Papers

Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.

A seminal result of Hastings showed that entangled inputs can increase the rate of classical communication through independent uses of a quantum channel. While Hastings' result proved the existence of this phenomenon, it has remained a major open problem to find an explicit channel with this property. In this work, we resolve this question by constructing an explicit family of finite-dimensional quantum channels for which entangling inputs between pairs of channel uses yields an unbounded separation from the achievable communication rate using unentangled inputs. We describe a circuit implementation of our channel that uses only deterministic qudit Clifford unitaries and a single binary measurement with classical feedforward. The key ingredient is a careful design of the measurement and feedforward process that yields the required one-copy and two-copy output entropy bounds from moment estimates alone, without requiring strong convergence of the underlying unitary family.

Families of explicit analytic solutions of the magnetohydrodynamic equilibrium equations are presented, equivalent to steady incompressible Euler flow. The solutions are non-axisymmetric and possess exact nested toroidal flux surfaces. No expansion is made in inverse aspect ratio or in the deviation from axisymmetry. The magnetic field and flux surfaces are given explicitly in Cartesian coordinates using elementary functions. The field, current density, and scalar pressure are smooth over the toroidal domain. The pressure gradient vanishes only on the magnetic axis. One family of solutions has uniform rotational transform $\iota=2$, while another family has a sheared $\iota$ profile. These counterexamples to Grad's conjecture are valuable for understanding the existence and regularity of 3D equilibria and for testing numerical codes.

We prove that the real $K$-theory functor is lax symmetric monoidal. Moreover, it is initial among those real functors $\mathsf{Wald}_\infty^\mathrm{gd}\to\mathsf{Sp}^{C_2}$ that are additive and lax symmetric monoidal.

The second sine harmonic $\Pz$ of longitudinal $\Lambda$ polarization is sensitive to the balance between the kinematic-vorticity contribution $\Pz^\omega$ and the kinematic-shear contribution $\Pz^\Xi$ at decoupling. We quantify this competition in a scan of 48 Pb-Pb initial states evolved with the (3+1)-dimensional Particle-In-Cell Relativistic (PICR) ideal-hydrodynamic code. With the isothermal local-equilibrium (ILE) prescription, the total harmonic follows $\Pz\simeq a\,\efo-b\,v_2$, with $a\simeq0.010$ and $b\simeq0.014$, where $\efo$ is the freeze-out eccentricity and $v_2$ the elliptic flow of $\Lambda$ hyperons. With coefficients fixed on these 48 states, the relation describes 86% of the variation of $\Pz$ in 12 states outside the fit. To identify the origin of the two terms, we decompose the ILE integrand exactly into an emission-gradient term and a curl term. On the PICR decoupling surfaces, the tangential emission gradient is positive on average, correlates most strongly with $\efo$, and admits a surface-curvature representation, whereas the normal emission gradient and the curl term are negative on average and correlate most strongly with $v_2$. A blast-wave calculation with the same integrand gives coefficients of comparable magnitude and shows how the cancellation point $\efo/v_2=b/a$ changes with the flow and the surface geometry; both PICR and blast-wave calculations predict that $b/a$ rises with hadron mass from the proton to the $\Omega$ when the decoupling surface is held fixed. Although the geometry inferred from ALICE data depends on the background, the measured polarization constrains the displacement of $\efo$ from the cancellation boundary $\efo=(b/a)v_2$. For the central ALICE measurement values at $5.36\,\mathrm{TeV}$, the backgrounds examined give $\efo\simeq0.2$–$0.4$ in 30–50% Pb-Pb collisions.

We develop an exact simulation algorithm for scalar diffusion paths and diffusion bridges when the Poisson potential is unbounded in both tails. The method reconstructs the realized range of a Brownian bridge proposal by sampling its maximum and location, together with the maxima and locations of the two adjacent restricted Brownian meanders. Conditional on this finite information, the remaining path decomposes into four conditionally independent interval-constrained Brownian bridges, which can be sampled exactly at the Poisson times required by the rejection test. In contrast to constructions based on an enclosing range layer, the proposed representation retains the exact extrema and their locations. Our algorithm returns an exact finite-dimensional skeleton without time-discretization error and permits exact post-acceptance refinement at arbitrary finite collections of times. Numerical experiments validate the resulting finite-dimensional laws and identify the restricted-meander extremum simulation as the principal computational cost in the nonlinear example.

In the 1980s, Stanley and J. Watanabe proved that over a field of characteristic zero every monomial complete intersection has the strong Lefschetz property. Determining which monomial complete intersections in positive characteristic have the weak Lefschetz property has been the subject of ongoing research. In this paper, we give a complete description of the locus of exponent vectors corresponding to monomial complete intersections over a field of positive characteristic that fail the weak Lefschetz property. We describe this locus explicitly in terms of translates of positive-orthant lattice balls in the taxicab metric.

Detection of overlapping communities is essential for modelling networks in which nodes participate simultaneously in multiple structural or functional groups. Existing graph neural network approaches commonly rely on local message passing, which can obscure community boundaries through smoothing and limit the representation of structurally relevant long-range dependencies. We introduce Diffusion-Induced Spatial Attention Community Detection (DISCO), a deep-learning framework that combines a structural prior derived from influence spreading dynamics, sparse multi-head attention, and non-negative community-affiliation learning. The prior identifies candidate interactions beyond immediate graph neighbours and biases attention according to their structural proximity, while a Bernoulli-Poisson edge-reconstruction objective enables overlapping community inference from node attributes and structural profiles, or both. Benchmark experiments show that DISCO performs competitively against established graph convolutional and graph attention approaches across different input configurations. To demonstrate its practical applicability, we present a proof-of-concept cybersecurity use case in which changes between community assignments inferred from consecutive communication-network snapshots provide an interpretable anomaly signal. Temporal community similarity identifies structural deviations, while node-level contributions help locate the devices associated with them. DISCO therefore provides both a flexible method for overlapping community detection and a foundation for analysing structural change in dynamic networks.

Modular quantum computing is a leading candidate for scalable fault tolerant quantum computation. Performing lattice surgery using surface code patches in different modules requires the use of inter-module gates that are noisier than the local gates. Furthermore, the merged layouts during lattice surgery expose multiple directions in which hook errors can propagate to reduce the fault distance. Existing hook-avoiding schedules either increase the syndrome extraction circuit depth, or let link faults propagate into the bulk. We introduce a seam construction that allows splitting arbitrary merged rotated surface code layout into smaller patches with uniform boundaries that could keep their own hook-avoiding N/Z shaped schedule. At the seam we split weight-4 stabilizers into pairs of weight-2 stabilizers that are mediated through a Bell pair and measured within a CNOT depth of 4. We also employ the use of three-qubit $\texttt{CXX}$ gates for parity measurements at the seam to maintain the minimum CNOT depth. Through this seam construction, we confine the elevated link error at the seam, protecting the bulk from higher error rate. We also give a custom detector annotator for such seams that can find the detector space, annotate detectors in the Stim circuits, and ensure that they are suitable to use by matching decoders. We implement our construction for the logical Pauli-$XX$ measurement lattice surgery and a cross-shaped spatial junction and sweep the link error rate from $10$ times to $100$ times a fixed bulk error rate, and show that our construction gives the lowest and most stable logical error rate than the other frameworks compared here.

We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.

Receptor-mediated endocytosis requires a particle to approach the cell membrane to within a few nanometers before ligand–receptor binding can occur. Existing continuum models often start from an already established contact and do not explicitly describe how crowding particles on the extracellular side influence the distribution of the particle near the membrane. We examine entropic depletion forces as one possible nonspecific contribution to this initial approach. For ideal depletants, the Asakura–Oosawa excluded-volume construction gives an exact depletion potential for the planar geometry before contact. The potential and force vanish continuously at the onset of excluded-volume overlap. This interaction provides a possible contribution to membrane proximity before specific binding, while its extension to curved wrapping geometries requires additional approximation. Within a reduced continuum model, we combine depletion attraction, ligand–receptor binding, membrane deformation, and cytoskeletal viscoelastic dissipation. The viscoelastic contact is formulated through a hereditary integral and a standard linear solid. The kinetic model gives a conditional minimum ligand density for complete engulfment, a finite particle-size window, and a stiffness-dependent upper limit. When the stationary radius lies inside the domain of finite positive wrapping times, the estimated wrapping time has a minimum at a radius that decreases with increasing binding energy density. At fixed viscosity and other independent parameters, the same time approximation predicts slower wrapping as cell stiffness increases. The two positive roots defining the size window merge at a limiting parameter value, which characterizes closure of the admissible size interval. Depletion attraction is interpreted as one possible contribution to particle-membrane association, alongside electrostatic interactions

The rapidly advancing capabilities of vision-language models (VLMs) present a systemic challenge to academic integrity. VLMs now allow students to bypass meaningful engagement by capturing and submitting graphical problems as singular images, a practice we define as trivial plagiarism. To provide educators with actionable data on VLM limitations, we investigate the efficacy of heuristic adversarial image transformations designed to degrade model performance while remaining human-interpretable. Through a two-phase evaluation of introductory assessments, we manually assess baseline VLM performance on circuit diagrams, followed by an automated large-scale evaluation of topological structures (logic gates) and coordinate geometry (Karnaugh maps). We find that while highly capable VLMs can exhibit appreciable robustness, all models suffer vulnerability to adversarial perturbations. We conclude that while visual perturbations act as a viable near-term stopgap, long-term assessment security requires educators to reapproach assessment design given continually increasing VLM performance.

We prove the Mumford–Shah conjecture, formulated in 1989, by completing the classification of blow-up limits of minimizers in the plane. The remaining case concerns limits whose discontinuity set is noncompact and has connected complement. In this setting, we rule out bounded connected components using second-variation inequalities obtained from combined inner and outer variations associated with two perpendicular translations of a compact subset separated from the rest of the discontinuity set. Established results then imply that the discontinuity set is connected, and the known classification for connected discontinuity sets completes the proof.

Ground-based planetary imaging suffers from atmospheric turbulence, sensor noise, and limited sampling, making restoration a joint denoising, deblurring, and super-resolution problem. We present ASTRA-SR, a blind single-frame restoration framework trained on a physics-grounded synthetic dataset. High-dynamic-range spacecraft RAW observations serve as clean sources, and paired LR inputs are synthesized using measured layer-integrated turbulence strengths, propagated moving phase screens, exposure-averaged spatially varying PSFs, and sensor noise.ASTRA-SR first estimates a noise-suppressed but blur-retaining LR image, then restores spatial structure through multiscale processing and reconstructs HR detail with serial spatial-amplitude refinement. It yields a 0.49 dB foreground PSNR gain over the strongest baseline approaches.

Development of novel devices often involve combining functional materials and substrates, each chosen for particular physical properties such as low-loss or thermal conductivity. This, however, can introduce a large lattice mismatch and high density of structural defects that may limit device performance and reliability by trapping charge or creating leakage pathways. Such is the case for integrating superconducting TiN with a low-loss sapphire substrate. Here we resolve and image the resulting misfit, screw, and threading dislocations in epitaxial TiN on sapphire using the depth-sectioning capabilities of multislice electron ptychography. We find misfit dislocations spaced ~1.6 nm apart and track out-of-plane crossings of misfit dislocations as well as their transitions to threading dislocations, details which are obscured in conventional defect-imaging methods. The ability to correlate specific dislocation types and their interactions in extremely small volumes and at high densities is valuable for understanding structure-property relations even in modern, scaled devices.

In this paper, we proposed GAD-MambaUNet, a lightweight medical image segmentation network that combines efficient local modeling, direction–group state-space interaction, and training-time foundation-model supervision. To improve contextual modeling in compact segmentation networks, we introduced Direction-Group Graph Selective Scan (DG-GSS), which treated scan-direction and channel-group responses as graph nodes and enabled structured information exchange before multi-directional fusion. We further incorporated DINOv3-GAD supervision, where a frozen DINOv3 teacher provided semantic guidance during training, and Gradient-Adaptive Distillation dynamically regulated the distillation strength. GAD-MambaUNet achieves a favorable accuracy–efficiency balance compared with representative lightweight and general segmentation methods. Ablation studies further verify the effectiveness of DG-GSS and training-time DINOv3-GAD supervision. In future work, we will explore more flexible teacher–student alignment strategies and extend the proposed framework to more diverse medical segmentation scenarios, such as multi-class and multi-modal segmentation tasks.

In this paper, using the equivalence of two extensions, we obtain an improved upper bound for the perturbation index, improving the bounds of Quy and N. V. Trung.

With the rapid advancement in data collection devices and storage capacities, we have access to increasing amount of spatial point pattern data where each event location is augmented by multiple, potentially non-scalar marks. Therefore, there is need for efficient analysis techniques to investigate the structural relationships between the marks. In this paper, we recall the distance covariance and distance correlation and adjust them to the marked point process setting. As a result, we introduce a novel class of mark characteristics for single real-valued marks as well as multivariate combinations of marks, including mixtures of integer- and real-valued quantities, and non-scalar marks.

Incentive design aims to guide the performance of a system towards a human's intention or preference. We study this problem in a multi-agent system with one leader and multiple followers. Each follower independently solves a mdp to maximize its own expected total return with the same state space and action space. However, the leader's objective depends on the collective best-response policies of all followers. To influence these policies of followers, the leader provides side payments as incentives to individual followers at a cost, aiming to align the collective behaviors of followers with its own goal while minimizing this cost of incentive. Such a leader-followers interaction is formulated as a bilevel optimization problem: the lower level consists of followers individually optimizing their MDPs given the side payments, and the upper level involves the leader optimizing its objective function given the followers' best responses. The main challenge to solve the incentive design is that the leader's objective is generally non-concave and the lower level optimization problems can have multiple local optima. To this end, we employ a constrained optimization reformation of this bi-level optimization problem and develop an algorithm that provably converges to a stationary point of the original problem, by leveraging several smoothness properties of value functions in MDPs. We validate our algorithm in a stochastic gridworld by examining its convergence, verifying that the constraints are satisfied, and evaluating the improvement in the leader's performance.

AI tools for digital product design now offer prompt-to-design capabilities, allowing designers and their non-designer colleagues to create prototypes through conversational workflows with large language models (LLMs). While these tools promise time savings, experimental evidence in product design remains limited compared with evidence from software engineering. We conducted a randomized controlled trial with 50 product designers and 50 product managers to evaluate prospective time savings from leveraging Figma Make in design work. Participants attempted three standardized design tasks with or without access to Figma Make. Among participants who completed the study tasks, access to Figma Make was associated with approximately 20% shorter completion times, with larger gains among product managers. Our findings suggest that prompt-to-design tools may enable product managers to further contribute to design work, while the benefits for professional designers may be task dependent.

Quasinormal modes (QNMs) of black hole perturbations in the large overtone limit, namely, asymptotic QNMs, are highly sensitive to a black hole's interior geometry. We probe the singularity structure of a class of exact (2+1)-dimensional quantum black hole solutions to semi-classical gravity by computing their asymptotic QNM spectra. We do this using methods of complex analysis in which the radial coordinate is analytically continued to the complex plane. The leading order QNM frequencies all fit the generic form $\omega=(\text{offset})+n(\text{gap})$, for overtone number $n$, which we also confirm numerically. Under a general assumption about the global Stokes topology of the complexified radial coordinate, we show how to reconstruct the scaling exponent of more general (spacelike or timelike) black hole singularities from the offset. We then compute subleading corrections to the asymptotic QNMs, from which we provide a robust method for extracting the scaling of the metric function near the singularity. Our findings exemplify the transition between "Kasner eons", successive regimes encountered on approach to a spacelike singularity, as non-perturbative quantum effects become dominant. For charged quantum black holes, the asymptotic QNMs exhibit a crossover between two regimes in which the neutral and charge contributions to the metric function dominate, respectively. Within the crossover region, the QNMs develop an oscillatory behavior that may be a signature of the inner horizon.

For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge4$. This number is optimal when $r=2^k+2$, $k\ge1$.

The Non-Equilibrium Green's Function (NEGF) approach is applied to a chain of Hubbard atoms in a helical geometry sandwiched between two leads, one of which is magnetic, to assess what role the electronic correlation effects within the molecule may play in explaining the Chirality Induced Spin Selectivity (CISS) effect. The correlation effects are treated at the level of the self-consistent second Born approximation. Two chiralities, Laevorotatory (L) and Dextrorotatory (D), of the helix, are discussed. Further, we account for spin-orbit coupling (SOC) within the chains using a simplified model of Kane and Mele that exploits next-nearest neighbour interactions intrinsic to a curved geometry. The interaction with the leads is considered within the Wide Band Approximation. We consider the ratio, U/t, of the Hubbard constant U to the hopping t between atomic sites along the chain of up to two, but found very little spin polarisation in our calculations. Even though the spin polarisation factor defined as a relative difference between the largest and smallest total current is found for both enantiomers practically the same, the orientation of the tip's polarisation required to reach the largest and smallest currents for the two enantiomers differed. It is also found that the qualitative agreement with observations can only be achieved if we account, self-consistently, for the second order of the perturbation theory. The distribution of the spin populations on the chain atoms for different U/t ratios is also discussed. This study concludes that even though accounting for electron-electron interactions are important in understanding spin selectivity within molecular junctions, they cannot be made solely responsible for the high spin polarisation factors observed in experiments.

In 2002, Montiel and Urbano conjectured that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus $g\geq 1$ in $\mathbb{C}P^2$ has $\mathcal{W}^-$-energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which $\mathcal{W}^-$ strictly decreases.

Parker Solar Probe, near its 23rd perihelion in March 2025, sampled an extended interval of sub-Alfvénic solar wind likely originating from a large equatorial coronal hole. At heliocentric distances of approximately 10 solar radii, with speed mostly above 400 km/s, this interval is a first example of “polar coronal hole-like (PCH-l) fast" solar wind observed in the sub-Alfvénic solar corona. We characterize the turbulence properties of this unique interval using Parker Solar Probe measurements. Despite being sampled well inside the nominal Alfvén surface, the turbulence appears to be already well developed while remaining strongly transverse and highly imbalanced, exhibiting a large cross helicity. These observations provide new constraints on the development and evolution of solar wind turbulence within the lower corona.

We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold $F_{\mathrm{bif}}$ separating intra-well and inter-well dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: the boundary between intra-well and inter-well motion converges to $F_{\mathrm{bif}}$ as $m\to0$, and the families of pitchfork bifurcations responsible for asymmetric inter-well states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as $m\to0$. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.

Long-context LLMs focus on retrieving distant evidence from extensive context, yet existing work has largely focused on overcoming distance alone. In this work, we identify the Proximity Trap, insufficient attention to distant evidence often arises less from distance itself than from cumulative competition with abundant, task-irrelevant proximal background. To address the Proximity Trap, we introduce LYRA (Long-context heavY-tailed Relevance Alignment), a t-distributed directional matching mechanism that reshapes the context retrieval distribution, directing more attention mass toward task-relevant evidence, while preserving the relative positional information encoded. Extensive experiments on LongBench-v2, RULER, and LongBench demonstrate consistent improvements across context lengths and task categories. We further introduce ProxBench, a multi-level fine-grained benchmark for evaluating distant evidence utilization under increasing proximal background interference. Project page: https://xiaoyuyoung.github.io/LYRA/

Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\,d V =8\pi\bigl(\chi(M)-V_M\bigr)\le8\pi(1-V_M), \] where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $\chi(M)\in\{0,1\}$. Thus positive asymptotic volume ratio gives the value $8\pi(1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8\pi\chi(N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.

We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly–Peternell–Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.

We present an analysis of dust kinematics in the local starburst galaxies M82 (NGC 3034) and NGC 253 using Polycyclic Aromatic Hydrocarbon (PAH) features observed with JWST/MIRI spectroscopy. We are able to produce high-quality velocity maps of the 5.2 $\mu$m, 6.2 $\mu$m, and 11.3 $\mu$m PAH features, as well as numerous lines of molecular gas via H$_2$ rotational transitions and ionized gas from [NeII] and H recombination lines. Given the field of view and inclination, we trace a rotating disk in M82 where we observe a steep rise in the velocities followed by flattening, typical of galaxy rotation curves. In NGC 253, however, the 6.2 $\mu$m and 11.3 $\mu$m PAH features trace the launching region of the outflow, firmly within the starburst region. We find the ionized gas also shows outflow contributions in NGC 253 while the warm molecular gas is dominated by rotation. Additionally, the molecular gas outflow velocities are lower than the ionized gas, with the 11.3 $\mu$m PAH feature consistent with the ionized gas rather than the molecular gas. We therefore suggest that PAHs are more closely associated with the ionized gas rather than the molecular at the base of the galaxy outflow, where larger scale imaging shows comparable morphology between PAHs and HI. The 6.2 $\mu$m PAH feature has an even higher outflow velocity for both galaxies, possibly suggesting preferential ionization of the PAHs within the faster-moving hot phase of the base of the outflow.