-allocation strategy under resource constraints, for which we show that equal allocation of consensus rounds across iterations is optimal under our stepsize rule, up to integer rounding. Numerical experiments corroborate the theoretical findings and demonstrate favorable communication-computation tradeoffs compared with existing multi-consensus decentralized optimization baselines.

For the critical random-cluster model on $\mathbb Z^2$ with cluster-weight $q=4$, we prove the convergence of loops to CLE(4) in the scaling limit. Conformal invariance of the four-state Potts model is then deduced via the Edwards-Sokal coupling.

Predictive simulation of human movement is a promising tool for studying “what-if” scenarios in human movement and its underlying motor control, yet its realism is often limited. To address this gap, we incorporate a learned state prior that is trained on a large-scale dataset of human gait kinematics and external forces into predictive simulations. Resulting gait simulations yield kinematics and kinetics across diverse walking and running speeds that better match experimental data than current physics-based simulations, achieving accuracy comparable to data-driven models that reproduce learned data. Furthermore, our method enables robust hypothesis testing by demonstrating how varying optimality assumptions, muscle weakness, and footwear choices influence predicted gait. We also show that this prior generalizes well beyond its training data, successfully reconstructing full-body kinematics for curved running and cutting maneuvers from sparse marker sets. Ultimately, these results suggest that state priors should be broadly integrated into predictive simulations.

Based on the analysis of moiré superlattice characteristics and interlayer hybridisation in small-angle twisted graphene multilayers with a monolayer hexagonal boron nitride (hBN) spacer, we predict spectral features and quantum transport effects in these systems. Here, we consider both a twisted bilayer and double bilayers with a spacer, discuss double bilayers in two stacking orders - with and without inversion symmetry - and also compare the results obtained with self-consistent Hartree calculations with and without in-plane periodic charge distribution, offering computationally efficient models for these graphene/hBN/graphene moiré systems. We note parametric intervals where the appearance of minibands can be enhanced by tuning with an out-of-plane displacement field, and identify the miniband edges and van Hove singularities across a broad range of densities and displacement fields. These system host Lifshitz transitions at which the Fermi lines form a network across moiré Brillouin zones, and to which Brown-Zak oscillations converge at low magnetic field and elevated temperatures. We analyse interlayer hybridisation across the same parameter space, discuss how hybridisation affects in-plane magneto-transport in ultra-high-mobility structures, and predict that semiclassical magneto-oscillations of interlayer hybridisation can lead to 1/B-periodic quantum oscillations of resistivity persistent to high temperature.

We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + $\varepsilon$)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.

We study how independently trained Transformer neural networks reconstruct a binary string from its local domain walls. Runs sharing the data and training protocol can realize different functions. We treat them as replicas and measure truth alignment $m$, prediction confidence $q_{\mathrm{self}}$, and cross-replica agreement $q_{\mathrm{cross}}$. Confident disagreement defines the finite-size replica fragmentation that we call glass-like. With small training sets, replicas predict all training examples correctly but remain confident in incorrect predictions for unseen inputs, a regime we call memorization. With larger sets, runs can generalize and then retreat. Retreat occurs when outputs start to deviate from truth while confidence remains high. The frontier between learned and unlearned outputs recedes toward shorter strings. Many later recover as the frontier advances again. The self–cross gap $q_{\mathrm{self}}-q_{\mathrm{cross}}$ clearly distinguishes the three learning regimes of memorization, retreat, and recovery. With overall and position-dependent truth alignment subtracted, their residual correlations also differ: memorizing replicas have weakly and uniformly correlated residuals, while retreat has the largest fraction of replica pairs whose residuals are anti-correlated. That is, on the inputs where one replica of such a pair does better than its average, the other tends to do worse. These finite-size observations distinguish persistent memorization from ongoing retreat–recovery dynamics. The thermodynamic and long-time limits, and extensions to other learning tasks, remain open.

Proactive power management systems reduce processor dynamic power through runtime power prediction and power-aware scheduling. Accurate, stable and low-overhead digital on-chip power meters (OPMs) are crucial for improving the prediction quality. Recent studies have explored various modeling methods, including using linear models, decision trees, and multi-layer perceptrons (MLPs) to construct OPMs. However, most current approaches train models end-to-end without analyzing the physical interpretability of features, affecting their ability to generalize to unseen workloads. Grounded in the design principles of synchronous digital VLSI circuits, X-OPM introduces a robust feature engineering framework that uses tree-based models to capture feature interactions and linear models for prediction. It also incorporates a human-in-the-loop workflow to balance model accuracy against modeling effort. Evaluated on a commercial C906 vector processor, X-OPM consistently achieves $R^2 > 0.93$ across all workloads with sampling window size set below $8$ cycles. In contrast, state-of-the-art methods including APOLLO, COBIT, and standard MLPs fail to generalize across all test cases. Layout with commercial EDA tools shows that X-OPM incurs an area overhead below $0.1\%$, which is on par with lightweight tree-based and linear models, and significantly smaller than MLP-based models.

Depression has no diagnostic blood test. Language models promise tireless, consistent assessment, but can accurate raters disagree about individuals? We pre-registered 880 language-model raters, crossing 11 open models with prompting and scoring choices, and applied them to 189 interviews against the eight-item Patient Health Questionnaire. Model choice explained 30.0% of summed-symptom score variance, stable participant differences 10.5%. Two randomly drawn raters with area under the receiver operating characteristic curve (AUC) >= 0.70 disagreed on screening decisions for 40% of participants, on average. Average over-rating governed how many were flagged, yet equal-capacity raters chose differently for about one participant in five. A locked analysis of 86 new interviews reproduced the main pre-registered findings. Exploratory recalibration with 40 labelled participants raised accuracy from about 60% to 75% and halved disagreement, leaving one participant in five decided differently. Calibration repaired much of the rater dependence without securing agreement about individuals.

We provide character expansions of certain specialisations of multiparameter Hall–Littlewood polynomials of types $\mathrm{B}_n$, $\mathrm{C}_n$, and $\mathrm{BC}_n$ for rectangular shapes. We use these expansions to establish the six Jacobi-Trudi-type identities conjectured by Ole Warnaar in 2025.

We prove a sheeting theorem for stable solutions to the Allen–Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures $\varepsilon|Du_\varepsilon|^2\,dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in $\mathbb R^N$, $N\le7$, with $O(R^{N-1})$ energy growth are one-dimensional. When $W''(-1)=W''(1)$, every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for $4\le N\le10$. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.

Clustering is a fundamental task in unsupervised learning. The goal is to partition unlabeled data into clusters such that similar elements are assigned to the same cluster, while elements in different clusters are dissimilar. We propose a quantum Monte Carlo approach to density estimation, a key routine in density peak clustering. Building on this approach, we develop quantum algorithms for the full clustering problem and three related tasks: decision clustering, cluster counting, and heavy cluster counting. For each one, we establish a polynomial quantum query speedup under assumptions on the scaling of dataset-dependent parameters. Numerical experiments suggest that these conditions are satisfied for several practically relevant settings.

Transfer functions of Partial Differential Equations (PDEs) are irrational and difficult to obtain. Unlike for rational transfer functions, only a few methods are available for the control and analysis of irrational transfer functions. Thus, for robust analysis and control of PDEs, we need to use a rational approximation of the transfer function, preferably with provable $H_\infty$ error bounds. In this paper, we extend Krylov subspace model reduction to infinite-dimensional systems modeled by Partial Integral Equations (PIEs) – an equivalent representation of PDEs. First, we propose a state transformation of the PIE system that allows us to compute a controllability matrix. Then, to obtain a rational approximation, we project the PIE states onto the vector space spanned by the truncated controllability matrix. Next, we show that sufficient conditions for exponential stability and well-posedness of the PIE system guarantee the exponential stability of the reduced-order system and provide constructive finite-frequency $H_\infty$ error bounds. Finally, we provide four illustrative numerical examples to demonstrate the performance of the proposed method.

Let $P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. For each $p\in\{4,5,6,7,8\}$, we prove that the coefficients of $P_n(q)^p$ in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coefficients are in fact strictly positive for $5\le p\le8$. This proves Conjecture 2.1 of Berkovich and Dhar in full. We further determine the limiting transition constants and a four-term asymptotic expansion for the transition centres. For $n\ge301$, the signs in residue class $2$ are determined outside an interval of length $2\varepsilon_p/n^2$ centred at $\alpha_p n^2+\beta_p n+\gamma_p+\delta_p/n$, where $\varepsilon_4=1500$ and $\varepsilon_p=150$ for $5\le p\le8$, and all constants admit explicit analytic definitions. The proof combines a corrected saddle relation, higher-order expansions resolving cancellation between the two dominant saddle contributions, and a rescaled positive-integral argument that uniformly controls the small-degree range.

Machine learning can accelerate molecular discovery by designing molecules and planning experiments. However, many scientific challenges demand molecules with very rare properties, and in this sparse setting, existing algorithms offer little gain over random guessing. We propose a method to efficiently search large regions of molecular space using algorithmically controlled stochastic synthesis. Rather than design, make and test individual molecules, we design and make complex mixtures, test them as a pool, then deconvolute the molecule-activity map. We optimize synthesis to encode maximal information. Theoretically, this approach can reduce the number of experiments required to find the optimal molecule among $d$ candidates from $\mathcal{O}(d)$ to $\mathcal{O}(\log d)$ or $\mathcal{O}(1)$. In simulation, on estimated protein fitness landscapes, it finds active molecules with an order of magnitude fewer experiments than existing Bayesian optimization methods.

Engineering magnetism in two-dimensional semiconductors through interfacial interactions offers an attractive alternative to substitutional doping and provides opportunities for integrating magnetic and optoelectronic functionality within the same heterostructure. Here, we investigate ultrathin VOx films deposited on monolayer MoSe2 and demonstrate a pronounced enhancement of the magnetic response upon formation of the VOx/MoSe2 interface. Combined structural, spectroscopic, magnetic, and first-principles analyses support a consistent picture in which finite VOx clusters possess intrinsic magnetic moments that are further enhanced by electron transfer from MoSe2 to the oxide. The predicted charge transfer is also consistent with the increased trion contribution observed in the photoluminescence response of the heterostructure. Beyond this interfacial enhancement, the magnetic behavior is strongly governed by the morphology of the ultrathin oxide. A crossover from discontinuous clusters to more continuous films produces qualitatively different temperature dependences: the continuous-film regime exhibits conventional thermal demagnetization and coercive softening, whereas the clustered regime displays an unusual increase in magnetization and nonmonotonic coercivity with increasing temperature. These contrasting responses are captured by a phenomenological model in which morphology determines the collective magnetic dynamics through a distribution of effective reversal barriers and their thermally activated accessibility. Together, the results establish a unified picture in which interfacial charge transfer enhances the local magnetic moments of VOx, while nanoscale morphology governs their collective magnetic dynamics, providing complementary routes for controlling magnetism in oxide/transition-metal dichalcogenide heterostructures.

In this paper, we propose an explainable artificial intelligence framework to explain the solution of optimal control problems predicted with graph neural networks (GNNs). The proposed approach first trains a GNN that predicts the solution of the optimal control problem for a given instance represented as a graph. Given the trained GNN, we use explainable AI algorithms to identify variables, constraints, variable-constraint connections, and parameters, i.e., nodes, edges, and features in the graph representation of the optimal control problem, that affect the prediction of the optimal solution the most. We apply the proposed approach to a case study regarding the optimal control of a continuous stirred tank reactor. First, the results show that the GNN model can accurately predict the optimal values of the manipulated variables. Application of explainable AI algorithms reveals equality and inequality constraints that are the most important for predicting the optimal solution. The inequality constraints correspond to ramping constraints, some of which are active at the optimal solution.

A constant line-of-sight acceleration (LOSA) of the centre-of-mass of a quasi-circular compact binary coalescence (CBC) leads to modulations in the gravitational-wave (GW) waveform at $-4$ post-Newtonian (PN) order. In our previous work, we calculated the phase and amplitude corrections only for the $(\ell,m) = (2,2)$ mode. The gravitational wave event GW190814 has recently attracted considerable attention because it has been suggested to provide evidence for a nonzero LOSA. However, several LOSA analyses of GW190814 employ higher-order-mode waveform models, such as IMRPhenomXPHM, while incorporating LOSA corrections only for the dominant (2,2) mode. This approximation assumes that their omission has a negligible impact on parameter estimation. In this paper, we present a self-consistent implementation of LOSA in a higher-mode waveform model by deriving the corresponding phase and amplitude corrections. We show that these corrections are essential for asymmetric binaries such as GW190814 and that neglecting them biases parameter inference. We also discuss inconsistencies in the derivation and implementation of LOSA corrections in waveform models in the literature that include higher-order modes.

The composition of planets and their atmospheres is largely determined by the formation environment of the planet. This environment is pre-determined during early stages when the protostar is still forming. Charting the structure of protostellar envelopes is therefore crucial for understanding how the chemistry is linked to the physical processes occurring in stellar nurseries. We use JWST NIRSpec IFU observations to extract spectra at every pixel in the IFU and map the distribution of $^{13}$CO$_2$,$^{12}$CO$_2$, CO, OCN$^-$ and H$_2$O ice. The 13CO2 ice feature, a known thermal tracer, is also used to identify high-temperature regions where the ices are thermally processed. The findings show that the distribution of ices in both L1527 and IRAS 20126+4104 is not uniform, with both envelopes showing localized temperature zones and substructures. In L1527, CO$_2$ ice segregation and OCN$^-$ enhancement with respect to CO$_2$ are observed in the warm central disk region close to the protostar. In IRAS 20126+4104, the warm temperature zones are observed mostly towards the shocked region in the NW outflow where dense material is interacting with the precessing jet as well as close to the central source. The CO$_2$ ice is segregated at these locations and we observe sublimation of CO ice. In addition, crystalline H$_2$O ice is detected at a few locations in the envelope. In both low mass and high mass protostar the findings indicate that the distribution of ices and their structure are linked to the temperature structure of the envelopes. The envelope of IRAS 20126 shows a more complex structure, where interacting material results in localized regions of high temperatures and denser regions that harbor cooler ices. L1527 in contrast appears to be a more structured system with heated warm ices observed mostly towards the central disk regions.

The solvable degree of a Schur algebra $D$ is the greatest common divisor of all integers $n$ such that $D^{n\times n}$ is a direct summand of the rational group algebra $\Q G$ for some finite solvable group $G$. The main result of this paper gives divisibility conditions on the solvable degree depending on the center and the rational primes that divide the discriminant of $D$.